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WORKS OF 
 PROFESSOR C W. MACCORD 
 
 PUBLISHED BY 
 
 JOHN WILEY & SONS. INC. 
 
 Doscriptive Geometry. 
 
 With Applications to Isometrical Drawing and Cavalier 
 Projection. 8vo. vi -|- 248 pages, 280 figures. Cloth, 
 $3.00. 
 
 Kinematics; or, Practical Mechanism. 
 
 8vo. xi-f335 pages, 306 figures. Cloth, $5.00, 
 
 Mechanical Drawing. 
 
 Part I. Progressive Exercises. 
 
 Part II. Practical Hints for Draughtsmen. 
 
 The two parts complete in one volume. 4to. 258 
 
 pages, 232 figures. Cloth, $4.00. 
 
 Velocity Diagrams. 
 
 Their Construction and their Uses. Addressed to all 
 those interested in Mechanical Movements. 8vo. iii-|- 
 116 pp., 83 figures. Cloth, $1.50. 
 
ELEMENTS 
 
 OP 
 
 DESCRIPTIVE GEOMETRY. 
 
 WITH APPLICATIONS TO 
 
 ISOMETRICAL DRAWING AUD CAVALIER PROJECTION. 
 
 BY 
 
 CHARLES WILLIAM MacCORD, A.M., Sc.D., 
 
 Emeritus Professor of Mechanical Drawing and Designing in the Stevens Institute of 
 
 Technoloffy, Hohoken, N. J. ; 
 
 Member of the American Society of Mechanical Engineers; 
 
 Formerly Chief Draughtsman for Capt. John Ericsson ; 
 
 Author of " Kinematics,'' " Mechanical Drawing : Progressive Exercises and Practictd 
 
 Hints,'' " Velocity Diagrams," and Numerous Monographs on Mechanism. 
 
 SECOND EDITION, REVISED. 
 EIGHTH THOUSAI^D. 
 
 NEW YORK: 
 JOHN WILEY & SONS, Inc. 
 London: CHAPMAN & HALL, Limited 
 1914 
 
M 3 
 
 Copyright, 1895, 
 
 BY 
 
 CHARLES WILLIAM MacCORD. 
 
 THE SCIENTIFIC PRESS 
 
 ROBERT DRUMMONO ANO COMPANr 
 
 BROOKLYN, N. Y. 
 
PREFACE. 
 
 Having been convinced, by a class-room experience of many 
 years at the Stevens Institute of .Technology, of the desirability of a 
 text-book on Descriptive Geometry different in some respects from 
 any previously existing, I have endeavored to produce a work suit- 
 able for use in colleges and scientific schools, and also by those who 
 may wish to acquire some knowledge of the subject without the aid 
 of an instructor. In the course of that experience many points 
 have arisen, leading to original work embodied in this treatise; in 
 the preparation of which, however, much benefit has been derived 
 from reference to the works of Olivier, Jullien, Church, Warren, 
 Watson, and others. 
 
 The study of Descriptive Geometry is not usually begun, nor 
 should it ever be, until some familiarity with the ordinary opera- 
 tions of Mechanical Drawing has been attained. But when the 
 former is taken tip its identity with the latter should never be lost 
 sight of, as it too often is : for this reason a departure has been 
 made from the stereotyped methods of treatment, which in fact 
 tend rather to conceal than to exhibit tliat identity. 
 
 At the outset considerable difficulty is often experienced in 
 forming clear conceptions of the relations between abstract things, 
 such as lines and planes, by the aid of orthographic projections 
 only. The power of doing so is of course essential; and it is 
 believed that the pictorial representations which have been intro- 
 duced will be of assistance in acquiring it. But that power will be 
 best developed, and greatly increased, by the instrumental construc- 
 tion of the problems — which indeed is absolutely necessary to the 
 
 iii 
 
 359450 
 
IV PREFACE. 
 
 attainment of such a mastery of the principles and processes as 
 alone would be of any practical value. 
 
 As a hint to those who may choose to dispense with an instruc- 
 tor, it may be stated that at the Stevens Institute of Technology it 
 is required that tlie diagrams shall be drawn with care, but not 
 required that they shall be drawn in ink. JS^or is the larter recom- 
 mended; the time required to ink in one diagram can be better 
 occupied in drawing another; moreover, work of this description 
 affords the best of practice in neat, effective, and accurate pen- 
 cilling, — an accomplishment which is becoming more and more 
 important to the practical draughtsman. 
 
 C. W. JVIacCord, 
 
 HOBOKEN, New Jersey, September 23, 1895. 
 
TABLE OF CONTENTS. 
 
 CHAPTER I. 
 
 PAGB 
 
 Definitions. — The Principal Planes of Projection. — The Four Dihedral 
 Angles. — The Profile Plane. — Representation of the Point, Right 
 Line, and Plane.— Geometrical Principles and Deductions. — Revolu- 
 tion and Counter-revolution. — Separate Construction of the Hori- 
 zontal and Vertical Projections. — Supplementary Planes and 
 Projections 1 
 
 CHAPTER II. 
 Elementary Problems Relating to the Point, Right Line, and Plane . . 32 
 
 CHAPTER III. 
 
 Generation and Classification of Lines and Surfaces. — Tangents, 
 Normals, and Asymptotes to Lines. — Osculation, Rectification, 
 Radius of Curvature. — Tangent, Normal, and Asymptotic Planes 
 and Surfaces 69 
 
 CHAPTER IV. 
 
 On the Determination of Planes Tangent to Surfaces of Single and of 
 Double Curvature 93 
 
 CHAPTER V. 
 
 Of Intersections and Developments 113 
 
 Intersection of Surfaces by Planes.— Development of Single-curved 
 Surfaces. — Tangents to Curves of Intersection before and after 
 Development. — Problem of the Shortest Path. — Intersections of 
 Single-curved Surfaces. — Infinite Branches. — Intersections of Double- 
 curved Surfaces. — Intersection of a Cone with a Sphere.— Develop- 
 ment of the Oblique Cone. 
 
 V 
 
VI TABLE OF CONTENTS. 
 
 CHAPTER VI. 
 
 PAGE 
 
 Of Warped Surfaces 152 
 
 The Hyperbolic Paraboloid ; its Vertex, Axis, Principal Diametric 
 Planes, and Gorge Lines.— The Conoid. — The Hyperboloid of Revolu- 
 tion. — The Elliptical Hyperboloid, and its Analogy to the Hyperbolic 
 Paraboloid. — The Helicoid of Uniform and of Varying Pitch. — The 
 Cylindroid. — The Cow's Horn. — Warped Surfaces of General 
 Forms. — Planes Tangent to Warped Surfaces. — Warped Surfaces 
 Tangent to Each Other. — Intersections of Warped Surfaces. 
 
 CHAPTER VII. 
 Isometrical Drawing, Cavalier Projection, and Pseudo-perspective . . . 230 
 
DESCRIPTIVE GEOMETRY. 
 
 CHAPTER I. 
 
 DEFINITIONS. THE PRINCIPAL PLANES OF PROJECTION. THE FOUR 
 
 DIHEDRAL ANGLES. THE PROFILE PLANE. REPRESENTATION OF 
 
 THE POINT, OF THE RIGHT LINE, AND OF THE PLANE. ■ 
 
 GEOMETRICAL PRINCIPLES AND DEDUCTIONS. REVOLUTION AND 
 
 COUNTER-REVOLUTION. SEPARATE CONSTRUCTION OF THE HORI- 
 ZONTAL AND VERTICAL PROJECTIONS. SUPPLEMENTARY PLANES 
 
 AND PROJECTIONS. 
 
 1. Descriptive Geometry treats of tlie methods of making, 
 Avitli nmthematical exactness, drawings for the representation not 
 only of geometrical magnitudes, but of the solutions of problems 
 relating to them. 
 
 2. This branch of science does not deal with the phenomena of 
 t)inocular vision, and for its purposes the eye is regarded as a single 
 point. 
 
 The surface upon which a drawing is made may be of any 
 form, as cylindrical, in panoramic painting, or spherical, in deco- 
 rating the interior of a dome. But in order to make correct draw- 
 ings upon such surfaces, it is necessary to be thoroughly familiar 
 Avith the methods of making them upon planes, which are usually 
 employed ; and to these our attention will be confined. 
 
 3. The object to be drawn may be placed between the eye and 
 the plane, or the plane may be placed between the eye and the 
 object. In either case, light is reflected from any point of the 
 object to the eye in a right line ; and the point in which that line, 
 produced if necessary, pierces the plane, is the representation of 
 
2 b'ESCRTPflVE GEOMETRY. 
 
 that point in tlie object from wliicli it came. A sufficient nnin])ei 
 of such points being found, the outhnes maj be fullj determined ; 
 and the drawing thus made will present to the eye, if placed in the 
 position originally assigned to it, the same appearance as the actual 
 contour of the object itself. 
 
 It may be said, then, that the representation of a point is found 
 by projecting it along a right line passing through the eye. Sucli 
 a line is called a projecting line, and all drawings thus made are 
 technically called projections. 
 
 4. If the eye is at a Unite distance, the drawing, on any surface, 
 is called a scenographic projection. If made upon a vertical plane, 
 against winch the eye is directed jperpendiciilarly , the drawing is 
 said to be in perspective; the plane is then called the picture pl&ne, 
 and the projecting lines, which converge, are called visual rays. 
 
 If the eye is removed to an infinite distance, the i3rojecting lines 
 become parallel to each other and to the axis of vision. The plane 
 upon which the drawing is made is called the plane of projection; 
 it may be perpendicular to tlie projecting lines, in wliicli case the 
 drawing is called an orthographic projection; or it may cut them 
 obliquely, and the drawing is then called an oblique projection. 
 
 5. Of these three systems of projection the second is the most 
 simple and the most extensively used, and a knowledge of it is an 
 essential preliminary to the study of the. others. We proceed then 
 at present to consider the methods of representing magnitudes and 
 the solution of problems in orthographic projection only. Evidently 
 the number of such projections, or views, necessary to tlie adequate 
 representation of an object of three dimensions, will depend mucli 
 on the form of the object itself. But, beginning with the least of 
 geometrical magnitudes, the point, considered as a visible and 
 material particle ; it can be located in space by giving its distance 
 from each of two fixed planes, and represented by its projections 
 upon them. 
 
 6. The Principal Planes of Projection. The most simple and 
 natural relation between two planes for this purpose, which is 
 universally adopted, is that shown in Fig. 1 ; the one being hori- 
 zontal, the other vertical. Moreover, these suffice for many, 
 though by no means all, of the ordinary operations of descriptive 
 
DESCRIPTIVE GEOMETRY. 
 
 geometry ; lience we may say that the principal planes of projection 
 
 are 
 
 1. The horizontal plane, usually designated simply as H for 
 brevity. 
 
 2. The vertical plane, usually designated simply as T for brevity. 
 
 These planes are supposed to extend indefinitely in each direc- 
 tion ; they intersect in a line called the ground line, designated and 
 referred to as AB. The eye is supposed to be at an infinitely re- 
 mote point, in front of the vertical plane and above the horizontal 
 plane ; whence it is directed either perpendicularly against T, as 
 
 V 
 
 2 
 
 1 
 
 •p 
 
 1 
 
 
 H 30 
 V 
 
 4 i 
 
 Fig. 
 
 ^ H 
 
 2 
 
 ip 
 
 Tig. 3 
 
 
 d' ^-^ 
 \ f 
 
 i d 
 
 E* 
 
 c 1 
 
 e' 
 
 Fig. 4 
 
 i. 2. 
 
 3. 
 
 4. 
 
 
 d\ 
 
 et 
 
 
 
 1 1 
 
 i 
 
 
 
 A i 
 
 1 
 
 ■et 
 
 B 
 
 c* 
 
 1 
 
 ft 
 
 
 
 /I 
 
 64 
 
 /i 
 
 
 Fig. 5 
 
 -■^E,, 
 
 Fig. 6 
 
 shown by the horizontal arrow, or perpendicularly downward upon 
 H, as sliown by the vertical arrow. 
 
 7. The Four Angles. Fig. 1 is a pictorial representation of a 
 model, such as can readily be made by cutting tAVO cards, each 
 through half its length, and "halving" them together. If this 
 model be lield so that the eye is directly in front of the point A, 
 and looking in the direction AB, it will appear as shown in Fig. 2 ; 
 the ground line appearing as the point 0, while each plane, being 
 
4 DESCRIPTIVE GEOMETRY. 
 
 seen edgewise, will appear as a mere line. Thus the two planes 
 form four equal dihedral angles, which are numbered 1, 2, 3, 4, in 
 the order shown; that in which the eye is placed, as above set 
 forth, being the Jirst angle. 
 Thus we have : 
 
 1st Angle Above H and in front of V„ 
 
 2d Angle Above H and behind V. 
 
 3d Angle Below H and behhid V. 
 
 4th Ano:le Below H and in front of V. 
 
 't5^ 
 
 8. In Figs. 1 and 2, P is a point in space, here taken in the 
 first angle ; the vertical line Pjp is its horizontal projecting line, and 
 jp is its horizontal projection. The line Pj)' perpendicular to V is 
 its vertical projecting line, and^' is its vertical projection. 
 
 This illustrates the notation adopted, the capital letter denoting 
 a point in space, and the small letters denoting its projections, that 
 npon the vertical plane being accented : thus we write, for ex- 
 ample, " the point J/," indicating the point whose horizontal pro- 
 jection is w, and whose vertical projection is m' . 
 
 These two projections suffice to detennine the position of the 
 point in space ; for if in Fig. 1 we suppose ^j> and 2^' only to be 
 given, a vertical line through _^ and a perpendicular to Y through 
 ^' will intersect in P, 
 
 9. In Fig. 1, draw through J9 a perpendicular to V, cutting AB 
 in c ; then jpc is parallel and equal to Pjp' . Completing the rect- 
 angle, ;p'g is parallel and equal to Pp. That is to say, the distance 
 of a point in space from the vertical plane is equal to the distance 
 of its horizontal projection from the ground line ; and its distance 
 from the horizontal plane is equal to tlie distance of its vertical pro- 
 jection from the ground line. 
 
 10. Hold the page in a vertical position, and looking perpeiN 
 dicularly against it at Fig. 3, suppose the paper to be the plane ] 
 of Fig. 1 ; the line AB will then represent the horizontal plane 
 seen edgewise, and the line p'c will be seen in its true length and 
 position. Next, hold the page in a horizontal position, and look- 
 ing vertically downward at the same figure, imagine the paper to 
 represent the plane II oi Fig. 1. The line AB will then represent 
 
DESCRIPTIVE GEOMETRY. 5 
 
 the vertical plane seen edgewise, and the line ^c will be seen in its 
 true length and position. Thus the single plane of the paper rep- 
 resents both H and V, and with a little effort can at pleasure be re- 
 garded as either the one or the other. Fig. 3, then, is a repre- 
 sentation of a point P, situated in the Urst angle, in orthographic 
 projection upon the tw^o principal planes. In reality, only the 
 ground line and the two points^ and^i' are absolutely necessary; 
 but we observe, tliat since in Fig. 1 jpc and p'c are both perpen- 
 dicular to ^^ at the same point <?, they will in Fig. 3 coincide in 
 one right line. That is to say, the two projections of a given point 
 mnst lie on the same perpendicular to the ground line. 
 
 11. The Profile Plane. In effect. Fig. 3 is both sl front yiew 
 and a toj? view of the model shown in Fig.l ; the space above AB 
 representing the upper part of Y and also that part of H which is 
 behind V, wliile the space below AB represents the lower part of V 
 and the front part of H. 
 
 ]^ow. Fig. 2 is an end view of the same model, the eye looking 
 (7) in the direction J. ^ ; and J.^ is seen in Fig. l.to be perpen- 
 dicular to tlie plane of the rectangle Pc : in other words. Fig. 2 is 
 an orthographic projection upon a plane perpendicular to the 
 ground line. 
 
 Such a plane is called a profile plane ; the projection upon it is 
 called simply a profile, and if made, as here, separate and distinct 
 from the projections upon the prmcipal planes, it is often of the 
 greatest use. 
 
 Evidently, a profile may be constructed, representing the model 
 as seen from the right, looking in the direction BA ; the first and 
 fourth angle will in that case lie on the left of V, and the order of 
 the numbers will be the reverse of that shown in Fig. 2. 
 
 12. Location of the Profile. If the profile is made on the first 
 supposition, (7) the view heingfro77i the left, it should be placed 
 at the left of the drawing showing the projections on the principal 
 planes ; if made on the second supposition, that the view is from 
 the right, the profile should be placed at the right of that drawing. 
 Thus, Fig. 5 shows the projections on H and V of four points, 
 one in each dihedral angle ; Fig. 4 is a profile showing the same 
 points, with their projecting lines, seen from the left ; and Fig. 6 
 
b DESCKirXIVE GEOMETRY. 
 
 is a profile, in wliicli the eve is at the right, looking in the direc- 
 tion jSA. 
 
 13. If in Fig. 4 we suj)pose the point C, for instance, to ap- 
 proach the vertical plane, Co' will be diminished, and c will ap- 
 proach ; w^hen the point reaches T , Ce will coincide with c' 0. 
 Eeasoning similarly vrith regard to H, we perceive that if a poiot 
 lies in either plane, it will be its own projection on that plane, and its 
 projection on the other plane will lie in the ground line. If it lies in 
 both planes, the point itself and both its projections coincide in one 
 point on the ground line. 
 
 In Fig. 5 it is observed that both projections of D, a point in 
 the second angle, lie above AB, while both those of J^^, which is in 
 the fourth angle, lie below AB. 
 
 Now, were D equally distant from H and V, its j^i-ojections d 
 and d' would fall together in one ]3oint; but the t ,o letters would 
 still be used. The same would in like case be true of the projec- 
 tions of J^; and conversely, if the two projections coincide, but do 
 not lie on the ground line, the point itself is equidistant from H and 
 V, and therefore lies in a jAeme bisecting the second and fourth 
 angles. 
 
 EEPRESENTATION OF THE EIGHT LINE. 
 
 14. The projection of any line upon any plane is determined l)y 
 projecting all its points upon that j^lane. In the case of a right 
 line, two points determine it in space, and the projections of these 
 two are sufficient. 
 
 Thus in Fig. 7, m and n are the projections of Jf and i\^upon 
 the plane JTY. Tlie two projecting perpendiculars determine a 
 plane containing the line J/iV^; this is called tlie projecting plane, 
 and it cuts xy in a right line wai, called its trace, which is the pro- 
 jection of the given line. 
 
 If the line is not parallel to the plane, it must pierce it when 
 prolonged. The point of penetration must lie on tlie given line, 
 and also in its projection ; it is therefore at their intersection P. 
 This point is sometimes called the trace of the line on the plane. 
 
 Draw, in the projecting plane J[/??, a line iVT perpendicular to 
 Mm ; it is also .perpendicular to JVn, consequently In is a, rectangle. 
 
DESCRIPTIVE GEOMETRY. 7 
 
 Now regarding IN as tlie given line, mn is its projection ; there- 
 fore, if a line be parallel to a plane, its projection on that plane will 
 be parallel and equal to the line itself. 
 
 Regarding MI as the given line, the projections of all its points 
 fall together at in\ that is to saj, if a line be perpendicular to a 
 plane, its projection on that plane is a point ; which point lies in the 
 line itself. 
 
 If a line be inclined to a plane, Us projection on that plane will be 
 shorter than the line itself; thus, MN^ being the hypothenuse of 
 the right-angled triangle -JZ/iV^, is greater than the base /iV, or its 
 equal nin. 
 
 1 5 . If two lines are parallel in space, their projections on the same 
 plane will also be parallel. In Fig. 7 let DE be parallel to MN\ 
 then its projecting plane De is parallel to the projecting plane Mn. 
 The plane XY cuts these two parallel planes in the lines m?i, de^ 
 which are therefore parallel to each other. 
 
 Prolong J/iYto any point L on the opposite side of XY^ of 
 which I is the projection ; also produce Mm and draw IK perpen- 
 dicular to it. Then mKr= 11^ and ml = nX, whence MI — Mm 
 — iW^, and MK— Mm -\- II. Xow MX is the hypothenuse of 
 the triangle MIX^ whose base IX is equal to ma., and MI is the 
 hypothenuse of the triangle MKI^ whose base is equal to ml. We 
 see, then, that the true length of a line in space is equal to the hy- 
 pothenuse of a right-angled triangle, whose base is equal to the pro- 
 jection of the line on any plane ; the altitude being equal to the 
 diiference of the projecting perpendiculars of the two extremities of 
 
8 
 
 DESCRIPTIVE GEOMETRY. 
 
 tlie line if tliey lie on tlie same side of the plane, and equal to their 
 sum if tbey lie on opposite sides. 
 
 16. If two lines intersect in space, their projections upon any 
 plane will either intersect each other, or thej will coincide. If the 
 plane of the given lines is perpendicular to the given plane, it will 
 be their common 23rojecting plane, and the projections will coincide ; 
 thus 7)in is the projection of both MJV and IW. But the lines 
 FC^GII^ which intersect at 7?, have not a common projecting 
 plane ; and the two planes Gh^ Fc^ intersect in a line Ri\ wdiich 
 must be perpendicular to XY^ and is therefore the project- 
 ing line of R ; that is to say, the intersection of the projections of 
 two intersecting lines upon the same plane is the projection of the 
 intersection. 
 
 But the projections may intersect although the lines themselves 
 do not ; it is evident that in the plane Hg many lines may be drawn 
 which would pass either under or over FC^ and in the jDlane Cf 
 many others which would not intersect Gil. 
 
 17. A line, like a point, is represented by its projections on 
 H and T; thus in Fig. 8, cd is the horizontal, and c'd' is the verti- 
 
 cal, projection of the line CD. This is shown pictorially in Fig. 9, 
 where Cd is the horizontal projecting plane, and Cd' is the vertical 
 projecting plane, of the given line. In general, a nne is fully de- 
 termined by its projections ; for if they are given the projecting 
 planes can be constructed, and since the line must lie in each of 
 them, it will be their intersection, if they have one. The limited 
 line CD is a portion of the line X Y of indefinite length, repre- 
 sented by indefinitely extending the projections, as xy, x'y'. 
 
 18. In general, then, any indefinite line a?y in H may be assumed 
 
DESCRIPTIVE GEOMETRY. 9 
 
 as the liorizontal projection, and any indefinite line x'y' in T as tlie 
 vertical projection, of a line whose position in space is thus defi- 
 nitely fixed. 
 
 This, however, is subject to the restriction that if either pro- 
 jection be perpendicular to AB, the other projection, whether it be 
 a point or a right line, must lie in the prolongation of that per- 
 pendicular. 
 
 Thus in Fig. 8, the horizontal projection Teg is perpendicular to 
 AB; but (10) the vertical and the horizontal projections of any 
 point must both lie in the same perpendicular to the ground line, 
 consequently h'g' will lie in lig produced. 
 
 In this case the indefinite projections on H and Y do not suffice 
 to determine the line ; the projecting planes coincide, being perpen- 
 dicular to A B at the same point, and have no line of intersection. 
 If, as in Fig. 8, the j)rojections of two points of the Kne are dis- 
 tinguished by letters, the line is determined, but such a representa- 
 tion is most unsatisfactory and difficult to read : the line lies in a 
 profile plane, and its projection thereon should always be added. 
 
 19. A line in space may lie in either plane of projection; its 
 projection on the other then falls in the ground line. If it lies in 
 neither plane, it may be parallel to one only, parallel to both, or 
 inclined to both. 
 
 If a line is perpendicular to one plane, it is parallel to the other ; 
 its projection on the first is a point, and its projection on the other 
 is perpendicular to AB. 
 
 If the line is parallel to one plane and inclined to the other, its 
 projection on the first is parallel to the line itself, and its projec- 
 tion on the other is parallel to AB. 
 
 If the line is parallel to both planes, the line itself and both its 
 projections are parallel to AB. 
 
 20. The doubly oblique positions may be divided into two 
 groups ; one including those which ascend as they recede, like the 
 ones drawn on the sloping plane J/Z, Fig. 10; and the other 
 including those which descend as they recede, like the ones on the 
 farther plane J/iT. 
 
 Those of the first group may cross the first angle, piercing the 
 front part of H and the upper part of V ; they may cross the third 
 
10 
 
 DESCRIPTIVE GEOMETRY. 
 
 angle, piercing tlie lower part of V and the rear part of H ; or tliey 
 may intersect AB, in wliicli case tliej lie wholly in the second and 
 fourth angles. 
 
 Those of the second group may cross the second angle, piercing 
 
 Fig. 10 
 
 the upper part of V and the rear part of H ; they may cross the 
 fourth angle, piercing the front part of H and the lower part of V ; 
 or they may intersect AB, in which case they lie wholly in the first 
 and third angles. 
 
 Lines of either group, as shown in Fig. 10, may inchne either to 
 the right or to the left, in which case their projections will be 
 inclined to AB; or they may do neither; lying then in profile 
 planes, their projections on H and Y are perpendicular to AB. 
 
 21. In Fig. 11 the projections on H and V consist merely of 
 
 d' g^e 
 
 B C 
 
 
 2 
 
 
 
 ! ' 1 
 
 
 d c 
 
 4 
 
 9 
 
 e 
 12 
 
 3 
 
 a 
 
 E 
 
 
 Fig. 
 
 
 
 Fig. n 
 
 two lines below AB and perpendicular to it, and two ]3oints above 
 AB. These representations being identical, it would be impossible 
 to distinguish between them, or to decide with certainty what either 
 was intended to show, were they not lettered. By the aid of the 
 letters we perceive that CD is a line of limited length, perpendicu- 
 lar to V, and lying in the first angle, while OE is a limited vertical 
 line situated in the third angle. These things are seen by a single 
 
DESCRIPTIVE GEOMETRY. 
 
 11 
 
 glance at Fig. 12; which illustrates the value of the profile, some- 
 times in even very simple cases.. 
 
 22. In Fig. 13 are given the projections of a horizontal line 
 lying, as pictorially represented in Fig. 14, in the second angle. 
 Being horizontal, its vertical projecting plane is also horizontal, and 
 therefore cut by T in a horizontal line ; that is to say, the vertical 
 projection c'd' is parallel to AB (19). 
 
 . Eiaae 
 
 Fig. 15 gives the projections of an inclined line, in the first angle 
 and parallel to V, as seen in Fig. 16. Its horizontal projecting plane 
 is therefore parallel to V, and since these two parallel planes are cut 
 by H in parallel lines, the horizontal projection cd is parallel to 
 AB (19). 
 
 23. The doubly oblique lines, not being parallel to either plane, 
 pierce them both. In relation to these, beginners sometimes find 
 difficulty in reading the drawings, — that is, in forming by aid of 
 the projections alone, clear perceptions of the actual positions of the 
 lines in space. This difficulty may perhaps be lessened by consider- 
 
12 
 
 DESCKIPTIYE GEOMETRY. 
 
 y\^ Fig. 23 
 
 Fig. 24 
 
DESCRIPTIVE GEOMETRY. 13 
 
 ing at first such as do not meet the ground line, and confining the 
 attention to the portions intercepted between H and V. 
 
 In Fig. 17 is given a pictorial representation of a line which 
 crosses the first angle. In order to draw the projections of such a 
 line we may assume <?, Fig. 18, as the horizontal projection of any 
 point lying in H in front of V ; its vertical projection c' will lie in AB : 
 also assume d ' as the vertical projection of a point in V above H ; its 
 horizontal projection d must also lie in AB ; therefore cd is the hori- 
 zontal and c'd' is the vertical projection of the required line. 
 
 24. So, by assuming the projections of a point in each plane, 
 it is easy to represent a line crossing any angle at pleasure. Thu& 
 Fig. 19 shows one which crosses the second angle; its representa- 
 tion in projection. Fig. 20, differs from Fig. 18 only in this, that g 
 lies above instead of below AB, being the horizontal projection of a 
 point in H behind T. 
 
 Completing the series, the four following figures represent, pic- 
 torially and in projection, lines crossing the third and the fourth 
 angles. And it is observed that the two projections of the inter- 
 cepts in the first angle. Fig. 18, and in the third angle. Fig. 22, 
 do not intersect each other. If the intercept lies in either of the 
 other angles, its projections cross each other, the intersection being 
 above AB if it lies in the second, as in Fig. 20, and helow AB, as in 
 Fig. 24, if it lies in the fourth. 
 
 25. To find the traces of a line whose projections are given. 
 The points in which a line pierces H and T are called respectively 
 its horizontal trace and its vertical trace. In constructing Fig. 18, 
 as explained in (23), the traces w^ere assumed, and determined the 
 projections of the line ; by merely reversing the j^rocess, the traces' 
 may be found if the projections are given. For if the line in space 
 be produced, its projections w411 be extended; the distance of any 
 point of the horizontal projection from AB shows the distance of the 
 corresponding point in the line from V, which becomes zero when 
 that projection meets AB ; and similarly the altitude, or distance 
 from H, becomes zero when the vertical projection meets AE. 
 Therefore, if in Fig. 18 the projections mn^ Tn'n' are given, pro- 
 duce mn to cut AB in d\ this will be the horizontal projection of 
 the vertical trace : the other projection must lie on a perpendicu- 
 
14 
 
 DESCRIPTIVE GEOMETRY. 
 
 lar to AB at d^ and also on tlie prolongation of the vertical projec- 
 tion m'li' ; therefore it is at their intersection d' . Prodnce vin' to 
 cut AB at g' ^ the vertical projection of the horizontal trace ; at c' 
 draw a perpendicular to AB, cutting mn produced in <?, the other 
 projection. TJie points C and D are the traces sought. 
 
 26. In attempting to find the traces of a line given in this man- 
 ner, it may happen that both its projections meet the ground line at 
 the same point. This means simply that the line pierces both H and 
 V at that point, and therefore intersects tlie ground line. Tliis is the 
 case in Fig. 25 ; and as the line JlfiTlies in the first angle, it must 
 when prolonged pass into the third angle after crossing AB, as more 
 distinctly seen in the profile, Fig. 26, the addition of which third 
 
 riG.27 
 
 projection greatly facilitates the reading in such cases. In Fig. 27 
 the given limited portion MN of the line lies in the second angle ; 
 when prolonged the line cuts AB at C^ and, as shown in the profile, 
 Fig, 28, passes on into the fourth angle. 
 
 27. OMique Lines with Coincident Projections. In Fig. 29 tlie 
 point M is as far behind V as it is above H, and in consequence m 
 and m! fall together. The point D is as far below H as it is in front 
 of V, whence d and d' also fall together. - Therefore the vertical 
 and the liorizontal projections of the line MD coincide in one line. 
 
DESCRIPTIVE GEOMETRY. 
 
 15 
 
 Every other point of tlie line will therefore be represented by coin- 
 cident projections, as ?i, 71' ; all such points lie in either the second 
 angle or the fourth, and since they are equidistant from H and V, 
 they and the line itself lie, as shown in the profile, Fig. 30, in a 
 plane which bisects those two angles (13). 
 
 Fig. 29 
 
 JFiaSS 
 
 Fig. 30 
 
 d' 
 
 e 
 n' 
 
 e' 
 
 
 iW 
 
 c' 
 
 
 n c 
 
 
 ) 
 
 m g 
 
 g' 
 
 c 
 
 Fig. 31 
 
 
 
 d 
 
 n' A 
 
 H^- 
 
 
 my 
 
 Gy\ : 
 
 / 
 
 X m 
 
 n/ 
 
 
 > 
 
 ' 
 
 Fig. 32 
 
 .iV 
 
 Fig. 34 
 
 28. Oblique Lines in Profile Planes. In Fig. 31 both projections 
 are perpendicular to the ground line. In tins case these projec- 
 tions upon the principal planes are utterly inadequate to convey a 
 clear idea of the position of a limited portion of the line, even with 
 the aid of the letters. The indefinite projections, even supposing it 
 to be known which is the vertical and which the horizontal, do not 
 suffice (18) to locate the line in space, and since their prolongations 
 coincide, it is impossible by their use to find the traces. The obvious 
 and the only sensible course is to make an independent and detached 
 drawing in profile, as shown in Fig. 32, which exhibits in the clear- 
 est possible manner the position of the line in relation to both H 
 
10 DESCKIPTIVE GEOMETRY. 
 
 and V, whether it crosses one angle or another, like MN^ or like 
 GE intersects the ground line. 
 
 29. Lines Parallel to Both Principal Planes. A line which is 
 parallel to botli H and V is parallel to the ground line. It may lie 
 in one of those planes; where.it is its own projection, the projec- 
 tion on the otlier plane falling in AB, If it does not lie in either 
 
 11 or V, both its projections are parallel to the ground line. The 
 projections of such a line upon the principal planes are sufficient to 
 locate the line in space, and, as shown in Fig. 33, they suffice to 
 represent it. Nevertheless, in this case also the reading of the 
 drawing is facilitated, and the locution of the line more clearly indi- 
 cated, as shown in Fig. 34, by adding a profile. 
 
 REPRESENTATION OF THE PLANE. 
 
 30. The intersection of a plane with V is called its yertical trace ; 
 its intersection with H is called its horizontal trace ; and the plane i& 
 represented by drawing these traces. 
 
 Any horizontal plane, being parallel to H, has no horizontal 
 trace, and its vertical trace is parallel to AB. Example : the ver- 
 tical projecting plane of 6ZZ>, Fig. 14. 
 
 If a plane is parallel to V it has no vertical trace, and its hori- 
 zontal trace is parallel to AB. Example : the horizontal projecting 
 plane of CD, Fig. 16. 
 
 If a plane is parallel to AB and inclined to H and V, both traces 
 will be parallel to AB ; or, they may coincide in the ground line 
 itself. These cases will be illustrated farther on. 
 
 If a plane is perpendicular to AB, i.e., if it is a profile plane, 
 both traces are perpendicular to AB. Example : the projecting 
 planes of KG, Fig. 9. 
 
 It will be perceived from the examples here quoted that the 
 projections of all lines are the traces of their projecting planes (14). 
 
 31. If a plane be inclined to AB, it will cut it in a point; the 
 traces must intersect at this point, and one or both of them will be 
 incHned to AB. Thus in Fig. 35, the oblique plane Jf Aleuts AB 
 at D ; its horizontal trace is dDc, and d'Dc' is its vertical trace. 
 
 Such a plane is represented in projection as in Fig. 36 ; it is 
 designated and referred to as the plane dDd' . If, as often hap- 
 
DESCRIPTIVE GEOMETRY. 
 
 17 
 
 pens, 'itteution is to be confined to tliat portion of the plane which 
 lies in the first angle, between H and V, the parts Dc^ Dc' of the 
 traces are omitted ; but it must be kept in mind that the plane is 
 capable of indefinite extension, and both traces can be indefinitely 
 produced. 
 
 It is to be observed, in regard to this notation, that d and d' are 
 not used to indicate the two projections of the same point ; d merely 
 designates a point in H, and d' designates a point in V. If the loca- 
 tion in space of a particular point in either trace is to be indicated. 
 
 Fig. 36 
 
 the two projections of that point are lettered in the usual manner ; 
 thus c, o' are the projections of a point in Dd\ and ^, r' those of 
 one in Dd ; these points are referred to as and R respectively. 
 
 32. For the purpose of aiding those who may at first find dif- 
 ficulty in reading the diagram, Fig. 36, there are placed above it 
 drawings, on a reduced scale, of the two cards with their slots 
 which form the planes of projection in the model. Fig. 35. 
 
 In looking at the card F", the paper is held, or supposed to be 
 held, in a vertical position ; while in looking at the card H^ it is 
 held, or imagined to be, in a horizontal position, and viewed from 
 above. The diagram represents both these cards in skeleton ; make 
 therefore the same suppositions in regard to the position of the 
 paper and the direction in which it is to be viewed, closing the 
 mental eye to one projection while studying the other. By persist- 
 ent eft'orts of this kind, the power may gradually be acquired of 
 
18 
 
 DESCRIPTIVE GEOMETRY. 
 
 reading tlie diagrams with ease — that is to saj, of forming by tlie 
 aid of tlie projections alone, clear mental images of the positions and 
 relations of the lines and planes which thej represent, so that they 
 will, as one may say, stand out in relief with stereoscopic dis- 
 tinctness. 
 
 33. In regard to these two cards, it is evident that the direc- 
 tions of the slots are entirely arbitrary and independent of each 
 otlier ; but when put together, the point D on one must coincide 
 with tlie point D on the other. Which is only another way of say- 
 ing that from the same point on AB we may draw one line in any 
 direction on H, and another in any direction on V, and these two 
 lines will determine a plane, of whicli they are the traces. 
 
 If the vertical trace is perpendicular to AB, the plane is verti- 
 cal, but may make any angle with V, as in the swinging of a com- 
 mon door upon its hinges. If tlie horizontal trace is perpendicular 
 to AB, the plane is perpendicular to V, but may make any angle 
 with H ; as in the opening of a trap-door whose hinges are perpen- 
 dicular to the wall. The plane dDcV illustrates the former, and 
 the plane tTif illustrates the latter, of these two cases, in Figs. 37 
 and 38. 
 
 .Fig. 37 
 
 ^ 
 
 
 y 
 
 6! 
 
 
 
 / 
 
 
 D 
 
 
 ' 
 
 T 
 
 
 
 
 / 
 
 t 
 
 
 
 \^, 
 
 > Pig; 38 
 
 34. In relation to the angle included between the parts of the 
 traces in front of V and above H, it is apparent that the angle pic- 
 torially represented by dDd' in Fig. 35 is in fact acute, while in 
 Fig. 37 dDd' and tTt' represent right angles. In Fig. 39 dDd' 
 
DESCRIPTIVE GEOMETRY. 
 
 19 
 
 represents an obtuse angle ; and in the diagram, Fig. 40, are given 
 the traces of the same plane, similarly lettered. Above are added 
 the small drawings of the cards with their slots, for forming the 
 planes H and V of the model. 
 
 These slots are inclined to AB in the same direction, though 
 the angles are somewhat different in the two cards. A moment's 
 reflection will show that these angles might be made exactly the 
 same, and also that if they were, the two traces dD, Dd\ instead 
 of forming an angle with each other in the diagram, would form 
 
 A 
 
 -^i^"' "" 
 
 R 
 
 
 ^ ! 
 
 
 
 
 
 A 
 
 Di>\ " 
 
 R 
 
 
 / 
 
 y 
 
 Fig. 39 
 
 Fig. 40 
 
 one continuous right line ; which is the case with the traces of the 
 plane tTt\ Clearly, the position of the plane shown in Fig. 39 
 would be but slightly changed by this modification. 
 
 35. In Fig. 41, the plane TT is parallel to AB; consequently 
 its horizontal trace tt and its vertical trace ff are both parallel to 
 the ground line. The plane DD passes through AB, which there- 
 fore constitutes both traces. 
 
 The diagram, representing these two planes is given in Fig. 42 ; 
 but it is very obvious that they are represented much more clearly 
 in the profile, Fig. 43. 
 
 Since any number of planes may be passed through the ground 
 line, the position of any one of them must be determined by some 
 other condition ; but when it is determined, its true relation to H 
 and V is at once shown by the detached profile. 
 
 In Fig. 44, the horizontal trace dd coincides with the vertical 
 trace d'd', the former being as far behind V as the latter is above 
 
20 
 
 DESCinrnVE GEOMETRY. 
 
 H. Again, tt is the liorizontal trace of a plane, and lies as far in 
 front of V as the vertical trace t't' is below H, so that these two 
 traces are also represented bj one line. Finally, mn^ nn'n' are the 
 projections of a line, parallel to AB, in the fourth angle, and equi- 
 distant from the principal planes. The superiority of the profile, 
 
 B 
 
 
 
 
 
 
 
 y^ 
 
 
 ' / 
 
 t' 
 
 r 
 
 
 ^ 
 
 
 d' 
 
 d' 
 
 
 ^ 
 
 A 
 
 ^ 
 
 d 
 
 B 
 
 
 t 
 
 t 
 
 
 
 
 Pig. 42 
 
 
 
 d' 
 
 n' t' 
 
 Fig. 44 
 
 Fig. 43 
 
 M* 
 
 Fig. 45 
 
 Fig. 45, in respect to distinctness and ease of comprehension, is too 
 obvious to require comment. 
 
 GEOMETRICAL PRINCIPLES AND DEDUCTIONS. 
 
 36. If two planes intersect, any line of either will pierce the 
 other in a point of their common line, if at all ; hence — 
 
 1. If a line lie in a plane, the traces of the line will be points 
 in the corresponding traces of the plane. 
 
 2. To draw a line in a given plane, join any point in one trace 
 w-ith any point in the other. 
 
 3. To draw a plane containing a given line, join the traces of 
 the line with any point on AB. 
 
 Any horizontal line in a given plane is parallel to the horizontal 
 trace, pierces V in a point of the vertical trace, and its vertical pro. 
 jection is parallel to AB. 
 
DESCRIPTIVE GEOMETRY. 
 
 21 
 
 If a line in a- given plane is parallel to T, it is parallel to the 
 vertical trace, pierces H in a point of the horizontal trace, and its 
 horizontal projection is parallel to AB. 
 
 If a plane contain any two lines, it will also contain any third 
 line which cnts those two. 
 
 3 7. In illustration of the above : Let it be required to draw a 
 line in the plane tTt\ Fig. 46. Assume c as the horizontal projec- 
 tion of a point in the horizontal trace ; its vertical projection is o' 
 in the ground line. Let d' be the vertical projection of a point in 
 the other trace, then its horizontal projection is d in the ground 
 line ; cd^ c'd\ are the projections of a line which lies in the giveu 
 plane. It follows from this^ that if one projection of a point in a 
 given plane be assumed, the other can be found by drawing through 
 the assumed one, the corresponding projection of a line in the plane. 
 Then the otlier projection of the point must lie on the other pro- 
 jection of the line. For example, suppose the horizontal projection 
 o in Fig. 46 to have been assumed. Join o with any point c of the 
 horizontal trace, and produce this horizontal projection to cut AB 
 
 in d. Since this line is to lie in the plane, its vertical projection is 
 cd\ upon which must lie the vertical projection o' , The point O 
 thus determined lies in the given plane. 
 
 Again, let it be required to draw in the plane tTt' ^ Fig. 47, a 
 horizontal line at a given distance above H. 
 
 Draw c'd' parallel to AB, at the given distance above it: this is 
 the vertical projection of the required line, and d' that of its verti- 
 cal trace, which is horizontally projected at d in AB. Therefore 
 dc parallel to Tt is the horizontal projection of the required line. 
 
22 
 
 DESCRIPTIVE GEOMETRY. 
 
 Let it be further required to draw in the same plane a line 
 parallel to V, at a given distance in front of it. The horizontal 
 projection is mn^ parallel to AB and at the given distance below it ; 
 the line pierces H at the point iT, whose vertical projection is n' on 
 AB, and n'm' parallel to Tt' is the vertical projection of the re- 
 quired line. 
 
 38. The two lines CD and JfiT evidently intersect; and since 
 they cannot intersect in more than one point, the test of the accu- 
 racy of the constructions lies in this, that the intersection o of the 
 horizontal projectioils, and the intersection o' of the vertical pro- 
 jections, lie on the same perpendicular to the ground line. 
 
 39. To draw two lines which shall intersect. This may be done 
 by assuming the point of intersection 6', Fig. 48 ; c and c must 
 necessarily lie on the same perpendicular to AB (10). The hori- 
 zontal projection of each line must pass through c, and its vertical 
 projection through c' ; but the directions of mc?i, in'e'n^ as well as 
 those of jpcr^ jp'c'r\ are entirely arbitrary, with the excej^tion that 
 if one projection of either line is perpendicular to AB, the other 
 projection of that line must be so likewise (18). 
 
 The two lines GL^ DE^ in Fig. 48, intersect at 0\ the hori- 
 zontal projections intersect at 6>, but the vertical projections coin- 
 cide. This merely shows that the plane determined by the two 
 
 Fig. 49 
 
 lines is perpendicular to V, and is their common vertical projecting 
 plane (16). 
 
 40. To draw two lines which shall not lie in the same plane. 
 
 ^Neither the vertical nor the liorizontal projections can coincide, 
 since if they did the two lines would have a common projecting 
 
DESCRIPTIVE GEOMETRY. 
 
 23 
 
 plane. The horizontal projections must therefore cross eac-h other, 
 and so must the vertical projections ; but these two points of inter- 
 section must not lie in the same perpendicular to AB. 
 
 Thus in Fig. 49, the horizontal projections of the lines CG^ 
 LO^ intersect; let this point of intersection be the horizontal pro- 
 jection m of a point M upon L 0^ then its vertical projection is m' 
 upon Vo'. Let the same intersection be the horizontal projection d 
 of a point upon CG^ tlien its vertical projection is d' upon eg'. 
 Similarly, the intersection of I'o' and c'g' is the common vertical 
 projection of two points, J^ upon L 0^ and E upon CG. 
 
 41. If two parallel planes are cut by a third plane, the lines of 
 intersection are parallel. Therefore, in order to represent a plane 
 parallel to one of which the traces are given, draw the vertical 
 trace of the second parallel to that of the first, and the horizontal 
 trace of tlie second parallel to the horizontal trace of the first. If 
 the new plane is required to be so located as to satisfy some special 
 condition, it is clear that the determination of one point in either 
 trace is sufficient. 
 
 For example : Let it be required to draw a plane parallel to the 
 given plane tTt\ Fig. 50, through the given point 0. Draw, 
 through the given point, a line parallel to the horizontal trace ; its 
 
 Horizontal projection passes through o and is parallel to^ Tt, its 
 vertical projection passes througli o' and is parallel to AB. This is 
 a line of the required plane, and pierces V at the point C^ whose 
 vertical projection c is a point in the vertical trace, which is d'c' D 
 parallel to Tt' . This trace cuts AB at />, and the horizontal trace 
 Dd is parallel to Tt. 
 
24 
 
 DESCRIPTIVE GEOMETRY. 
 
 42. Let ZJf, LN^ Fig. 51, be two planes, and Zi^ their line 
 of intersection. From any point P let fall upon these planes the 
 perpendiculars PR^ P0\ these two lines determine a plane which 
 is perpendicular to both the others, and therefore to LF\ it also 
 cuts them in the lines CRD^ COE^ which meet at C on LF. But 
 LF is perpendicular to the plane OPR^ and therefore to the lines 
 CEy CD^ which pass through its foot in that plane. Now, regard- 
 ing ZJf as a plane of projection, andZT^as the trace upon it of 
 any plane LIi\ then CR is the projection of PO^ a perpendicular 
 to LN^ and the trace LF is perpendicular to the projection CR. 
 
 43. Therefore, if a line be perpendicular to a plane, the yertical 
 projection of the line will be perpendicular to the vertical trace of 
 the plane, and the horizontal projection will be perpendicular to the 
 horizontal trace. 
 
 And conversely : if the projections of the line are respectively 
 perpendicular to the traces of the plane, the line itself is perpen- 
 dicular to the plane. 
 
 In illustration, let it be required to draw through the point 0, 
 Fig. 52, a line perpendicular to the plane tTt. Since the projec- 
 
 / 
 
 p 
 
 '1 
 
 / 
 
 
 V" y 
 
 Fig. 52 
 
 Fig. 53 
 
 tions of the line must pass through those of the point, we have 
 merely to draw m'o'n' perpendicular to Tt' ^ and mon perpendicular 
 to Tt ; then MN is the required line. 
 
 Or, having given the line MN and the point 6^, let it be re- 
 quired to draw a plane through that point and perpendicular to the 
 line. Here again, since the directions of the traces are known, the 
 determination of one point in either trace suffices to locate the 
 plane ; and this is effected by drawing through the point a parallel 
 to the other trace. Thus, to find a point in the horizontal trace, draw 
 
DESCRIPTIVE GEOMETRY. 25 
 
 tlirongli C a line parallel to the vertical trace ; its vertical projec- 
 tion is o'd' perpendicular to mn ^ its horizontal projection is cd 
 parallel to AB, and it pierces ^in the point d^ d' . The horizontal 
 trace of the required plane is therefore tdT perpendicular to mn^ 
 and tlie vertical trace is Tt' perpendicular to rrin' . 
 
 44. In Fig. 53, let ZZ be a plane of projection, iTiV^ a pro- 
 jecting plane perpendicular to it, of which dd is the trace, andP(? 
 a line perpendicular to iViVand consequently parallel to LL. Then 
 po^ the projection of PO^ is parallel to that line itself, and therefore 
 perpendicular to NN and to its trace dd. Now P (9, being per- 
 pendicular to ^iY, is perpendicular to all right lines drawn through 
 its foot in that plane, as OR^ 0S\ and the projections of all these 
 lines fall in the trace dd^ which is perpendicular to po\ conse- 
 quently we have, that the projection of a right angle will be a right 
 angle, if one of its sides is parallel to the plane of projection. 
 
 REVOLUTION AND COUNTER-KEVOLUTION. 
 
 45 . The axis of a circle is a right line passing through its centre, 
 and perj)endicular to its plane. A point is said to revolye about a 
 right line as an axis, when it describes the circumference of a circle 
 whoso centre is in the axis, and whose plane is perpendicular to the 
 axis. 
 
 When all the points of any geometrical magnitude move in this 
 manner, without change of relative position and therefore with tlie 
 same angular velocity, the whole magnitude is said to revolve about 
 the right line as an axis. 
 
 It will be found in subsequent operations, that magnitudes 
 under consideration can often be thus revolved into positions in 
 which certain processes can be more conveniently executed ; after 
 which they are revolved back again into their original positions; 
 and this restoration is called counter-revolution. 
 
 46. One of the simplest- examples of this kind of manipulation 
 is the following : Given ^ a line lying in a plane of projection, and 
 a point not in the plane ; Pequired^ to show where the point will 
 fall, when revolved about the line into the plane. 
 
 This is pictorially illustrated in Fig, 54, where EF is the line, 
 lying in the plane LL^ and P the given point. The plane of rota- 
 
26 
 
 DESCRIPTIVE Geometry, 
 
 tion is WW^ perpendicular to EF and therefore to ZL ; it contains 
 the projecting line P/? of the given point, and^ lies in the trace 
 dd^ which is perpendicular to EF and cuts it at C, the centre of 
 the circular patli of P. 
 
 Consequently tlie point will fall in the plane LL either at r or 
 
 Fig. 55 
 
 at s on the trace dd^ at a distance from 6^ equal to PC i\\Q radius 
 of the circle. And it is seen that PC is the hypothenuse of a tri- 
 angle, of which the altitude Pp is the distance of the point from the 
 plane, and the base ^ 6^ is the distance of the projection of the point 
 from the axis. The axis may pass through this projection ; that is, 
 P may coincide with (7, in in which event the projecting line Pj) is^ 
 the radius of the circle. 
 
 47. The construction in projection is shown in Fig. 55, where 
 is the given point. First : to revolve this point about CD into 
 the horizontal plane. Draw through <?, the horizontal projection 
 of the point, an indefinite perpendicular to cd^ cutting it in m ; the 
 point will fall somewhere upon this line, which is the trace of the 
 plane of rotation. The distance of the point from the plane is o'ic^ 
 the projecting line, and om is the distance of the projection from 
 the axis. Set off xy = om, then o'y is the true distance of the 
 point from m; therefore set off mo'' = o'y, and o" is the position 
 of after revolution. 
 
 Second : to revolve about PR into the vertical plane. Draw 
 through o an indefinite perpendicular toj^V, cutting it at /?-, and 
 on this perpendicular set off no'" , equal to the hypothenuse of a 
 triangle whose base is on, and whose altitude is equal to xo, the 
 
DESCRIPTIVE GEOMETRY. 
 
 27 
 
 distance of from V : this triangle may be conveniently constructed 
 by setting off oiij)'?'', ne = xo^ giving o'e as the true distance of O 
 from n. The line o'e is of course not actually to be drawn, but its 
 length being taken in the dividers, is set off from n as no'" ^ and o'" 
 thus located is tlie required position ; similarly, o'y is measured, 
 but not drawn. 
 
 48. Since a right line and a point outside of it are sufficient to 
 determine a plane, the point in Fig. 55 may be regarded as lying 
 in a plane of w^hich cd is the horizontal trace ; and when this point 
 has been revolved about that trace into the position o" ^ it is clear 
 that the whole plane just mentioned will coincide with H, so that 
 any lines or figures which it had contained originally, or which may 
 now be drawn upon it, will be seen in their true forms, dimensions, 
 and relative positions. 
 
 In like manner, the same point O lies in another plane of which 
 f'r' is the vertical trace, and when reaches o" ^ that plane coin- 
 cides with V. x\ny plane, then, can be revolved about either trace 
 into the corresponding plane of projection ; and it is clear that by 
 revolving it about a line parallel to one of those traces, it may be 
 made parallel to the corresponding plane. 
 
 49. Counter-reYolution of a Plane. In Fig. ^^^ let the point/* 
 be revolved about the axis EF into the j^lane of projection LL as 
 in Fig. 54, ^'' being its revolved position, and G the centre of its 
 
 circular path. The plane PCEwqtn coincides with LL\ assuming 
 o" to be the revolved position of any point therein other than P, 
 
28 DESCRIPTIVE GEOMETRY. 
 
 it is required to find its position wlien the plane is revolved back 
 again. 
 
 Draw ]p"o" and produce it to cut the axis in D ; then the re- 
 quired original position of o" must lie on the line PD^ whose pro- 
 jection on LL is j?Z). The plane of rotation of the second point is 
 parallel to that of the first, and its trace is o" G perpendicular to 
 F.F\ which cuts jpD in 6>, the projection of the point sought. 
 Therefoi-e erect at r? a j)erpendicular to LL^ cutting PD in 0^ the 
 required point. 
 
 50. This construction, in projection, is shown in Fig. 57, 
 where EF in the horizontal plane is the axis, P the first point in 
 its original position, andj^'' its revolved position determined as in 
 Fig. 55. Take o" as the revolved position of tlie second point ; 
 draw j?"(>'' and produce it to cut the prolongation Gi fe in d. This 
 is the horizontal projection of a point in H, therefore its vertical 
 projection is d' in AB, and j9<^, jp'd\ are the projections of the line 
 PD whicli contains the point sought. In the counter-revolution, 
 o" describes a circle in a vertical plane perpendicular to the axis, of 
 which the horizontal trace is perpendicular to <?/", and cuts^^ in 6>, 
 the horizontal projection of the required point; o' \v^ i^'d' \^ its 
 vertical projection. 
 
 SEPARATE CONSTRL^CTION OF THE VERTICAL AND HORIZONTAL 
 PROJECTIONS. 
 
 51. The combination of the projections on H and V in one 
 diagram, and the use of AB to represent one plane in reading tlie 
 vertical and the other in reading the horizontal projection, is often 
 a source of perplexity at first, even to those thoroughly familiar 
 with the various views which represent solid objects in ordinary 
 mechanical drawing. Moreover, this combination is often a cause 
 of excessive and needless obscurity ; the various lines used in repre- 
 senting magnitudes, operations of revolution, counter-revolution, 
 and what not, upon one plane, becoming so interwoven with those 
 of the projections upon the other, as to present a bewildering maze 
 €ven to tlie expert in reading such diagrams. 
 
 Now, the fact that these two projections can be and always have 
 been thus combined is not at all a good reason why they always 
 
DESCRIPTIVE GEOMETRY. 
 
 29 
 
 ^lioiild be. Tliey may be constructed separately as sliown in Fig. 
 58, where in'n' is the vertical projection of a line, i/7/ represent- 
 
 3/ 
 
 VERTICAL PROJECTION 
 OR FRONT ELEVATION 
 
 H H- 
 
 PROFILE OR 
 END ELEVATION 
 
 .Fig. 58 
 
 HORIZONTAL PROJ. 
 
 PLAN, OR TOP VIEW \m, 
 
 ing the horizontal plane; mn is the horizontal projection, YY 
 representing the vertical plane ; and at the left of the vertical pro- 
 jection is the profile, where the two planes are shown in their true 
 relative positions. These three views correspond to what in working 
 drawings are called the front elevation or side view, the top view 
 or plan, and the end view or end elevation. For the purpose oi 
 comparison, the combined diagram of the projections of the same 
 line is sliown in Fig. 59 ; and there are no doubt many to whom 
 the latter will seem far less clear, and more difficult to read, even 
 in regard to so simple a magnitude as this oblique line. 
 
 In what follows, both methods wdll be used, as circumstances 
 may indicate one or the other to be the more convenient. The 
 student, of course, may use either in any case at pleasure. It is 
 desirable that he should become familiar with both, but in any 
 given construction he alone can tell which method seems to him 
 the clearer, and that is tlie one for him to adopt. 
 
 SUPPLEMENTARY PLANES OF PROJECTION. 
 
 52. Thus far but three planes of projection have been con- 
 sidered, viz., H, V, and the profile plane. It will, however, fre- 
 quently be found convenient to make use of others, which may be 
 called supplementary planes ; upon these the object is projected, 
 remaining fixed in respect to the principal planes. 
 
 The positions of such supplementary planes are determined 
 wholly by conditions of convenience, and therefore depend upon 
 
30 
 
 DESCKTPT[YE GEOMETRY. 
 
 the nature of the object; but thej are in the great majority of 
 cases such that the j)lanes are j^erpendicular to one of the principal 
 planes ; indeed, it may be said tliat they are probably more often 
 vertical than otherwise. 
 
 .53. The use of such a plane is j^it^torially represented in Fig. 
 60, in which OR is, let us say, an oblicpiely placed wire, suj^ported 
 
 Fig. 60 
 
 by two vertical ones fixed in the horizontal plane. There are 
 shown the projections of these not only upon H and T, but uj^on a 
 supplementary plane S, in this case parallel to the horizontal jiro- 
 jecting plane Or. The three lines are therefore projected upon S 
 in their true lengths and relative positions, while upon II and V tliey 
 are not ; and it is for tlie purpose of thus convejdng directly in- 
 formation which the other views do not give explicitly, that such 
 supplementary projections are chiefly employed. 
 
 It is obvious that in viewing S perpendicularly, as indicated by 
 the arrow, the axis of vision is parallel to H, which, being thus seen 
 edgewise, will in projection be represented by a line bearing the 
 same relation to this new drawing that the original ground line 
 bears to the vertical projection. 
 
 54. This is shown in Fig. 61, where <?r, o'r' are the projections 
 upon H and V of a limited oblique line, like OR in Fig, 60. Be- 
 low is drawn a supplementary view, looking perpendicularly 
 against the horizontal projecting plane, as shown by the arrow a. 
 
 The horizontal plane is represented by the line II' H' perpen- 
 dicular to the arrow, and the points o andr, being projected per- 
 pendicularly toward this line, appear in the new view at distances 
 
DESCRIPTIVE GEOMETRY. 31 
 
 above it equal to those of o' and r' above AB. To use a geo- 
 grapliical illustration, if the projection upon V be regarded as a 
 view from the south, the observer looking due north, this supple- 
 mentary jDrojection is a view of the same object from the south- 
 west, the observer looking northeast ; bj this change of position on 
 the part of the spectator tlie altitudes of the various parts of the 
 object are clearly neither increased nor diminished. Also, the line 
 OR is now seen in its true length. 
 
 55. But again, the object still retaining its original position, 
 the eye may be sup|)Osed to be above and at the same time either 
 to the right or to the left of it, and to be directed, not vertically, but 
 obliquely dowuAvard, yet still in a line parallel to V. In this case 
 the vertical plane will be seen edgewise, but the vertical projecting 
 lines will remain unchanged in length, so that all the points of the 
 object will appear just as far from that plane as in the original 
 horizontal projection. 
 
 In illustration, a supplementary view is given in the upper part 
 of Fig. 61, looking in the direction shown by the arrow h. The 
 line y F', perpendicular to the arrow, represents the vertical 
 plane toward which the points o' ^ r' are projected, their distances 
 from y y being equal to those of the horizontal projections 6>, r, 
 from AB. 
 
 56. Such supplementary projections, like the profiles, should 
 always be constructed as detached and independent view^s; their 
 precise location is of course arbitrary, but should always be such as 
 to j^revent the possibility of confounding the lines with those of the 
 other views. 
 
d2 DESCRIPTIVE GEOMETRY. 
 
 CHAPTER II. 
 
 ELEMENTARY PROBLEMS RELATING TO THE POINT, RIGHT LINE, AND 
 
 PLANE. 
 
 57. It is necessary to make a clear distinction between tlie 
 solution of a problem and tlie representation of that solution. 
 
 Tlie solution is effected by abstract reasoning : one link after 
 another being added to a cliain of logical ai-guments until a detinite 
 conclusion is reached which demonstrates that the object sought 
 can be accomplished in a certain way. This is a purely mental 
 process ; clear conceptions can be formed in the dark, or by a blind 
 man, of the magnitudes involved, of their relations to each other, 
 of the various steps to be taken and their results — in short, of the 
 complete solution of any problem ; which is wholly independent of 
 its representation and of any graphic operation whatever. 
 
 The processes of descriptive geometry, on the other hand, are 
 purely graphic. And it is the province of this science to explain 
 the methods, not of solving j^roblems, but of exactly representing 
 the data, steps, and results of solutions already effected by mathe- 
 matical reasoning. This distinction is natural and inevitable, be- 
 cause before a thing can be represented it must be known what 
 that thing is. 
 
 58. Analysis and Construction. A problem being enunciated, 
 then, its treatment will consist of two distinct parts. First^ a clear 
 statement of the principles and reasoning employed in the solution 
 and applied to the magnitudes in space ; this is the analysis. Second y 
 an explanation, in due order, of the lines employed in representing^ 
 on paper, the problem and its solution ; this is called the construction 
 of the problem. 
 
 59. Method of Study. The same processes may in general be 
 applied to magnitudes under widely varying conditions; and in the 
 
DESCKIPTIYE GEOMETRY. 
 
 33 
 
 nature of tilings but a limited number of eases, and often only one, 
 can be worked out in illustration. Consequently great care should 
 be taken to avoid associating the solution of any problem with tlie 
 aj^pearance of the figure, because the assumption of different data 
 may result in the production of a totally dissimilar diagram. This 
 is of especial importance in regard to these elementary problems, 
 because they are subsequently to be used as mere steps in the solu- 
 tion of more complex ones, and the conditions thus fixed may be 
 quite unlike those previously met with. The best course therefore 
 is to di823ense, as soon and as far as possible, with all reference to 
 the illustrations ; first mastering the analysis and fixing in mind the 
 successive stej)s, and then making an original construction by apply- 
 ing them to assumed data. 
 
 60. Problem 1. To find the true distance between two imlnt^ 
 given hy their jyrojections. 
 
 Analysis. The required distance is the length of the right line 
 joining the two points. If either projecting plane of this line be 
 revolved about its trace into the corresponding plane of projection, 
 the line will be seen in its true length. 
 
 Construction. This is represented pictorially in Fig. 62, and 
 Fig. 63 shows it in projection, M and N being the given points. 
 Revolving the horizontal projecting plane of MN about nm into H, 
 
 Fig. 62 
 
 liie point M goes to m" ; mm" being perpendicular to mn and equal 
 to tn'x^ the horizontal projecting line. Similarly, N goes to n" ^ the 
 distance nn" being equal to n'y\ and m"n" is the distance required. 
 The required distance may also be ascertained by means of a 
 supplementary projection, as shown in Fig. 61, or by means of an 
 
34 
 
 DESCRIPTIVE GEOMETRY. 
 
 independent construction, apart from tlie drawing, as explained in 
 (15). 
 
 JS^.B. In this case the distances tnm"^ nn'\ are set off in op23osite- 
 directions, because the points M and N are on opposite sides of the 
 axis. Had tliej been on the same side, as M and P are, these dis- 
 tances would have been set off in the same direction. In either 
 case, the line joining the two points, if not parallel to the plane 
 into which it is revolved, will pierce it if j)rolonged, as at o^ o' in 
 Fig. 63. This point of penetration, being in the axis, remains 
 fixed, and the given line must pass through it in its revolved as 
 well as in its original position; thus 7irh"n" passes through o. 
 
 61. If it is required to set off from Jf a distance along J/iV^ 
 equal to a given line cd^ first revolve the line MN into H as above, 
 and then lay off m!'p" equal to cd. In the counter-revolution, j)" 
 goes, in a direction perpendicular to mn^ to the position^, ^vhich is 
 vertically projected at p' on m'n\ This operation is identical with 
 that represented in Figs. 56 and 57. 
 
 62. Problem 2. To jpass ajplane through three given points 
 tiot in the same right line. 
 
 Analysis. Through either tw^o of tlie points draw a right line. 
 Through the third point draw another right line, either parallel to 
 the first or intersecting it at any point. These two Ihies determine 
 the plane, and their traces wiU be points in the corresponding traces 
 of tlie plane. 
 
 Construction. In Figs. 64 and 'oOy C\ J), and J^ are the given 
 
 Fig. 64 "^ Fig. 65 
 
 points. Draw CD and produce it to pierce II at 6>, and V at Jf ; 
 
 then <9 is a point in the horizontal and m' a point in the vertical 
 trace of the required plane. Join the third point E with any point 
 
DESCRIPTIVE GEOMETRY. 35 
 
 G of OM^ and produce EG to pierce H and V at 7? and N\ then r 
 is another point in tlie horizontal and n! another point in the verti- 
 cal trace. Therefore n'm' ^ or are tlie required traces, which, 
 when produced, must meet in the ground line, unless tliej are 
 parallel. 
 
 Note. — The direction of the second line EG should be so chosen 
 that the distance between o and r^ and also that between /?/ and m\ 
 shall be as great as possible. 
 
 63. The problems of drawing a plane through one right line 
 and parallel to another, and of drawing a plane through a given 
 point parallel to two given right lines, are scarcely more than varia- 
 tions of the preceding one ; for, in the first case, we have already 
 one line of the required plane and know the direction of another, 
 w^hich may be drawn through any given point of the given line ; 
 and in the second case, we know the directions of two lines of the 
 required plane and have merely to draw them through a given 
 point and find their traces. If either line be parallel to AB, the 
 plane itself and both its traces will be parallel to the ground line. 
 In this case a profile should be drawn, in addition to the projections 
 on II and V. 
 
 64. Problem 3. To draw through a given jpoint a plane per- 
 'pendieular to a given right line. 
 
 Analysis. Tlie directions of the traces are known, being respec- 
 tively perpendicular to the projections of the line (43). Draw 
 through the given point a line parallel to either trace ; this wiU 
 be a line of the plane, and will pierce the other plane of projection 
 in a point of the required trace upon that plane. This trace, being 
 perpendicular to the corresponding projection of the line, may now 
 be drawn ; it will cut the ground line in a point of the remainiDg 
 trace, of which the direction is also known. 
 
 Construction. Let P, Figs. QQ and 67, be the given point and 
 MN the given line. Draw through P a line parallel to the hori- 
 zontal trace of the required plane; its horizontal projection isj[?o, 
 perpendicular to mn^ and its vertical projection \s, j[>'o\ parallel to 
 AB. This is a line of the plane, and its vertical trace is a point 
 in the vertical trace of the plane. Therefore t'o'T^ perpendicular 
 
36 
 
 DESCRIPTIVE GEOMETRY. 
 
 to m'n',^ is that vertical trace, Avliicli cuts AB at T\ and Tt^ per* 
 pendicular to inn^ is the horizontal trace. 
 
 N. B. If the projections of the given hne coincide, as in Fig. 29, 
 the traces of the plane will also coincide, like those of the plane 
 tTt' in Fig. 40, If the given line lie in a profile plane, the re- 
 quired plane will be ]3arallel to the gronnd line. Thus in Fig. ^'^^ 
 P is the given point, MN the given line ; these are seen in their 
 true relations to H and V in the prohle, Fig. 69, where a per- 
 
 n 
 
 
 
 \ 
 
 
 *P 
 
 m 
 
 Fig. 68 
 
 Fig. 69 
 
 pendicular to MN through P represents the required plane, cutting 
 V in ^' and H in ^; these points are the profile projections of the 
 traces tt^ t't^ in Fig. ^'^. 
 
 65. The reasoning in the analysis of this problem is precisely 
 the same as that used (41) in reference to the drawing of a plane 
 through a given point and parallel to a given plane, the construc- 
 tion of which was shown in Fig. 50. The gist of the argument is 
 Bimply this, that when the directions of the ti-aces are known, the 
 location of a single point in either trace determines the plane, ex- 
 cept when it is parallel to the ground line \ in that case a point in 
 each trace must be found. 
 
DESCRIPTIVE GEOMETRY. 
 
 37 
 
 66. Problem 4. To find tJie intersection of two flanes. 
 
 Analysis. The intersection of the vertical traces will be one 
 point, and the intersection of the horizontal traces will be anotiier, 
 in the required line, which is determined by those two points. If 
 both planes are perpendicular to either of the principal planes, their 
 traces on the other jDlane will be parallel to each other and to the 
 required line, which Avill pass through the intersection of the other 
 two traces. If both planes are parallel to the ground line, the re- 
 quired line will be so likewise ; it is determined by tlie intersection 
 of the profile traces of the given planes. 
 
 Construction. In Figs. 70 and 71, sSs\ tTt' are the two planes. 
 
 Fig. 72 
 
 Fig. 74 
 
 Fig. 73 
 
 The horizontal traces intersect at c, whose vertical projection is <f 
 in AB, and the vertical traces intersect at d' ^ whose horizontal pro- 
 jection is d in AB; therefore cd is the horizontal and c'd' the ver- 
 tical projection of the required intersection. 
 
 AYhen both planes are vertical, their intersection is vertical and 
 passes through the intersection of the horizontal traces, as shown in 
 
 Fig. 
 
 <2, at the left; when the planes are perpendicular to V, as 
 
38 
 
 DESCRIPTIVE GEOMETRY. 
 
 shown in the same figure at the right, tlieir intersection is- also per- 
 pendicular to y and passes through the intersection of the vertical 
 traces. 
 
 When both planes, and consequently their intersection, are 
 jDarallel to AB, the detached profile. Fig. 73, shows the condition of 
 things with perfect distinctness ; but the projection on the princi- 
 pal planes. Fig. 74, is bj itself simply useless as a means of impart- 
 ing information. 
 
 67. Some Special Cases of the Above Prchlein. — In Fig. 
 75 the horizontal traces do not intersect within the limits of the draw- 
 ing ; but one point, i>, of the required Kne is determined by the 
 intersection of the vertical traces. In order to ascertahi its direc- 
 tion, draw an auxiliary plane ILL \ parallel to tTt' and cutting sSs' 
 
 Fig. 77 
 
 Fig. 78 
 
 Fig. 79 
 
 in the line JfiT, found as in Fig. 71. Tliis intersection is parallel 
 to the one sought, of which, therefore, the vertical projection is c'd\ 
 parallel to r)i'n\ and cd, parallel to mn^ is its horizontal projection. 
 In Fig. 76 the intersections of the vertical traces and of the 
 horizontal traces are both inaccessible. Draw an auxiliary horizon- 
 tal plane, of which Jc'h' is the trace. This cuts the plane tTt' in a 
 
DESCRIPTIVE GEOMETRY. 3.9 
 
 line of which one point is r' on Y, horizontally projected at r on 
 AB ; this line, being horizontal, is parallel to the horizontal trace 
 and its horizontal projection is therefore drawn through r and par- 
 allel to Tt. The auxiliary plane also cuts sSs' in a line whose hori- 
 zontal projection is drawn through o, parallel to Ss. These two 
 lines, one in eacli given plane, cut each other in a point of which 
 the horizontal projection is <?, and the vertical projection is c' on 
 k'k' . Thus one point in the required line is determined, and its 
 direction is ascertained as in Fig. 75. 
 
 In Fig. 77 the traces of the plane tTt coincide as in Fig. 40, 
 and sSs' is a profile plane cutting the ground line at the same point ; 
 in Fig. 78 both planes are oblique, with coincident traces. Draw- 
 ing in each case an auxiliary plane parallel to sSs^ as in Fig. 75, 
 the line MN cut from the j)lane tTt' is parallel to the required 
 intersection. And in each case this line pierces H behind Y, and 
 Y above H, in points equally distant from AB ; it therefore crosses 
 the second angle, as shown in the profile. Fig. 79, and is equally 
 inclined to H and Y. The required line CD^ being parallel to MN 
 and intersecting AB, therefore lies in a profile plane and bisects the 
 first and third angles ; its projections in Fig. 77 coincide in S8\ and 
 in Fig. 78 they coincide in a line through T, perpendicular to AB. 
 
 68. Problem 5. To find the ^oint in which a gwen right line 
 'pierces a given plane. 
 
 Analysis. Pass any plane through the given line and find its 
 intersection with the given plane. This line will cut the given line 
 in tlie required point. 
 
 Construction. In Figs. 80 and 81, MN \s> the given line, tTt' 
 the given plane. Since any plane containing MN will serve our 
 purpose, we use for convenience one of its projecting planes; in 
 this case the horizontal. Its horizontal trace coincides with the 
 horizontal projection of the line, and its vertical trace is perpen- 
 dicular to AB ; it intersects tTt' in the line (7i>, whose vertical pro- 
 jection c'd' cuts the vertical projection m'n' in <?', the vertical pro- 
 jection of the required point ; the horizontal projection is 6>, on cd. 
 
 N. B. Had the intersection at o' been very acute, the deter- 
 mination would have been less reliable ; "and a better result might 
 have been obtained by using the vertical projecting plane, thus de* 
 
40 
 
 DESCRIPTIVE GEOMETRY. 
 
 terminiiig first the horizontal projection o of the required point. It 
 is not certain that this would happen, since if the line were but 
 slightly inclined to the plane, both these intersections would be 
 acute ; in which case both determinations should be made, and if 
 
 JFiG.Sl 
 
 Fig. 80 
 
 they do not agree, a mean between them may be taken as the cor* 
 rect result. 
 
 69. The preceding construction involves the use of the traces 
 of the given plane : but if two lines of a plane are given, it is not 
 necessary to dnd the traces in order to determine the j)oint in which 
 it is pierced by a third line. Thus in Fig. 82, let it be required to 
 
 Fig. 82 
 
 Fig. 83 
 
 find the point in which the line MN pierces the plane determined 
 by the two intersecting lines Kl^ Gl. Using again for conven- 
 ience the horizontal projecting plane mdd' of the given line, it cuts 
 
DESCiUPTlVE GEOMETRY. 
 
 41 
 
 KI in E^ and GI in F\ EF therefore lies in both planes, and tlie 
 point in which it intersects MN^ is the required point. The 
 traces of the given j)lane are shown in this pictorial representation, 
 for tlie purpose of calling attention to the fact that EF\^ merely a 
 portion of the same line of intersection GD^ which was determined 
 in Fig. 80 by means of the traces of the two planes. 
 
 The construction in projection is given in Fig. 83, where GR^ 
 KL^ intersecting at /, determine a plane, and it is required to find 
 the point in which this plane is pierced by the line MN. The 
 horizontal trace mn^ of the horizontal projecting plane, must con- 
 tain the horizontal projections of all lines and points that lie in it, 
 because the plane is vertical. And mn cuts gr at/*, which is the 
 horizontal projection of a point on GR^ whose vertical projection 
 is/" on g'r' . Similar Iv, the line KL is seen to pierce the project- 
 ing plane in a point whose horizontal projection is e^ and whose 
 vertical projection is e' on h'V . Consequently e'f is the vertical 
 projection of the portion of the line of intersection thus determined ; 
 it cuts m'n in o\ the vertical projection of the required point, and 
 the horizontal projection is o on mn, 
 
 70. Some Sjyecial Cases of the Above Problem. In Fig. 84, 
 the given plane is parallel to AB, and the projections of the 
 
 Fig. 86, 
 
 V 
 
 i 
 
 
 
 o' 
 
 
 
 N 
 \ 
 
 H 
 
 /y 
 
 
 Fig. 87 
 
 • V 
 
 
 
 given line MN coincide. The horizontal trace of the horizontal 
 projecting plane cuts tt at <?, vertically projected at g' in AB ; its 
 
42 DESCRIPTIVE GEOMETRY. 
 
 vertical trace cuts t't' at d\ of whicli d is the horizontal projection; 
 and c'd' intersects Tn'n' in o\ the vertical projection, which coin- 
 cides with 6>, the horizontal projection of the required point. 
 
 In Fig. 85, the traces of tTt' coincide, and MN is parallel to 
 AB. The horizontal projecting plane cuts tTt' in a line parallel to 
 y aivd therefore to Tt' \ one point of this line is determined by the 
 intersection of the horizontal traces at 6', vertically projected at c' 
 in AB, and its vertical projection c'o' cuts m'n' at o\ of which o on 
 7nn is the horizontal projection. 
 
 In Fig. '^^^ the two projections of J/LZTare nearly perpendicular 
 to AB. In such cases the direct determinations by the method* 
 before explained are apt to be very unreliable on account of the 
 acuteness of the intersections : and the profile may be used to great 
 advantage in the manner here i]lustrated. In this instance the 
 vertical projecting plane of the given line has been used ; its verti- 
 cal trace cuts Tt' at x' ^ and the ground line at L ; its horizontal 
 trace is perpendicular to AB and cuts Tt at ?/. Tlie line of inter- 
 section will therefore pass through x' on V and y on H ; but its 
 projection on the latter is not drawn. In drawing the profile, Fig. 
 87, x' is projected horizontally across from Fig. 86, and the distance 
 of ?/ from FF is equal toZy in the horizontal projection; then x'y 
 represents the line of intersection. In this particular case MN 
 pierces V ati\'^, therefore n' is projected directly across to W\ 
 the altitude of M is the same in both views, and so is its distance 
 from V ; and MN in the profile intersects x'y in 0^ which being 
 projected back to in'n! in Fig. 86, determines o' the vertical projec- 
 tion of. the required point. Tlie distance of from T is seen in 
 the profile ; and drawing in Fig. ^^^ a parallel to AB at that dis- 
 tance from it, the horizontal j)rojecti()n o is determined much more 
 accurately than it could be by drawing through o' a perpendicular 
 to AB. 
 
 Should the given line lie in a plane perpendicular to AB, tlic 
 construction of a profile is of course a necessity. 
 
 71. Problem 6. To find the distance of a given ^oint from 
 a given jplane. 
 
 Analysis. ]. Draw through the point a perpendicular to tlie 
 
DESCRIPTIVE GEOMETRY. 
 
 43 
 
 plane. 2. Find tlie point in wliicli it pierces tlie plane. 8. Find 
 the distance between tins point and the given point. 
 
 Construction. In Fig. 88, let P be the given point, tTt' the 
 given plane. Draw through P a perpendicular to tTf as in Fig. 
 
 4 
 e 
 
 „ .^ — 7*" 
 
 c/' >' t/ 
 
 / 
 
 d 
 
 \JrKf 
 
 V \p FiG.89 
 
 
 Fig. 88 
 
 52. Find the point in which it pierces the plane as in Fig. 81. 
 Find the true length oi PO as in Fig. 63. 
 
 In Fig. 89, the direction of the vertical trace Tf is the same as 
 in Fig. 88, but that of the horizontal trace 2Y is different. In con- 
 sequence of this change, the vertical trace of the horizontal pro- 
 jecting plane cuts that of the given plane at a point d' below AB in- 
 stead of above it as before, and d^c' must be produced to detennine 
 g\ PO is here revolved into V instead of H; it pierces Y at a 
 point of which the horizontal projection is ^, and the vertical pro- 
 jection is e' on p'o^ produced ; and since e' is on the axis, it re- 
 mains fixed, and the prolongation oi p"o" passes through it. 
 
 When the given plane is parallel to AB, the required distance is 
 found directly by constructing a profile. 
 
 72. Problem 7. To jproject a given right line tipon a given 
 plane. 
 
 Analysis. Through any point of the given line, draw a per- 
 pendicular to the given plane : these two lines determine a second 
 plane, perpendicular to the first. The intersection of these two 
 planes is the required projection. 
 
 Construction. In Fig, 90, let KG be the given line, tt the hori- 
 zontal trace, and t't' the vertical tra(;e of the given plane. From any 
 point P on KG^ draw a perpendicular to the plane ; the traces of 
 
44 
 
 DESCJtiPTiVE geomp:try. 
 
 this perpendicular are X and Z, and the traces of the given Hne are 
 TJ and Y. Therefore 112 is the horizontal and x'y' is the vertical 
 trace of tlie plane sSs ^ determined by the given line KG and the 
 projec ing perpendicular PX. This plane cuts the given plane in 
 the line CD^ which is the required projection. 
 
 73. This intersection CD evidently must contain the point TT, 
 in which the given line pierces the plane tTt' ^ and also the point (9, 
 which is the foot of the perpendicular let fall upon the plane from 
 the point P ; and ON is the projection of the hypothenuse PN of 
 the right-angled triangle PON. 
 
 The points and N might have been found as in Problem 5, 
 without determining the traces of sSs . And if the projection of a 
 definite portion of the line, as for instance PG in the figure, is 
 required, two perpendiculars, as PO^ GA, may be drav/n, and 
 the points of penetration, and A^ found in the same way; 
 indeed, this may be necessary, if the line is but slightly inclined to 
 the plane. The three methods are identical in j^i'inciple, and the 
 selection must depend upon considerations of convenience, deter- 
 mined by the given conditions in any particular case. 
 
 If the giTen line be parallel to the given plane, its projection on 
 
DESCRIPTIVE GEOMETRY. 
 
 45 
 
 that plane will be parallel to the line itself (14). Therefore the 
 required projections will be parallel to those of the given line, and 
 the determination of one point in each is sufficient. 
 
 74. Some Special Cases of the Above Problem. In Fig. 91, 
 tTt' is the given plane; and the given line KG is parallel to 
 AB. Draw througli any point P on GK 2, perpendicular to tTt' \ 
 it pierces the horizontal plane in Z and the vertical plane in 
 X. The plane of these two lines is parallel to AB, therefore its 
 traces are sz and s'x ^ also parallel to AB ; and it cuts tTt' in the 
 line CD^ the required projection. 
 
 In Fig. 92, tTt' is the given plane ; it is required to project the 
 ground line on it. From any point P on AB draw Pr perpen- 
 dicular to Tt and Pr' perpendicular to Tt' ; these are the pro- 
 jections of a line perpendicular to the plane. The vertical pro- 
 jecting plane of this line cuts tTt' in the line (72>, which intersects 
 PR in (9, the projection of the point P upon the given plane. 
 That plane cuts AB in the point T\ consequently To is the hori- 
 zontal and To' is the vertical projection of the required line. 
 
 In Fig. 93, the given line is inclined to both planes, piercing H 
 in the point ^and.V in the point Y. The given plane being par- 
 allel to the ground line, the perpendicular to it from the point P 
 
 Fig. 94 
 
 will lie in a plane perpendicular to AB ; its traces are readily deter- 
 jnined by drawing the profile, Fig. 94 ^ then setting off z and x' in 
 Fig. 93 at the distances from AB thus found, we have, as before, 
 
46 
 
 DESCRIPTIVE GEOMETKY. 
 
 uz for the horizontal and x'y' for the vertical trace of the plane 
 sSs\ which cuts tTt' in the Hue CD^ the required projection. 
 
 75. Problem 8. To find the distance of a given point from a 
 given line. 
 
 First Method. Analysis. 1. Through the given point pass a 
 plane perpendicular to the given line. 2. Find the point in which 
 the given line pierces this plane. 3. Find the distance between 
 this point and the given point. 
 
 Construction. In Fig 95, let P be the given point, KG the 
 given line. Draw through P a plane tTt' perpendicular to KG^ as 
 
 Pig. 96 
 
 in Fig. 67. Find the point 0, in which KG pierces tTt\ as ip 
 Fig. 81. Find the length of PO as in Fig. 63. 
 
 Special Case. In Fig. 96, the two projections of P coincide, 
 as do those of KG ; consequently the traces of the perpendicular 
 plane tTt' also coincide. The horizontal projecting plane of KG 
 cuts tTt' in the line CD^ which intersects KG in 0. The line KG 
 lies in a plane bisecting the second and fourth angles, and cuts AB 
 at iV"; the line OP also lies in that bisecting plane, and since it is 
 at the same time a line of the plane tTt\ it will when produced cut 
 AB at T. Therefore NT is the hypothenuse of the right-angled 
 triangle TOK\ and when this triangle is revolved about AB into 
 either V or H, will fall at o" on the circumference of a semicircle 
 of which NT \& the diameter, and /-* falls at/)" on o"T. 
 
DESCRIPTIVE GEOMETRY. 
 
 47 
 
 76. Second Method. Analysis. Througli the given point draw 
 a line either parallel to or intersecting the given line. Revolve the 
 plane of these two lines about one of its traces into tlie corresponding 
 plane of projection ; the line and point will then be seen in their 
 true relative positions. A perpendicular from this revolved position 
 of the point to that of the line will be the required distance. 
 
 Construction. In Fig. 97, P is the given point, KG the given 
 line. Draw through P a parallel to KG ; it pierces Y in iV, and 
 KG pierces it in M. Eevolve this plane about its vertical trace 
 
 Fig. 98 
 
 Tim' into Y\ P goes to p" ^ and as n' remains fixed, being in the 
 axis, jp"n' is the revolved position of the second line. And since 
 the two lines are parallel, a line through m', parallel to p"n\ is the 
 revolved position of the given line, and Jp"o" ^ perpendicular to it, 
 is the actual distance required. 
 
 In the counter-revolution, jp" returns to jp\ and o" goes, in a 
 direction perpendicular to the axis mn\ to the position o' on Tn'h' . 
 This is the vertical, and o on hm is the horizontal, projection of <?, 
 the foot of the perpendicular drawn from the given point P to the 
 given line. 
 
 Special Case, In Fig. 98, the projections of the given line 
 KG coincide, and the given point P is in the vertical plane. There 
 is a point on KG whose projections, c and c\ coincide with p' . The 
 vertical projecting line of this point therefore passes through the 
 given point, and, with the given line, determines a plane of which 
 c' N \'& the vertical trace. Revolving this plane about this trace 
 
48 
 
 DESCRIPTIVE GEOMETRY. 
 
 into V, C goes to g'\ c" N is the revolved position of the given hne, 
 and J?' is stationary. Therefore j^'o" perpendicular to c" N is the 
 required distance. In the counter-revolution g" returns to c\ o" 
 goes to o' on Tt'g'^ and jpo^ jp'o\ are the projections of the perpen- 
 dicular. 
 
 77. Problem 9. To find the angle hetween two lines ^ and ta 
 divide it. 
 
 Analysis. If the plane of the two lines be revolved about one 
 of its traces, or a line parallel thereto, until it coincides with or iz 
 parallel to the corresponding plane of projection, the angle will be 
 seen in its tnie size and may be subdivided. 
 
 Otherwise : If a supplementary projection be made upon a. 
 plane parallel to that determined by the given lines, the angle will 
 appear in its true size. 
 
 Construction. In Fig. 99, the two lines which intersect in 
 pierce H in the points P and N. Eevolving them about the 
 
 horizontal trace 7i/p of their plane 
 into H, their intersection goes- 
 to o'\ the distance xo" being equal 
 to the hypotlienuse of a triangle of 
 which ox is the base and o'y the 
 altitude. Tlius the angle is seen 
 in its true size as no"2y ; if it be 
 now required to divide it, for in- 
 stance into two equal parts, the 
 bisecting line cuts njp at r^ which 
 in the counter-revolution will re- 
 main stationary : therefore 07\ o'r\ 
 are the projections of the bisector.. 
 
 78. In Fig. 100, the horizontal trace of OP is inaccessible. 
 Draw therefore an auxiliary horizontal plane AA, cutting the given 
 lines in C and .Z>, and revolve the lines about cd until they are 
 parallel to H ; in this revolution goes to o'\ the distance xo" be- 
 ing equal to the hypotlienuse of a triangle liaving for its base ox^ 
 and for its altitude o'y^ the distance of tlie point from tlie plane 
 hh. Then co"d is the angle in its true size, the bisector cuts cd in 
 Sy and its projections after the counter-revohition are os^ o's' . 
 
 Fig. 99 
 
DESCRIPTIVE GEOMETRY. 
 
 49 
 
 In Fig. 101, the line OP is but slightly inclined to AB, so that 
 it is inconvenient to find where it pierces either of the principal 
 
 Fig. 102 
 
 planes. In this event, draw a plane ZZ, perpendicular to AB, cut- 
 ting the given lines in the points P and N. Constructing the pro- 
 iile, Fig. 102, the vertex of the angle appears at ^,, and n^jp^ is the 
 trace of the plane of the two lines upon the plane LL. Revolve 
 the lines about this trace into that plane ; 6>, goes to o'\ the distance 
 xo" being equal to the hjpothenuse of a triangle whose base is o^x^ 
 the distance of the projection of the point from the axis, and the 
 altitude is o'y^ the distance of the point from the plane in which 
 the axis lies ; nfi"p^ is the angle in its true size. 
 
 79. In Fig. 103, the line OP is perpendicular to V; and so, 
 consequently, is the plane of the two given lines. A supplement- 
 ary projection upon a plane parallel to this plane at once shows the 
 angle in its true size. In this projection, the vertical plane will 
 appear (55) as a line FT^ parallel to o'li'^ and OP as a line o^p^ 
 perpendicular to VV. Draw any line NM cutting both the given 
 lines: the points uj^on OP will all be vertically projected in o\ 
 and the vertical projection of iV^ is n\ These points are all pro- 
 jected perpendicularly toward W^ and the distances of ??z,, 7i,, o^^ 
 from VY^ in the new projection, are equal to the distances of 7/z, n^ 
 and from AB in the horizontal projection. The angle, being now 
 seen in its true size, may be bisected as before, the bisector cutting 
 mi?i, at r, : this point is projected back to 7'\ and thence to r, giv- 
 ing 07' as the horizontal projection of the bisecting line. 
 
60 
 
 DESCKIPTIVE GEOMETRY. 
 
 80. Such projections as this are not only of great convenience 
 in many of tlie operations in abstract descriptive geometry, but they 
 are of every-day occurrence in making mechanical drawings for in- 
 dustrial purposes. Indeed, the construction in Fig. 103 is identi- 
 
 Fig. 104 
 
 cal with that used in making the three views of the draughtsman's 
 triangle, shown in Fig. 104. Tliis implement is seen edgewise in 
 the front view, wliere the side CD is perpendicular to the paper ; and 
 it is foreshortened in the top view. Now in order to exhibit the 
 true form and size, no practical man will " revolve it about its side 
 until it becomes horizontal," or otherwise disturb it: on the con- 
 trary, leaving it at rest, a view is drawn, looking perpendicularly 
 against it as indicated by the arrow. There is no need of using a 
 reference plane, like YY'wl Fig. 103, in ordinary cases, because 
 its place is supplied by lines or planes of the object itself : thus, the 
 distances from EC in this instance are the same in the original top 
 view and in the third or supplementary view. 
 
 81. Special Case of the Above Problem. To find the angle 
 included between the vertical and horizontal traces of a given plane : 
 Kevolve the vertical trace about the horizontal trace into H. In 
 Fig. 105, let be any point on the vertical trace of the plane 
 tTt' ; its vertical projection is o on Tt' ^ and its horizontal projec- 
 tion is o on AB. In the revolution, goes to o'\ on a perpendicu- 
 lar to 215 through o, the distance xo" being equal to the hypoth- 
 
DESCRIPTIVE GEOMETRY. 
 
 51 
 
 enuse of a triangle of which ox^ oo\ are the base and the altitude. 
 Also, To" is equal to To\ since the distance To' is seen in its true 
 length, and remains unchanged during the revolution: and tTo" is 
 the required angle. 
 
 
 t' 
 
 v^ 
 
 CC/ 
 
 1 \ 
 
 ; r/ 
 
 \ \ 
 
 T/ 
 
 /' 
 
 
 \\ 
 
 !o 1 
 
 A 
 
 
 
 4 Fig. 105 
 
 82. Problem 10. To find the angle between a gwen line and 
 a given plane. 
 
 Analysis. The angle which a line makes with a plane is the 
 same as that included between the line itself and its projection on 
 the plane. 
 
 From any point of the line, draw a perpendicular to the plane. 
 From any other point of the given line, draw a perpendicular to 
 the second line. This third line will be parallel to the projection 
 of the given line upon the plane, and the angle which it makes with 
 the given line is the one required. 
 
 Construction. The pictoral representation. Fig. 106, illustrates 
 the analysis ; the given line MN pierces the given plane tTt' at iT, 
 
 Pig. 106 ^ FiG. 107 
 
 the perpendicular from P pierces it at (?, WO is the projection of 
 j^P, and PNO is the required angle. But PC^ perpendicular to 
 
62 
 
 DESCRIPTIVE GEOMETRY. 
 
 jP(9, is parallel to NO^ and the angle PDC is equal to the angle 
 PNO. Consequently it is not necessary to find either the point 
 iV^or the point O^ but DC md^y be drawn anywhere in the project- 
 ing plane determined by MN and PO. 
 
 Now in Fig. 107, from any point P on the given line 3IN^ 
 draw a perpendicular to the given plane tlY ; it pierces H at 7?, 
 and MN pierces H at E. Eevolve the plane of these two lines 
 into H about its horizontal trace er\ P goes to p" ^ and rjy'e is the 
 angle at P in its true size. From any point d oiip''e^ draw dc 
 perpendicular to p"r^ the revolved position of the projecting line 
 PP\ tliQnp'dc is the required angle. 
 
 Note. Should the given line be inconveniently situated, any 
 parallel to it may be used instead. 
 
 83. The determination of the angle may, however, sonietiiues 
 he facilitated by finding the actual projection of the line upon the 
 plane. Thus in Fig. 108, the given plane is parallel to AB; and 
 
 t' 
 
 L 
 
 
 1 
 
 
 y 
 
 
 /f\ 
 
 
 
 
 t 
 
 Fig. 108 ^^ 
 
 t 
 
 p 
 
 Fig. 109 
 
 the construction is as follows : Draw a plane ZL perpendicular to 
 AB; the given line J/ jY pierces this plane at P, and the given 
 plane at N. Construct the profile. Fig. 109 : the given plane is 
 here seen edgewise as the line t,t,\ to which n' is projected at n, , 
 and the point P is found at j?,. From p, let fall upon t^t/ the per- 
 pendicular p^o^ , and project a, back to o' ; the distance of o below 
 AB in Fig. 108 is equal to the distance of o, in the proiile in front 
 of V. "We have thus the projection JVO of the line NP upon the 
 given plane; and it is seen that the line PO lies in the plane ZZ. 
 
DESCKIPTIVE GEOMETRY. 
 
 53 
 
 Revolve N about p^o^ into this profile plane ; N falls at n'\ the dis- 
 tance o,n" being equal to the hypothenuse of a triangle whose base 
 is o^ti^ , and altitude n'y ; and p{ifi"o^ is the angle sought. 
 
 84. Special Cases of the Above Problem. In Fig. 110, it is 
 required to tind the angle made by the plane tTt' with the ground 
 
 Tig. 110 
 
 ^•x? 
 
 Fig. Ill 
 
 line. From any point P on AB, draw Pe^ Pd\ respectively per- 
 pendicular to Tt and Tt' ; these are the projections of a perpendic- 
 ular to the plane, and is the point in which it pierces the plane. 
 Therefore TO is the projection of AB upon tTt\ and PO is per- 
 pendicular to it. Revolve the plane of these two lines about AB 
 into V ; falls at o'\ and PTo' is the required angle. 
 
 The points <?, d' ^ and o" must lie upon the circumference of a 
 circle whose diameter is TP ; and the construction may be facil- 
 itated by drawing the circle first. 
 
 Ill Fig. Ill, the traces of a plane tTt\ and the projections of a 
 line MN^ coincide in one line inclined to AB ; it is required to find 
 the angle between the line and the plane. From any point P on 
 MN^ draw a perpendicular to the plane ; its projections coincide, 
 therefore this perpendicular cuts AB at R. Revolve the plane of 
 PT and PR about AB into H ; P falls at j?'', the distance xp" be- 
 ing equal to the hypothenuse of a triangle whose base and altitude 
 are each equal iQ> px\ draw Tt*'' perpendicular iop"P^ 1:hGnp"To" 
 is the required angle. 
 
 Evidently o" is also the revolved position of 0^ the foot of a 
 
54 
 
 DESCRIPTIVE GEOMETRY. 
 
 perpendicular let fall from R to tlie plane tTi\ and To" tlie re- 
 volved position of tlie projection of AB upon that plane. There- 
 fore RTo" is the angle made by the ground line with the given 
 plane, and RTjp" is the angle between AB and the given line. 
 
 85- Problem 11. To find the angle hetween two given planes. 
 
 Analysis. Any plane perpendicular to the intersection of the 
 given planes will be perpendicular to both, and will cut each of 
 them in a line. These two lines w^ill be perpendicular to the inter- 
 section at the same point, and the angle between tliem is the 
 required angle. 
 
 Construction. This might be accomplished by ajij^lying the 
 preceding problems, thus: 1. Find the intersection of the given 
 planes as in Problem 4. 2. Through any point of that line draw 
 a plane perpendicular to it, as in Problem 3. 3. Find the inter- 
 section of this plane with each of the given planes, as in Problem 4. 
 4. Find the angle between these two lines, as in Problem 9. 
 
 But a neater and less laborious process is pictorial ly represented 
 in Fig. 112, where CKD^ CID^ are the given planes, and CED is 
 the liorizontal projecting plane of CD their line of intersection. 
 The plane MFN is perpendicular to CD^ therefore its horizontal 
 
 Fig. 112 
 
 trace MN is perpendicular to the horizontal projection CE^ and 
 cuts it at 0. MN is also perpendicular to the vertical line through 
 O^ which lies in the projecting plane. Therefore MN is perpen- 
 dicular to the plane CED^ and consequently to OF^ whicli is thus 
 shown to be the true distance from to the vertex of the angle. 
 
DESCRIPTIVE GEOMETRY, 
 
 55 
 
 Eevolving JlfPiT about JO^into H, P falls at F on CE, OF he- 
 ing equal to OP^ and MFN is the required angle. 
 
 86. This construction in projection is given in Fig. 113, where 
 tTt\ s.Ss\ are the given planes, and CD is their intersection. Con- 
 struct a supplementary view, looking perpendicularly against the 
 liorizontal projecting plane of CI), as shown by the arrow ; in this 
 view the horizontal plane is seen as M^S^, in which e lies : the al- 
 titude dd' is tlie same as in the vertical projection, and the line 
 CD is thus seen in its true length and inclination. Draw ZL per- 
 pendicular to cd^ in this view; it is the plane MPJVin Fig. 112, 
 seen edgewise, and cuts the horizontal plane in the line appearing 
 as 7nn, perpendicular to cd, in the horizontal projection. It also 
 cuts cd' in a point p, which may be projected back to cd, thus de- 
 termining j9m, pn, the horizontal projections of the lines including 
 the angle sought. But for the purpose of measuring the angle, 
 this is not necessary, since op in the supplementary view is the true 
 distance fi-om to the vertex; and setting this distance off as of on 
 cd in the horizontal projection, we have mfn as the angle in its true 
 size. 
 
 In Fig. Ill, the traces of one plane, tTt\ are coincident, and 
 the other plane, sSs\ is parallel to AB. The construction is the 
 
 Fig. 115 
 
 V^-^ ^ Fig. 114 
 
 same as in Fig. 113, and the two diagrams being lettered to corre- 
 spond throughout, no further explanation is required. 
 
 In Fig. 115, both planes are parallel to AB; the profile tells 
 the whole story, and the projections on H and V are simply useless. 
 
66 
 
 DESCRIPTIVE GEOMETRY. 
 
 87. To find the angle made by a gUen plane with either plane 
 of projection. 
 
 In Fig. 116, tTf is the given plane; to measure its inclina- 
 tion to H, draw a j)lane sSs^ perpendicular to tlie horizontal trace, 
 as in the diagram at the left, and revolve its intersection CD with 
 
 Fig. 116 ^ 
 
 the given plane, about cd into H ; D falls at d'\ the distance dd'^ 
 being equal to dd\ and dcd'^ is the required angle. To find the 
 angle made with the vertical plane, draw sSs' perpendicular to the 
 vertical trace, as in the diagram at the right, and revolve the inter- 
 section into T about its vertical projection : O falls at c'\ c'c" being 
 equal to c'c^ and c'd'c" is the required angle. 
 
 Conversely : Oiven one trace and the angle with the corresponding 
 plane of projection, to find the other trace. This is done by simply 
 reversing the preceding operation. Thus, let \\\ii horizontal trace 
 and the angle with H be given ; then in the diagram at the left, 
 draw sS perpendicular to Tt and Ss perpendicular to AB : make 
 the angle Scg equal to the given angle, draw at aS' a perpendicular 
 to sS^ cutting eg in d" . Then set up Sd' equal to 8d'\ and Tt' 
 drawn through d' will be the required vertical trace. 
 
 88. Special Case of the Above Problem. In Fig. 117, it is 
 required to find the angle made by the oblique plane tTt' with the 
 profile plane sSs' . Drawing the profile. Fig. 118, the line of in- 
 tersection CD is seen in its true length and inclination as cd' ^ and 
 tlie plane LL perpendicular to it is parallel to AB. The traces of 
 this plane in Fig. 117 are ZZ, I'V \ it cuts the plane tTt' in the line 
 PN. The lines P6>, PM^ of Fig. 112, in this case coincide in one 
 line, cut from the profile plane s8s' by the plane LL^ and seen in 
 its true length as o'jp^ in Fig. 118. Revolve LL about its vertical 
 trace into T; P falls at/*, and Sfn' is the required angle. 
 
DESCRIPTIVE GEOMETRY. 
 
 57 
 
 Fig. 118 " Fig. 117 
 
 89. Problem 12, To jhid the cominon perpendicula/r of two 
 lines not in the same plane. 
 
 Analysis. If the two lines be projected upon a plane parallel 
 to both, the pi-ojections will be respectively parallel to the lines 
 themselves, and will intersect in a point. A perpendicular to the 
 plane at this point, being perpendicular to both projections and 
 therefore to each line, will cut them both ; the portion intercepted 
 between them is the required connnon perpendicular. 
 
 This is illustrated in Fig. 119, where CD^ MN^ are the two 
 lines ; their projections cd^ mn, upon the parallel plane, intersect at 
 J^ ; the intersection of their projecting planes is the perpendicular 
 at ^, and the intercept PO is the required lea^t distai:ce between 
 the given lines. If the plane approach the given lines, the project- 
 ing perpendiculars J/m, JVn^ will be reduced in length, until, as in 
 Fig. 120, they disappear, and J/iVlies in the plane, which is paral- 
 lel to CD^ and its intersection with cd at once determines the point 
 P. Upon this is based one method of construction, which consists 
 of tlie following steps, viz. : 
 
 Construction. 1. Through any point of one line, draw a 
 ])arallel to the other, and find the traces of the plane thus de- 
 termined. 
 
 2. Through any point of the second line, draw a perpendicular 
 to this plane, and find the point in which it pierces the plane. 
 
 3. Through this point draw a parallel to the second line; it is 
 the projection of that line upon the plane. 
 
58 
 
 DESCKIPTIVE GEOMETRY. 
 
 4. At the intersection of this projection with the first line, 
 erect a perpendicular to the plane. It will cut the second line, 
 and is the common perpendicular. 
 
 5. Determine the length of the intercept. 
 
 Fia.l22 
 90. In the execution of the above processes, the representa- 
 tions are all made on the principal planes of projection. But 
 
DESCKIPTIVE GEOMETRY. 59 
 
 the result may be attained in a far more direct and practical man- 
 ner by tlie use of other planes. As a preliminary to the explana- 
 tion, let c'd\ Fig. 121, be the vertical projection of a vertical line 
 in V ; its horizontal projection is o in AB : and let MN" be an 
 oblique line in another plane. Let (?/?, perpendicular to mn, be 
 the liorizontal projection of a horizontal line intersecting CD and 
 MN \ itS' vertical projection is ^'(9' parallel to AB, The line P 6^, 
 being horizontal, is perpendicular to CD ; and being perpendicular 
 to the horizontal projecting plane of MN^ it is also perpendicular 
 to that line ; moreover, it is seen in its true length as op in the 
 horizontal projection. 
 
 91. Now in Fig. 122, the group (1', 2) contains the vertical 
 and the horizontal projections of two lines CD^ MN^ not in the 
 same plane; it is required to find their common perpendicular- 
 Make a supplementary projection, 3', looking perpendicularly, as 
 shown by the arrow a?, against the liorizontal projecting plane Y' Y' 
 of the line CD, In this projection the horizontal plane appears as 
 H'W parallel to Y' F', and the distances of <?/, ^/, above H'H* 
 are the same as those of c' ^ n'^ above AB in the original vertical 
 projection. 
 
 Now, retaining the same vertical plane, make a projection, 4, 
 looking in a direction parallel to CD^ as shown by the arrow y. 
 For this purpose draw a new ground line, A^B^ , perpendicular to 
 c^d^ \ in this projection CD will appear as a point o^ in the 
 new ground line, and the distances m^r, , n^s^^ from A^B, , are 
 equal to the distances mr, ns^ from Y' Y' in the original horizontal 
 projection. Holding the page so that A^B, is horizontal, it will be 
 perceived that the group (3', 4) is identical with Fig. 121 ; and 
 o^jp^ , perpendicular to m,7i,, is the true length of the required line, 
 which is projected in 2l as o/jf?/. The points ^/, ji?/, are pro- 
 jected back from 3' to o, j?, in 2, and thence to o\ p\ in l', thus, 
 determining the projections of OP^ the common perpendicular, in 
 the original position. 
 
 92. Going one step farther, make another projection, 6', look 
 ing, as shown by the arrow s, perpendicularly against the horizon- 
 tal projecting plane m,^, in 4. In this last view the horizontal 
 plane appears as II" H" parallel to m^n^ ; and the distances above 
 
60 DESCRIPTIVE GEOMETRY. 
 
 it, as s^n^\ r^m,^^ etc., are the same as the distances 5,/?/, Tjn^y 
 etc., above A^B^ in the vertical projection 3^ 
 
 In this view tlie line CD appears as c^cL^ perpendicular to 
 H" 11" ^ and both it and the line MN are seen in their true lengths. 
 What is practically of equal importance, the angle between the pro- 
 jections of the two lines on a plane parallel to both is also seen in 
 its true size. Whereas, after executing the construction given in 
 (89), we should be no nearer to the determination of this angle 
 than we were before ; and without knowing it, the relative posi- 
 tions of the tw^o given lines is not fully deiined : in fact, for prac- 
 tical purposes it is absolutely essential that it should be known. 
 
 93. By turning the page so that the lines AB, H' H\ A,B, , 
 H" H'\ in succession, are brought into a horizontal position, it 
 will be perceived that each of the groups (1', 2), (3', 2), (3', 4), 
 and (5', 4) consists of a vertical and a horizontal projection, in 
 which are represented the given and required lines, and no others ; 
 in this respect also this construction possesses a marked advantage 
 over the one first described. 
 
 The analysis, however, applies equally well to both; it is 
 obvious that if a plane be drawn parallel to both lines, not only 
 that plane but also one of the lines can be placed in a vertical posi- 
 tion; and that, in eifect, is what is done in Fig. 122. 
 
 This has been accomplished by changing the positions of the 
 planes of projection ; assuming, 1st, a new vertical plane contain- 
 ing CD ; 2d, a new horizontal plane perpendicular to CD ; and 3d, 
 another vertical plane containing MN. 
 
 It might have been done by revolving the given lines, first^ 
 about a vertical axis until CD became parallel to T ; second^ about 
 an axis perpendicular to V until CD became vertical ; and third ^ 
 about CD itself until MN became parallel to V. Both methods are 
 extensively used, and that one should be adopted which is best 
 suited to the case in hand ; in the present instance, the successivo 
 rotations of the lines would have resulted in a very obscure and be- 
 wildering diagram. 
 
 94. Problem 13. To draw a plane mahing given angles with 
 the princijyal jplanes. 
 
 Analysis. This will be best explained by the aid of Fig. 123. 
 
DESCRIPTIVE GEOMETRY. 
 
 61 
 
 Suppose the problem solved, and let tTt' represent the plane. 
 Through any point P in the ground line draw two planes; one 
 perpendicular to the horizontal trace, cutting the plane tTt' in the 
 line MN^ the other perpendicular to the vertical trace, intersecting 
 tT*J in CD. These lines intersect at O^ and OP is perpendicular 
 to both. Therefore in the triangle CPD^ the angle GPO is equal 
 to the angle PDO^ which is given; so that if OP be determined, 
 the length oi PC can be found, thus fixing the point G in the hori- 
 zontal trace. And OP can be determined if PN is assumed ; for 
 
 Fig. 123 
 
 Fig. 124 
 
 since the angle at M is given, the triangle PMN can be con- 
 structed, and OP is perpendicular to MN, But PM being per- 
 pendicular to the horizontal trace, tT must be tangent to a circle 
 described in the horizontal plane about P as a centre, with radius 
 PM\ and Tt' must pass through the assumed point N. 
 
 Construction. About any point P in AB, Fig. 124, describe a 
 circle with any convenient radius PM. Through M draw a line, 
 making the angle PMO equal to the given angle A with the hori- 
 '2:ontal plane, and cutting at N the indefinite perpendicular to AB, 
 drawn through P. Draw PO perpendicular to MN\ make the 
 angle OPE equal to the given angle v with the vertical plane, and 
 draw PE cutting MN in E. Set off on the perpendicular through 
 P^ tlie distance PC equal to PE^ and draw, through (7, tT tan- 
 gent to the circle ; it is the required horizontal trace, cuts AB in T^ 
 and TNt' is the required vertical trace. 
 
 95. Should the assigned value of v be equal to 90° — A, the 
 
p55 DESCRIPTIVE GEOMETRY. 
 
 intersection E will fall at /, G will fall at K^ and tlie plane will be 
 parallel to AB, as shown in the small profile at the left. If v — 90°, 
 as OPF^ the intersection is at infinity, the horizontal trace is tan- 
 gent to the circle at J!/, and the vertical trace coincides with MN, 
 If 'y is greater than 90°, as OPG^ then GP produced cuts the pro- 
 longation of iO/ at ^ below AB, and the distance PII is to be set 
 up as PZ above AB ; the horizontal trace is ZSs tangent to the 
 circle, and the vertical trace (not shown) will pass through the 
 points S and JV, If v = 90° -|- ^) ^s OPP^ then PP coincides 
 with AB, its prolongation traverses M, and P W being equal to the 
 radius, the horizontal trace through IF, and therefore the vertical 
 trace through iV, will be parallel to AB, and the position of the 
 plane is shown in the small profile at the right. 
 
 Should the assigned value of v be less than 90° — A, or greater 
 than 90° 4- A, it is clear that the determining points of intersection 
 with 3IJV will fall between I and 31 ; and since a tangent to a 
 circle cannot be drawn through a point within the circumference, 
 the conditions cannot be satisfied. 
 
 96. The principles and methods developed in the discussion of 
 the preceding problems are sufficient for the solution of all others 
 relating solely to the point, right line, and plane. The geometrical 
 principles involved are simple enough, but thorough mastery of 
 the manipulations can be acquired only by applying them to a great 
 variety of cases ; the examples already given illustrating the fact 
 that shght changes in the data may cause the resulting constructions 
 to differ widely in appearance. 
 
 A very beneficial exercise for those desii'ous of attaining profi- 
 ciency, will be found in working the problems backward ; that is to 
 say, in assuming the points and lines of the construction to be ar- 
 ranged in such relations as may be desired, and then, by reversing 
 the steps of the operation, ascertaining the conditions which would 
 lead to that development. Indeed this inverted procedure is al- 
 most necessary in many cases to the production of a clear and ex- 
 planatory illustrative diagram, and has been used in numerous 
 instances in preparing the figures for this volume. In practical 
 operations, of course, the conditions must be taken as they are, let 
 the result be what it may. 
 
DESCRIPTIVE GEOMETRY. 
 
 63 
 
 For ready reference, we add here a few applications and con- 
 structions which may subsequently be found useful. 
 
 MISCELLANEOUS EXAMPLES. 
 
 97. Example 1. Given a right line in a given jplane. To 
 draw in that ;pl(me another line cutting the first in a given jpoint 
 at a given angle. 
 
 Construction. In Fig. 125, MN is the given line, the given 
 point. Eevolving the given plane tTt' about Tt into H, N falls at 
 
 W\ Tn" being equal to Tn' ; M remains fixed, and falls at o" 
 on mn" . Draw Go"d" making with mn" the given angle co"n" ; 
 it is the required line in the revolved position of the plane. In the 
 counter-revolution, <?, being in the axis, remains fixed, d" goes to 
 d' ^ the distance Td' being equal to Td" ; c' and d are found in AB, 
 and cd^ ed' ^ are the projections of CD the line required. 
 
 98. Example 2. Given a right line in a given plane. To 
 draw another plane cutting the first in the given line^ and maTcing 
 a given angle with the given plane. 
 
 Construction. In Fig. 126, CD is the given line in the plane 
 tTt' . Through any convenient point o draw an indefinite perpen- 
 dicular to cd^ cutting Tt in m. Make a supplementary projection 
 
64 
 
 PESCRIPTIVE GEOMETRY. 
 
 on the horizontal projecting plane of CD\ the horizontal plane 
 here appears as II'II\ in which c^ lies, d^d\ is equal to dd\ and 
 CD is seen in its trne length as c^d^ ; also, o and m fall together at 
 <?,. Draw 6>,^, perpendicular to c^d^^ and on H' II' set off o^g^ 
 equal to it ; project g^ back to g on cd^ draw ?7z^, and make the 
 angle nign equal to the given angle. Then gn produced cuts mo^ 
 produced if necessary, in n^ a point in the horizontal trace required ; 
 c is a point given, and scnB is the horizontal trace, aSV^'^' being the 
 vertical trace, of the second plane. 
 
 Note, By reference to Figs. 112 and 113, it will be perceived 
 that this operation is nearly the converse of the process of measui*- 
 ing the angle between two planes. And it should now be readily 
 understood that, instead of making a supplementary view^, the hori- 
 zontal projecting plane of CD might be revolved into either H or Y, 
 and after the determination of 0,^1, revolved back to its original 
 position. 
 
 99. Example 3. To draw a vlane jparallel to a given plane y 
 at a given distance front it. 
 
 Constraction. In Fig. 127, let tTt' be the given plane. Draw 
 a plane ILL' perpendicular to tT^ and revolve it into H about its- 
 horizontal trace; cd" is the revolved position of its intersection 
 with the given plane. Make cg^ perpendicular to Gd'\ equal to tlie 
 
 given distance, and draw e''h parallel to cd" ; it is the revolved po- 
 sition of the line cut from the plane ILo'hy the required plane, and 
 cuts LI in ^, a point in the horizontal trace *aS'; which is parallel to 
 
DESCRIPTIVE GEOMETRY. 
 
 65 
 
 iT. Tlie vertical trace Ss is parallel to Tt' ; also, it cuts LV in 
 e\ making de' equal to de" , 
 
 100. Example 4. To draw the projections of a circle m a 
 given jjlane. 
 
 Argument. Let G on the horizontal line CO in the plane tTt\ 
 Fig. 128, be the centre, about which is to be drawn a circle of a 
 given diameter. Through c draw an indefinite perpendicular to 
 Tt^ and revolve the plane about Tt into H, where C falls at c" ; 
 draw the circle, and circumscribe it by a square, whose sides are 
 
 1' 
 
 P' mf 
 
 o'X-^ 
 
 ^,yi 
 
 
 
 \ 
 
 0^5? 
 
 V 
 
 tX di 
 
 i , 
 
 
 
 ■s 
 
 
 oVC 
 
 ^ / 
 
 ^m 
 
 
 ^ 
 
 / 
 
 Fig. 128 
 
 parallel and perpendicular to Tt. At points on the diameter <3^"J'', 
 as V\ 2'\ draw ordinates to the circle: during tlie counter-revolu- 
 tion, the lengths of a''b" and of its subdivisions will remain un- 
 changed, and the sides of the square which are parallel to tho axis 
 will remain parallel to it and to each other. So that in the projec- 
 
66 DESCRIPTIVE GEOMETRY. 
 
 tion the square will appear as a rectangle, and the circle as a curve 
 inscribed within it ; and, since tlie radius and the ordinates which 
 are perpendicular to 2t are foreshortened in the same proportion, 
 it follows that this curve is an ellipse, whose major axis ah is equal 
 to the given diameter of the circle. 
 
 Let do and oo' be the traces of a plane, perpendicular to Tt, 
 cutting the given plane in the line DO', revolving this line about 
 do into H, it falls at do^ : on this set off o^e^ = ac, then on counter- 
 revolution e^ falls at^. This gives oe as the projected length of the 
 radius ; set oft* cr and cp equal to it, and the length ^'p of the minor 
 axis of tlie ellipse is determined. 
 
 101. Construction. Draw through c a parallel to Tt, and on it 
 set oft" ca, ch, equal to the given radius. Draw through e a per- 
 pendicular to Tt, cutting it at ^ ; on Tt set oft' grk equal to so', the 
 distance of from H ; draw Jcc, produce it, and on it set oft" e7i, 
 em, equal to the given radius. Tlirough n and m draw parallels to 
 Tt, thus determining rp the minor axis. It is evident that ho is 
 parallel to do, of the preceding argument, because oo^ is equal to 
 oo' , and that again to sc' . 
 
 A similar argument would apply to a square circumscribing the 
 circle, with its sides parallel and perpendicular to the vertical trace. 
 Therefore in the vertical projection draw through c' a parallel to 
 that trace, and on it set off the major axis equal to the given diam- 
 eter. Draw through c' a perpendicular to the trace, cutting it at 
 i ; set off it equal to so, the distance of C from V : draw a line 
 through I and c' , and on it set off c'u, c'v, equal to the given radius. 
 Through u and "o draw parallels to Tt' , determining x and y the 
 extremities of the minor axis. The two ellipses may now be drawn 
 by any of the usual methods. 
 
 102. It is obvious that p is the horizontal projection of the 
 highest point in the curve, and mpq of the tangent to the curve at 
 that point ; the vertical projection of this tangent is g'm! parallel 
 to AB, andjt>' lies on g^wl \ in like manner, ux, yv, being parallel 
 to the vertical trace, will be horizontally projected as lines parallel 
 to AB, which will also be tangents to the horizontal projection of 
 the circle. 
 
 Moreover, the ellipses will be limited at the right and left by 
 
DESCRIPTIVE GEOMETRY. 
 
 67 
 
 two common tangents perpendicular to AB. The exact location of 
 these can be found if desired by regarding each as the intersection 
 of tTt by a proiile plane. Draw through O a plane perpendicular 
 to AB; it cuts the given plane in a line which, when revolved into 
 H about Tt. takes the position o^o" . Parallel to this draw two tan- 
 gents to the circle, cutting Tt in f and li ; perpendiculars to AB 
 thr(jugh these points are the tangents required. 
 
 103. The fact that in either projection, lines perpendicular to 
 the trace are all equally foreshortened, while those parallel to it are 
 not foreshortened at all, can be advantageously applied in many 
 other cases. Referring to the figure, it is seen that in the horizon- 
 tal projection the apparent length, cr^ of the radius is to the actual 
 length as the base do of the triangle doo^ is to the hypothenuse do^^ 
 
 Now let two scales be made, whose units are to each other in 
 this proportion, and divided into the same number of equal parts; 
 then by measuring the ordinates, as V ^ 2'', with the larger scale, 
 and setting them off, as 1, 2, with the smaller scale, the projection 
 of any plane curve can be constructed, often more rapidly than by 
 anv other means. 
 
 Fig. 129 
 
 104. Example 5. To revolve a given point about a given line 
 
68 DESCRIPTIVE GEOMETRY. 
 
 which does not contain the jpoint^ into a given jplane which contains 
 neither. 
 
 Analysis. The plane of rotation passes throngli the given point, 
 is perpendicular to the given line, and cuts it in the centre of the 
 circular path. It also cuts the given plane in a line ; and if the 
 given point reach the given plane at all, it will be at some point of 
 this line. 
 
 Construction. In Fig. 129, 7^ is the given point, J/iYthe given 
 line, tTt' the given plane. Draw through P the plane sSs perpen- 
 dicular to MN\ it cuts tTt' in the line DE^ and MN in the point 
 C. Revolving the plane sBs a,bout its horizontal trace into H, DE 
 falls in the position de'\ C falls at c" and P at p" . About c" 
 describe a circular arc through jf)'^ ; it cuts de" in the points r" and 
 a". Making the counter-revolution, c" and p" return to their 
 original positions, and r" and o'^ are found respectively at B and 
 on the line DE: these are the required points in the giv^en plane, 
 at one of which P must fall when revolved about the given line 
 into that plane. 
 
DESCKIPTIVE GEOMETRY. 
 
 CHAPTER III. 
 
 Generation and Classification- of Lines and Surfaces.— 
 Tangents, JS^ormals, and Asymptotes to Lines. — Oscula- 
 tion, Eectification, Radius of Curvature. — Tangent, JS'or- 
 MAL, AND Asymptotic Planes and Surfaces. 
 
 GENERATION AND CLASSIFICATION OF LINES. 
 
 105. Eyery line may be generated by the motion of a point, 
 
 whicli is regarded as a material particle. Any two successive 
 positions of the generating point, Laving no assignable distance be- 
 tween them, are called consecutive points of the line ; practically, 
 they may be considered as coincident. But in going from one of 
 these positions to the next, the generating point moves in a deter- 
 minate direction, whicli cannot be conceived to differ from that of 
 the right line joining those consecutive points. This infinitely short 
 right line is called an elementary line : and every line may be regarded 
 as made up of an infinite number of these rectilinear elements. Any 
 two consecutive elements, since they have one point in common, 
 must lie in one plane ; but they may have different directions. 
 
 A Hue thus generated is called the locus of the successive po- 
 sitions of the moving point. 
 
 106. Lines are divided into classes, according to the law of 
 the motion of the generating point, as follows : 1 . Right Lines. 
 2, Single-curved Lines. 3. Double-curved Lines. 
 
 If the point moves always in the same direction, all the elements 
 lie in the same direction, and the line is a right line. 
 
 If the point in moving continually changes its direction, no two 
 consecutive elements have in general the same direction, and the 
 line is a curye. 
 
 If all the elements of a curve lie in one plane, the line is of 
 single curyature. 
 
70 DESCRIPTIVE GEOMETRY. 
 
 If no three consecutive elements lie in tlie same plane, the line 
 is of double curvature. 
 
 This last may be more clearly seen by the aid of Fig. 130, where 
 m, n, 0, p, r, are tlie horizontal, and m', n\ o\ p', r\ are the ver- 
 tical, projections of five consecutive points, the elements MI^^ OP^ 
 
 Fig. 131 
 
 etc., being enormously magnified. The three points iT, (?, Py 
 and therefore the two elements NO and OP^ lie in the horizontal 
 plane, but If lies above and P lies below it. The plane of MN 
 and NO is oblique, on being its horizontal trace, and it does not 
 contain OP : in like manner, op is the horizontal trace of the plane 
 determined hj PP and PO, which does not contain JVO. Tims 
 it is seen that any three, but no four, consecutive points — or, wliat 
 is the same thing, any two, but no three, consecutive elements — - 
 lie in the same plane. 
 
 REPRESENTATION OF CURVES. 
 
 107. A curve is represented by its projections; which contain 
 the projections of all its points, as shown in Fig. 131. And the 
 curve is in general fully determined if its projections on the princi- 
 pal planes are given. For the projections d, d\ for instance, 
 sufiice to locate the point D in space (8), and the same is true of all 
 the other points. 
 
 The traces of a curve are found in the same manner as those of 
 a right line (25). Thus, in Fig. 131, let the horizontal projection 
 deg be produced to cut the ground line at Ic ; then a perpendicular 
 to AB at that point, cutting the vertical projection in Tc\ will lie iu 
 V, and ^is the vertical trace of the curve. ' 
 
DESCRIPTIVE GEOMETRY. 71 
 
 JN^o projection of a double-curved line can in any case be a right 
 line. But if tlie plane of a single-curved line be perpendicular to 
 any plane of projection, the projections of all points of the curve 
 will lie in its trace, which is a right line. If the plane of the 
 curve be parallel to any plane of projection, its projection on tnat 
 plane will be similar and equal to the curve itself ; because the pro- 
 jection of each element will be parallel and equal to itself (14). If 
 the plane of the curve be perpendicular to the ground line, its pro- 
 jections on both the principal planes will be right lines perpendicu- 
 lar to AB ; the curve is not undetermined, but is seen in its true 
 form when projected on a profile plane. 
 
 108. A curve may also be generated by the motion of a right 
 line whose successive positions intersect in a series of points. 
 Thus, in Fig. 131^, the lines Z, Jf, intersect at m; Jfcuts iV^at n^ 
 
 and so on. l^ow if the points m, ^, <?, ^, etc., are consecutive, 
 the line joining them will be a curve, called the envelope of the 
 various positions Z, M^ iV, etc. , of the moving line. 
 
 If in this figure the lines Z, M^ N, etc. , lie in the plane of the 
 paper, the curve will also lie in that plane. But if they be re- 
 garded as the projections of lines inclined to that plane, equally or 
 unequally, it is clear that the envelope will be a line of double 
 curvature. In this case, the lines Z, M^ determine one plane ; thQ 
 lines J/, iV^, determine another, and JSTis the intersection of these 
 two planes. Since the like is true of any three of the lines taken 
 in order, a double-curved line may also be generated by the motion 
 of a plane whose successive positions intersect in a series of lines. 
 These lines will in general intersect each other two and two in con- 
 
72 DESCRIPTIVE GEOMETRY. 
 
 secutive points, and tlieir envelope will be a line of double curva- 
 ture. 
 
 TANGENTS, NORMALS, AND ASYMPTOTES TO LINES. OSCULATION, 
 
 RECTIFICATION, INVOLUTE AND EVOLUTE. RADIUS OF CURVATURE. 
 
 109. The Tangent to any line, is a right line drawn through 
 two of its consecutive points. Tliese two points determine the 
 direction of the tangent, and practically coincide in the point of 
 contact. 
 
 If the given line be of single curvature, the tangent will lie in 
 its plane ; for it contains two points of the curve, and they lie in 
 that plane. If one right line be tangent to another right line, the 
 two lines wull coincide; for they have two points in common. 
 
 Two curves are tangent to eacli other when they have two con- 
 secutive points in common ; that is to say, when they have a com- 
 mon tangent at a common point. 
 
 In Fig. IZla^ the lines Z, J/", iT, etc., will be tangents to the 
 enveloping curve when the points m, n^ <?, etc., become consecu- 
 tive. Suppose now a series of curves, respectively tangent at these 
 points to the same lines ; then the curve rtmopr will be tangent to 
 them also. And in fact the term envelope is used in a general sense, 
 to designate the line tangent to any given series of lines, straight or 
 curved. 
 
 If two lines are tahgent to each other, their projections on any 
 plane will also be tangent to each other. For the lines have two 
 consecutive points in common, and the projections of these will be 
 consecutive points, common to the projections of the lines. 
 
 The converse of this is not necessarily true. But if the verti- 
 cal projections are tangent to each other, and also the horizontal 
 projections, and the points of contact are the corresponding projec- 
 tions of a point common to the two lines, the lines themselves are 
 tangent to each other. Because the projecting perpendiculars at 
 the common consecutive points, will determine two consecutive 
 points in space, whicli will be common to the two lines. 
 
 The angle made by a curve at any point with a plane or a right 
 line, is the same as that made by its tangent at that point ; because 
 
DESCRIPTIVE GEOMETRY. 73 
 
 tlie taiiirent contains tlie rectilinear element of tlie curve. Sim- 
 ilarlj, tlie angle between two curves is the same as the angle be- 
 tw^een tlieir tangents at the common point. 
 
 110. Osculation. Osculating Plane. A right line can in gen- 
 eral contain onlv two consecutive points of a curve. But two 
 curves maj have three or more consecutive points in common ; in 
 that case the contact, which is more intimate than that of simple 
 tangency, is called osculation. 
 
 Through the tangent to a curve an infinite number of planes 
 can be passed, and all these planes are tangent to the curve. Each 
 will in general contain only two consecutive points of tlie curve ; 
 but one of them may be so placed as to contain tkree^ and this is 
 called the osculating plane. Thus, in Fig. 131«^, the line JT con- 
 tains the point m^ the line N contains the points n and o ; therefore 
 the plane determined by M and N passes through m^ n^ o. The 
 osculating plane of a single-curved line is, evidently, the plane of 
 the curve itself. 
 
 A Normal to any line is a perpendicular to its tangent, at the 
 point of contact. An infinite number of such perpendiculars can 
 be drawn, all of which lie in one normal plane. When not other- 
 wise specified, ' ^ the normal ' ' to any curve is understood to be that 
 one which lies in the osculating plane. 
 
 The rectification of a curve means the determination of a right 
 line equal to it in length. If the curve be so moved upon a fixed 
 tangent that the consecutive points of the curve in their order, 
 come into coincidence with those of the right line in their order, 
 the motion is one of rolling contact ; the curve measures itself off 
 upon the tangent, and the part rolled over is equal in length to the 
 arc of the curve whicli has rolled over it. 
 
 111. Involute and Evolute. If the tangent roll upon a fixed 
 curve, any point of it will describe a second curve, w^iich is called 
 an involute of the first ; and the fixed curve is called the evolute of 
 the second. 
 
 In Fig. 131«^, suppose a thread, wound upon the broken line 
 knr^ to be kept taut, and unwound as indicated by the arrow. 
 Then it is clear that the path of 5, a point upon the position Z, will 
 be made up of a series of circular arcs, st^ tu^ etc. , of whicli the 
 
74 DESCRIPTIVE GEOMETRY. 
 
 centres are tlie points m, n, o, etc. When these points become 
 consecutive, the change of centres and radii goes on continuously, 
 the broken hue becomes the evolute, to which the positions of the 
 thread, Z, M^ iT, etc., are all tangent; and the path svx becomes 
 an involute of Tcnr, But the nature of the motion is not changed ; 
 the tracing point at any instant is describing a circle about the 
 point of contact, and therefore moving in a direction per];>endicular 
 to the radius. Consequently, the tangent to the evolute is always 
 normal to the involute. 
 
 What is true- of the motion of s is equally true of the motion 
 of any. other point/*, situated upon the same lineZ; whose path 
 fgh also becomes ultimately an involute of the curve knr. Hence 
 a single-curved evolute may have an infinite number of involutes 
 in its own piano. These are not in general similar curves ; but 
 they SiVQ parallel^ in the sense that the normal distance between 
 any two of them is constant. On the other hand, since the inter- 
 sections of consecutive normals to any curve determine an evolute, 
 a single-curved line can have but one evolute in its own plane. 
 And this, being the envelope of all the normals to the given curve, 
 whether consecutive or not, can be drawn with much precision, if 
 the curve is such that the direction of its tangent, at any point 
 assumed at pleasure, can be determined. 
 
 112. Centre of Curvature. In Fig. 131a, let 7io, o-p, be two 
 consecutive elements of a curve; .these, if produced, form two 
 successive tangents iV, 0, whose included angle, called the angle of 
 contingence, is a measure of the rate of curvature at o. Draw a 
 perpendicular to each of these elements at its middle point ; these 
 perpendiculars intersect at <?, the centre of a circle passing through 
 n, (?, and^. 
 
 At n and ^, draw perpendiculars to iV, and at o and ^, per- 
 pendiculars to ; the second pair of perpendiculars will cut the 
 first pair at the points d^ e^ and the three angles at ^, c, ^, will be 
 equal to each other and to the angle of contingence. All these 
 perpendiculars are understood to lie in the osculating plane. Now 
 in the case of the actual curve here represented by a broken line, 
 the points d and e will coincide with c ; for the distances on^ op^ 
 being infinitely small, the three perpendiculars to N will form one 
 
DESCRIPTIVE GEOMETRY. 75 
 
 single line, and tlie three perpendiculars to will form another. 
 The angle of contingence is also infinitely small ; nevertheless the 
 point 6', then the intersection of two consecutive normals, is in gen- 
 eral at a finite distance from ^, and is the centre of a circle whose 
 circumference contains the three consecutive points n^ (?, ^, of the 
 curve. 
 
 This, then, is the osculating circle ; and since its circumference 
 has the same rate of curvature as that of the curve itself at <9, its 
 centre c is called the centre of curvature, and go is called the radius 
 of curvature, of the given curve at that point. 
 
 The intersections of successive normals to any curve will be a 
 series of consecutive points, determining in general another curve, 
 which is the locus of the centre of curvature. In the case of a 
 ])lane curve^ this locus will be the evolute of the given curve. 
 Thus, in Fig. IZla^ iV^and 6>, consecutive normals to svx, meet at 
 o on the curve hnr^ and o is the centre of curvature at v. This 
 does not hold true of double-curved lines. 
 
 113. An Asymptote to a line is another line which the given 
 line, during a portion of its course, continually approaches, becom- 
 ing tangent to it only when its own length becomes infinite. 
 
 Ordinarily, the asymptote to a plane curve is a right line, which 
 becomes tangent to the curve at an infinite distance. But this is 
 not essential : for example, the evolute of the Archimedean spiral 
 has a circular asymptote of finite diameter, and the curve lies 
 wholly within the circumference ; again, two conjugate hyperbolas, 
 having common rectilinear asymptotes, are asymptotic to each other. 
 Also, a right line may be an asymptote to a line of double cur- 
 vature. 
 
 GENERATION AND CLASSIFICATION OF SURFACES. 
 
 114. Every surface may be generated by the motion of a line. 
 
 This moving line is called the generatrix, and its difiierent positions 
 are called elements, of the surface. Any two successive positions 
 of the generatrix, having no assignable distance between tliem, are 
 called consecutive elements ; practically they may be regarded as 
 coinciding. 
 
76 DESCRIPTIVE GEOMETRY. 
 
 Surfaces may be separated into two grand divisions, according 
 to the form of the generatrix, viz, : 
 
 I. Ruled Surfaces, which contain rectihnear elements. 
 
 II. Double-Curved Surfaces, which have no rectilinear 
 elements. 
 
 In other words, the surfaces of the first division can be gener- 
 ated by the motion of right lines, while those of the second division 
 cannot. The former may also be generated by the motion of curved 
 lines ; the latter cannot be generated without it. 
 
 115. A right line may move so that all its positions lie in the 
 same plane. It may move otherwise ; in which case any two con- 
 secutive positions either will lie in the same plane, or they w^ill not. 
 According to the law of the motion of the rectilinear generatrix, 
 then, ruled surfaces are subdivided into three classes, as follows : 
 
 1 . Plane Surfaces All the rectilinear elements lie in 
 
 the same plane. 
 
 2. Single-curved Surfaces. .Any two consecutive rectilinear 
 elements lie in the same plane. 
 
 3. Warped Surfaces No two consecutive rectilinear ele- 
 ments lie in the same plane. 
 
 plane surfaces. 
 
 116. The Plane Surface is unique. That is to say, there is 
 but one form of plane, and there neither are nor can be any differ- 
 ent kinds. All planes are flat, and one is no flatter than another. 
 
 The rectilinear generatrix may move so as to toach another 
 right line, remaining always parallel to its first position ; so as to 
 touch two other right lines which are parallel to each other, or 
 which intersect; or it may revolve about another right line to 
 which it is perpendicular. 
 
 Acquaintance with the nature and properties of planes was 
 necessarily assumed at the outset; tlie methods of representing 
 them, and of assuming points and lines in them, have already been 
 described. 
 
DESCRIPTIVE GEOMETRY. 77 
 
 SURFACES OF SINGLE CURVATURE. 
 
 117. Single- curved surfaces are of three varieties, viz. : 
 
 1. Cones In which all the rectilinear elements intersect in 
 
 a common point. 
 
 2. Cylinders . . In which all the rectilinear elements are parallel 
 to each other. 
 
 3. Con volutes.. In which the consecutive elements intersect 
 two and two, no three having a common point. 
 
 CONICAL surfaces. 
 
 118. In generating a cone, the right line moves so as always 
 to touch a given curve called tlie directrix, and also to traverse a 
 given jDoint called the vertex. Since the generatrix is indefinite in 
 length, the surface is divided at the vertex into two parts, called 
 respectively the upper and lower nappes. It is clear that in the case 
 of a given cone, any line drawn upon the surface so as to cut all 
 tlie elements may be taken as a directrix, and any element as the 
 generatrix. 
 
 The cone may also be generated by the motion of a curve which 
 always touches a given right line, and changes its size according to 
 a proj^er law. 
 
 119. The portion of the cone usually considered, is included 
 between tlie vertex and a plane which cuts all the elements; the 
 curve of intersection is called the base, and its form gives a distin- 
 guisliing name to the surface — as a cone with a circular, a para- 
 bolic, an elliptical, or a spiral base, as the case may be. If the 
 base has a centre, a right line drawn through this centre and the 
 vertex is called the axis of the cone. The point in which any ele- 
 ment pierces the plane of the base is called the foot of the element. 
 
 A definite portion of either nappe, in«.Juded between two paral- 
 lel planes which cut all the elements, is called a frustum of the 
 cone, the limiting curves being called respectively the upper and 
 lower bases. 
 
 A secant plane through the vertex cuts the cone in rectilinear 
 elements intersecting the base. 
 
'8 
 
 DESCKIPTIVE GEOMETRY. 
 
 120. A ri^ht cone is one all of whose rectilinear elements make 
 equal angles with a right line passing through the vertex, which is 
 called the axis. This is also called a cone of reyolution, since it 
 can be generated by revolving the hypothenuse of a right-angled 
 triangle about one of its sides as an axis. 
 
 If the directrix of a cone be changed to a right line, or if the 
 vertex be placed in the plane of a single- curved directrix, the cone 
 will degenerate into a plane. 
 
 If the vertex be removed to an infinite distance, the rectilinear 
 elements will be parallel to each other, and the cone will become a 
 cylinder. 
 
 121. Representation of the cone. A cone is represented by the 
 projections of the vertex, one of the curves of the surface (usually 
 its plane base), and the principal rectilinear elements. Thus, in 
 Fig. 132, let O he the vertex; draw mxny, the horizontal projec- 
 
 riG.132 
 
 ria.133 
 
 tion of the base, and ox, oy, tangent to that curve : this completes 
 the horizontal projection of the cone. The j^lane of the base in 
 this case is perpendicular to V ; its vertical projection is therefore Vj 
 right line, limited at m' and n' by tangents to the horizontal pro- 
 jection perpendicular to AB; the vertical projections m'o\ n'o\ of 
 the extreme visible elements, complete the representation of the 
 surface. 
 
 To assume a rectilinear element, assume a point on the curve of 
 the base, as C or D, and draw through it a right line to the vertex. 
 
DESCRIPTIVE GEOMETRY. 79 
 
 To assume a point on the surface, assume one of its projections, 
 say the horizontal, as jp ; througli jp draw the horizontal projection 
 of an element, o'p ; this element intersects the base at 2>, and ^' 
 must lie on d'o\ the vertical projection of the element. The hori- 
 zontal projecting plane oi DO cuts the cone in another element, 
 {7(9, having the same horizontal projection ; upon this lies another 
 point 7?, whose horizontal projection r coincides with p. 
 
 122. Particular attention is called to the fact that the cone ic 
 here shown with its base not situated in the horizontal plane. 
 And it is so shown for the purpose of illustrating and emphasizing 
 another fact, viz., tliat the projection of the base must not be con- 
 founded with the base itself. It is very natural and proper to place 
 the base of a cone or a cylinder upon the horizontal plane ; but if 
 it be always so placed, experience has proved that the above neces- 
 sary distinction is very apt to be lost sight of, wdiich may lead to 
 serious errors in the subsequent applications of problems relating to 
 these surfaces. 
 
 CYIJNDRICAL SURFACES. 
 
 123. The Cylinder, as intimated in (120), is merely that limit- 
 ing form of the cone in which the vertex is infinitely remote ; and 
 it may be generated by a right line Avhich moves so as always to 
 touch a given curve, and have all its positions parallel. In the 
 case of a given cylinder, any line of the surface which cuts all the 
 rectilinear elements may be taken as a directrix, and any one of 
 those elements as the generatrix. 
 
 A cylinder may also be generated by a curvilinear generatrix, 
 all of whose points move in the same direction and with the same 
 velocity. 
 
 124. A plane cutting all the rectilinear elements, intersects the 
 cylinder in a curve called its base ; whose form, as in the case of 
 the cone, gives a distinguishing name to the cylinder. If the base 
 has a centre, a right line drawn through the centre, parallel to the 
 elements, is called the axis. 
 
 If a definite portion of the surface included between two paral- 
 lel planes is considered, the two curves of intersection are called 
 the upper and lower bases. 
 
80 DESCRIPTIVE GEOMETRY. 
 
 A plane parallel to the rectilinear generatrix cuts the cylinder, 
 if at all, in rectilinear elements intersecting the base. 
 
 125. A ri^ht cylinder is one whose rectilinear elements are per* 
 pendicular to the plane of the base ; and the base itself is then said 
 to be a right section. 
 
 A right cylinder with a circular base is also called a cylinder of 
 revolution, since it may be generated by revolving one side of a rec- 
 tangle about the opposite side as an axis. 
 
 If the curvilinear directrix of any cylinder be changed to a 
 right line, the surface will degenerate into a plane. 
 
 126. The projecting lines of the various points of a curve, as 
 seen in Fig. 131, are rectilinear elements of a riglit cylinder, whose 
 base, in the plane of projection, is the projection of the curve. 
 Thus the curve in space is determined by the intersection of two 
 cylinders, called respectively the horizontal and vertical projecting 
 cylinders. 
 
 127. The representation of the cylinder differs from that of the 
 cone only in this, that the projections of the rectilinear elements 
 are parallel instead of convergent. If a limited portion is to be rep- 
 resented, the projections of both bases must be drawn; if not, the 
 projections of the extreme vibible elements may terminate indefi- 
 nitely, as shown in Fig. 133. 
 
 To assnme a rectilinear element, assume a point on the curve of 
 the base, as C or Z^, and draw through it a right line parallel to 
 the rectilinear generatrix. 
 
 To assume a point on the surface, assume one of its projections, 
 say the vertical, as p ; through this draw the corresponding pro- 
 jection o'p' of an element ; this element intersects the base at 2>, 
 and p must lie on do^ the horizontal projection of the element. Tlie 
 vertical projecting plane oi DO cuts the cylinder in another ele- 
 ment, CK^ having the same vertical projection ; and upon this ele- 
 ment lies another point, 7?, of the surface, whose vertical projec- 
 tion coincides w^itli that of P. 
 
 CONVOLUTE SURFACES. 
 
 128. The Convolute may be generated by a right line which 
 moves so as always to be tangent to a line of double curvature. Any 
 
DESCRIPTIVE GEOMETRY. 
 
 81 
 
 two consecutive rectilinear elements, but no three, will lie in the same 
 plane ; for thej are the extensions of the elements of the directrix, 
 of which (106) any two, but no three, consecutive ones intersect 
 each other. 
 
 Suppose a piece of paper cut in the form of a right-angled tri- 
 angle to be wrapped around a regular polygonal prism, Fig. 134:, 
 its base becoming the perimeter of the base of the prism. Then 
 the hypothenuse will become a broken line, each portion lying in a 
 
 Fig. 136 
 
 }.131 
 
 face of the prism and being equally inclined to its edges. In un- 
 winding, the paper turns about each edge in succession as upon a 
 hinge, until it coincides with the plane of the next face ; when the 
 free portion of the hypothenuse will be an extension of, and there- 
 fore tangent to, the element of the broken line which lies in that 
 face. And, considering the successive positions of the hypothenuse, 
 it is seen that No. 1 intersects No. 2 at 7>, 'No. 2 intersects Xo. 3 
 at ^, No. 3 cuts No. 4 at J^, and so on ; but No. 1 does not inter- 
 sect No. 3, nor does No. 2 intersect No. 4. Now let the sides of 
 the polygon be indelinitely increased in number ; the base will ul- 
 timately become a circle, the prism will become a cylinder ; tlte 
 broken line will become a helix, and its tangents, then consecutive, 
 
82 DESCRIPTIVE GEOMETRY. 
 
 will lie in a continuous surface, of wliicli a limited portion is sliown 
 in Fig. 135. 
 
 The point M of the unwinding paper will always lie in the curve 
 myn in the horizontal plane ; this curve, which is the involute of 
 ih^:^ circular base, is the horizontal trace of the surface, which, ex- 
 panding as it rises, winds around the cylinder in convolutions like 
 those of a sea-shell. The cylinder itself has no connection with the 
 surface ; it is introduced merely in order to throw the nearer por- 
 tion of the convolute into stronger relief. 
 
 As in the cases of the cone and the cylinder, the curve of inter- 
 section with any plane which cuts all the rectilinear elements may 
 be taken as the base of the convolute. 
 
 129. The Edge of Kegression. The generatnx, in Fig. 135, is 
 shown as limited in length ; thus, the elements J/0, NG^ termi- 
 nate at O and ^, their points of contact with the directrix. 
 
 But they may be continued past those points, as indicated in 
 d<>tted lines ; and their extensions, 6^7? and GII^ lie in a continua- 
 tion of the surface, which expands in other successive whorls : and 
 these two portions, or nappes, of the convolute have in common 
 the helical directrix. 
 
 But it is to be noted that this is not a curye of intersection ; for, 
 as shown in Fig. 130, wdiich represents a detached portion of the 
 surface on a larger scale, the rectilinear elements which lie upon 
 the lower nappe neither pierce the upper nappe nor cut the helix. 
 Indeed, they are tangent to the helix by hypothesis ; and the sur- 
 face is continuous and unbroken, although reflected sharj^ly upon 
 itself, and forming at the helix what is called an edge of regression. 
 This is a limiting line, at which a surface terminates abruptly by 
 the law of its generation : it is always formed by the intersection 
 of consecutive generating lines, whether they are right lines, as in 
 this case, or curved ones. 
 
 A surface may also be reflected, or bent back, in an analogous 
 manner, along a line which is not thus formed ; the limiting line in 
 that case is called a gorge line. 
 
 Since there is an infinite number of double-curv^ed lines, a great 
 variety of convolutes may also exist, with peculiarities depending 
 upon those of their directrices ; but the one above described will 
 
DESCRIPTIVE GEOMETRY. 83 
 
 suffice for illustration. It has been selected for the reasons that it 
 is not only as simple as any, but possesses some interesting proper- 
 ties, which will be noticed in due course, and render it practically 
 more important than others. The methods of constructing and rep- 
 resenting it will also be subsequently discussed in connection with 
 problems relating to it. 
 
 GENERATION OF SINGLE-CURVED SURFACES BY MOVING PLANES. 
 
 130. Observing that any two consecutive elements of the con- 
 volute determine an osculating plane of the directrix (110), and 
 that each element is the intersection of two successive positions of 
 the osculating plane (108), it will now be seen that this surface 
 may be generated by the motion of a plane subject to the condition 
 that it shall always be oscillatory to a given line of double curva- 
 ture. Since three points not in one right line suffice to locate a 
 plane, this single condition will in general control absolutely the 
 motion of the plane generatrix, and determine the form of the 
 resulting surface. An exceptional case occurs when the directrix 
 is reduced to a point, the motion of the plane being, then indeter- 
 minate unless governed by another condition, which may be de- 
 duced as follows : 
 
 If, on any single-curved surface, any curve be drawn which 
 cuts all the rectilinear elements, any two consecutive ones will inter- 
 cept an element of that curve ; the plane of those two elements will 
 therefore be tangent to the curve (110). In the case just men- 
 tioned the surface becomes a cone; which rnay, consequently, be 
 generated by the motion of a plane which always passes through a 
 given point and also remains tangent to a given curve. If the vertex 
 be infinitely remote, the cone will become a cylinder ; all of whose 
 elements, being parallel, are perpendicular to one and the same 
 plane. Hence a cylinder may be generated by the motion of a 
 plane which is always tangent to a given curve and also always per- 
 pendicular to a given plane. 
 
 WARPED SURFACES. 
 
 131. The absolute motion of a right line in space is fully deter- 
 mined when the simultaneous motions of any two of its pomts are 
 
84 DESCRIPTIVE GEOMETRY. 
 
 given in direction and Telocity ; the form of the surface generated by 
 the moving hne will be determined if the directions and relative 
 velocities of these two motions be known. 
 
 These directions may be determined by reqniring that the rec- 
 tilinear generatrix shall always touch two other given lines, either 
 straight or curved ; but some third condition is necessary in order 
 to esta])lisli a definite ratio between the velocities. 
 
 By whatever means this is accomplished, it is clear that if upon 
 the resulting surface any other line be drawn which cuts all the 
 rectilinear elements, that line may be taken as a third directrix; 
 and the same surface will be produced if tlie rectilinear generatrix 
 move so as always to touch this last line and the two at first 
 assumed. 
 
 Any ruled surface whatever, then, may be generated by a right 
 line so moving as always to touch three given linear directrices. 
 
 132. Cone Directer and Plane Directer. Suppose the three 
 directrices to be so chosen that tlie resulting ruled surface is neither 
 plane nor single-curved ; then through any given point in space let 
 a series of consecutive right lines be drawn, parallel in tlieir order 
 to the consecutive rectilinear elements of the given warped surface 
 in their order. These w^ill be elements of a cone, called the cone 
 directer of the surface : and it is clear that if any two lines which 
 cut all the rectilinear elements be taken as directrices, a right line 
 moving so as always to touch those lines and have its consecutive 
 positions parallel to the consecutive elements of the cone directer, 
 will re-generate the same surface. 
 
 It is easy to see that the warped surface may be such that the 
 series of lines, drawn through the assumed point parallel to its ele- 
 ments, shall lie in one plane — which is a limiting form of the cone ; 
 and this plane is called the plane directer. 
 
 Any warped surface whatever, therefore, may be generated by a 
 right line moving so as always to touch two given lines, and have its 
 consecutive positions parallel either to a given plane directer, or to 
 the< consecutive elements of a given cone directer. 
 
 133. Since every warped surface is curved, it is possible also to 
 conceive it as being generated by a curve, .which moves, and at the 
 same time changes its form, according to some definite law. 
 
DESCRIPTIVE GEOMETRY. 85 
 
 There is a great variety of warped surfaces, with peculiarities 
 depending on the laws of their formation. The methods of repre- 
 senting them, of assuming points and lines upon them, etc. , require 
 for ready apprehension a familiarity with some matters not yet dis- 
 cussed, and are accordingly reserved for subsequent consideration. 
 
 DOUBLE -CURVED SURFACES. 
 
 134. A Double-curved Surface is one which contains no rectilin- 
 ear elements, and can be generated only by a curve which moves in 
 such a manner as not to generate a surface of either of the preced- 
 ing classes. 
 
 Double-curved surfaces may be either double convex, that is, 
 convex in all directions, as the surface of a sphere or an egg ; or 
 concavo-convex, that is, convex in some directions but concave in 
 others, as the surface of a bell or of the groove in a pulley. Both 
 these peculiarities may exist in a single unbroken surface, as in the 
 case of a cylindrical ring, or annular torus. 
 
 SURFACES OF REVOLUTION. 
 
 135. A Surface of Revolution is one which may be generated by 
 the revolution of a given line about a right line as an axis. 
 
 Tlie intersection of such a surface by a plane perpendicular to 
 the axis is, therefore, the circumference of a circle. Consequently 
 the surface may also be generated by a circle which, moving with 
 its centre in the axis and its plane perpendicular to it, changes its 
 radius according to a definite law. 
 
 This second mode of generation is the one to which the greater 
 pi-actieal interest attaches ; since it is in this manner that such sur- 
 faces, of extensive application in the mechanic arts, are actually 
 produced in the lathe. 
 
 A plane traversing the axis is called a meridian plane, and its 
 intersection with the surface is called a meridian line : all meridian 
 lines of the same surface are obviously identical, and any one of 
 them may be taken as the revolving generatrix. 
 
 136. If the revolving line be straight, it either will lie in the 
 same plane with the axis, or it will not. If it does, it will either 
 
86 
 
 DESCRIPTEVE GEOMETRY. 
 
 be parallel to it, or intersect it ; in the former case the surface will 
 be a cylinder, in the latter a cone ; and these are the only sin^le- 
 cnryed surfaces of reyolution. 
 
 If the revolving right line does not lie in the same plane witli 
 the axis, then, 1 : Every point of it moves, therefore the consecu- 
 tive positions do not intersect ; and, 2 : Every point more remote 
 from the axis moves faster than one nearer to it, therefore the con- 
 secutive positions are not parallel. The surface must, then, be 
 warped ; its meridian line, as will subsequently appear, is an hy- 
 perbola : and this is the only warped surface of reyolution. It may 
 also be generated by revolving an hyperbola about its conjugate 
 axis ; and the surface being unbroken, it is also known as the liyjper- 
 holoid of revolution of one nappe. 
 
 With the exception of the three just considered, all surfaces of 
 revolution are of double curvature. 
 
 137. If two surfaces of revolution, having a common axis, cut 
 or touch each other at any point, they will do so all around the cir- 
 cumference of the circle described by that point. Thus in Fig. 
 137, the meridian lines macn, oacp^ are tangent to each otlier at a^ 
 
 "Y 
 
 Y 
 
 f 
 
 \ 
 
 
 T 
 
 >_ 
 
 cj 
 
 \ 
 
 
 '/ 
 
 p 
 
 Fig. 137 
 
 and intersect each other at c ; they will maintain these relations 
 throughout the revolution, and the circles ab^ cd^ will be common 
 to the two surfaces. 
 
 138. Representation of Surfaces of Revolution. These surfaces 
 are represented by drawing two views, viz. : 1, a side view, show- 
 ing the meridian contour, and should this be a broken line, such 
 
DESCRIPTIVE GEOMETRY. 87 
 
 circles as are described by the intersections ; and 2, an end view, in 
 wliicli are drawn tlie largest circle of the surface, and such others as 
 niaj be necessary in order that the drawings may be clear and 
 easily read. 
 
 For all ordinary practical purposes, and for most of the pur- 
 poses of descriptive geometry as well, any reference to a ground 
 line is useless, if not worse.' As shown in Fig. 138, a line, con- 
 tinuous centre line, 7d, containing the axis, should be drawn 
 through and heyond both views ; another one, gh^ should be drawn 
 at right angles to the first, through the centre of the end view : 
 since these lines are imaginary, they should never terminate in any 
 outline, lest they be supposed to represent lines actually existing on 
 the surface. 
 
 The axis, when there is no reason to the contrary, is supposed 
 to be parallel to the paper, and either liorizontal or vertical as may 
 be more convenient ; in the latter case the end view is a " liorizon- 
 tal projection " — but whether placed above or below the side view, 
 this horizontal projection is invariably a top view, and represents 
 the object as seen from ahove^ never as seen from below. 
 
 139. To assume a point on the surface, assume one of its pro- 
 jections, for instance c in the end view.' Then the point nmst lie 
 on the surface of a cylinder whose radius oc is known ; draw at that 
 distance from the axis a parallel to it, in the side view ; this is the 
 outline of the cylinder, and cuts the meridian line in m and n ; 
 these points describe circles, to one or both of which c is projected^ 
 as at c' or c^' . 
 
 If the projection in the side view be assumed, as at d\ then 
 the point lies on a circle whose radius pr can be found. With thi& 
 radius describe an arc about o in the end view ; the other projec- 
 tion must lie on this arc, as at d or d^, 
 
 TANGENT, NORMAL, AND ASYMPTOTIC PLANES AND SURFACES. 
 
 140. If on any surface, any number of lines be drawn through 
 a given point, then the tangents to all these lines at the common 
 point will in general lie in one and the same plane. 
 
 Such a plane is said to be tangent to the surface ; and the point 
 is called the point of contact. 
 
88 DESCRIPTIVE GEOMETRY. 
 
 If the surface is a single- curved one, no plane can be tangent 
 to it at anj point, which does not coincide with some position of 
 the plane generatrix. That generatrix contains two consecutive 
 rectilinear elements^ and is tangent (130) to every curve of the 
 surface which cuts them both. An infinite number of such curves 
 can be drawn through any point of either ; hence the plane is tan- 
 gent all along the line in which those two elements practically coin- 
 cide. 
 
 In the case of a double-convex surface, the demonstration is as 
 follows: Let M^ Fig. 138(3^, be a section of the surface by any 
 
 plane, and T a tangent to it at any point jp ; if the plane be re- 
 volved about T as an axis, this line will also be tangent to any suc- 
 cessive section, as iT or 0\ for it contains two consecutive points 
 of iH/, and they remain fixed during the revolution. Through ]) 
 draw on the surface any other curves, as K^ Z, cutting M 2>X «, h. 
 The section N cuts these curves at a' ^ V \ the section cuts them 
 at a" ^ y ; therefore the secants ^a, jpb ; jpa\ jpV ; jpa" ^ ph'% al- 
 w^ays lie in the revolving plane. Ultimately, the section of the 
 surface will cut K and L at points consecutive to j?, and these 
 secants will become tangents, still lying in the same plane through 
 T. If the surface is concavo-convex, whether warped or double- 
 curved, the same argument applies, although the forms of the sec- 
 tions J/, iV^, etc. , wull be different. 
 
 To pass a plane tangent to any surface at a given point, there- 
 fore : Draw through the point any two intersecting lines of the 
 surface, and at the point draw a tangent to each line ; the plane 
 determined by these two tangents is the plane required. 
 
 141. A plane tangent to any ruled surface must in general 
 
DESCRIPTIVE GEOMETRY. 89 
 
 contain the rectilinear elements which pass through the point of 
 contact; for each rectilinear element is its own tangent (109), and 
 therefore lies in the tangent plane by the preceding definition. 
 The vertex of a cone is an exceptional case ; through that point an 
 infinite number of planes tangent to the surface may be passed, 
 each of which contains two consecutive rectilinear elements, but no 
 others. 
 
 Since a plane tangent to any single-curved surface is tangent all 
 along a rectilinear element (140), it follows that if the base of the 
 surface lie in any plane of projection, the corresponding trace of 
 the tangent plane will be tangent to the base, at the point in which 
 the element of contact pierces that plane of projection. And in any 
 case, the right line cut from the plane of the base by the tangent 
 plane will be tangent to the curve of the base. 
 
 142. A plane tangent to a warped surface contains the rectilinear 
 element which passes through the point of contact ; but since it 
 does not contain the consecutive one, it cannot in general be tan- 
 gent along the element ; but there are some cases in which it is. 
 If the surface has two sets of rectilinear elements, the tangent 
 plane will contain both those which pass through the point of 
 contact. 
 
 A plane containing one rectilinear element of a warped surface, 
 And not parallel to the consecutive ones, will cut each of them in a 
 point. The curve joining these points will cut the given element, 
 and the given plane will be tangent to the surface at the point of 
 intersection ; for it contains the given element, which is its own 
 tangent, and also the tangent to the curve of intersection at the 
 point mentioned. If the given plane be parallel to a plane directer 
 of the surface, there will be no such curve, and the plane will not 
 in general be tangent to the surface. 
 
 Consequently, unless the projecting planes of the rectilinear 
 elements are parallel to a plane directer, each of them will be tan- 
 gent to the surface at some point. The projecting lines of these 
 points form the projecting cylinder of the surface ; and the pro- 
 jections of the elements, being the traces of the projecting planes, 
 will all be tangent to the base of this cylinder, which lies in the. 
 plane of projection. 
 
90 DESCRIPTIVE GEOMETRY. 
 
 143. A plane tangent to a surface of reyolution is perpendicular 
 to the meridian plane passing through the point of contact. Be- 
 cause, it contains the tangent to the circle of the surface at that 
 po^nt ; and this tangent, lying in a plane perpendicular to the axis, 
 is perpendicular to the radius at its extremity, and also to any line 
 joining that extremity with the axis ; and both this radius and that 
 line lie in the meridian plane. 
 
 144. Two curved surfaces are tangent to each other when they 
 have, at a common point, a common tangent plane. Evidently, 
 the sections of the two surfaces made by any one plane passing 
 through the point of contact will be tangent to each other at that 
 point. 
 
 If two surfaces of revolution having a common axis are tangent 
 to each other at a point, they will be tangent all round tlie circum- 
 ference of the circle described by that point in the generation of 
 the surfaces (137). 
 
 If two single- curved surfaces are tangent to each other at a 
 point of a common rectilinear element, they will be tangent all 
 along that element. Because the plane tangent to either surface at 
 any point is tangent to it all along the rectilinear element passing 
 through that point (140). 
 
 It is possible also to construct two warped surfaces which shall 
 be tangent to each other all along a common rectilinear element. 
 They must then have, at every point of that element, a common 
 tangent plane ; the methods by which this condition can be satisfied 
 will be explained farther on. 
 
 145. Normal Lines, Planes, and Surfaces. A right line is normal 
 to a surface at any point when it is perpendicular to tlie tangent 
 plane at that point. 
 
 A curbed line is normal to a surface at any point when its tan- 
 gent at that point is normal to the surface. "When not otherwise 
 stated, ''the normal" to a surface is understood to be rectilinear. 
 
 A plane is normal to a surface at any point when it is perpen- 
 dicular to the tangent plane at that point. Thus there may be an 
 infinite number of planes normal to the surface at a point, while 
 there can be but one normal right line, common to all these planes. 
 
 If at the consecutive points of any line upon a given surface ' a 
 
DESCRIPTIVE GEOMETRY. 91 
 
 series of normals to that surface be erected, they will be elements 
 of a ruled surface normal to the giyen surface. If these normals are 
 tangent at those consecutive points to lines lying upon any other 
 surface, then that surface is also normal to the given surface. 
 This relation is mutual, and the two surfaces are said to be normal 
 to each other. 
 
 146. Asymptotic Planes and Surfaces. The relation between 
 asymptotic surfaces is analogous to that between asymptotic lines. 
 Thus if an hyperbola be made the base of a right cylinder, the 
 plane containing its asymptote and parallel to the elements of the 
 cylinder will be asymptotic to the surfacCc Again, if two conju- 
 gate hyperbolas, together with their common asymptotes, be re- 
 volved about either axis, the two hyperboloids of revolution will be 
 asymptotic to each other and to the cone generated by the asymp- 
 totes. 
 
DESCHIPTIVE GEOMETEY. 
 
 CHAPTER ly. 
 
 On the Determination of Planes Tangent to Surfaces of 
 Single and of Double Curvature. 
 
 planes tangent to single-curved surfaces. 
 
 147. In the construction of these problems, and of many others, 
 there is frequent occasion to draw tangents to curves of unknown 
 properties. A sufficient degree of accuracy for j^resent purposes, 
 and indeed for most practical purposes, may be ol)tained by means 
 of approximating circular arcs, as follows : Let it be required to 
 draw a tangent to the curve DJ^, Fig. 139, at the point P. By 
 Q trial and error a centre O and radius 
 
 CP can be found, sucli that the cir- 
 cular arc described about will 
 sensibly coincide with the given 
 curve for a short distance on each 
 Tig. 139 gj^^ ^f ^]^q given point ; the required 
 
 tangent is then drawn, perpendicular to CP. 
 
 If, on the other hand, it be required to draw a tangent in a given 
 direction, or through a given point, as : In tliis case the tangent 
 is drawn mechanically ; the ruler being set, not so as to coincide 
 with either the point or the curve, but at a small distance from 
 each, the eye being able to judge with perfect precision as to the 
 equality of these distances. The point of contact is then deter- 
 mined by dropping a perpendicular upon the tangent from the 
 centre O, found as aboye. 
 
 147a. The following constructions are more laborious, but 
 give more precise determinations : 
 
 1. In Fig. 139<^, to draw a tangent to the curve A^Z at any 
 pointy. Through ^ draw chords from any points of the curve, 
 produce them all in one direction (say to the left), then with j? as 
 
DESCRIPTIVE GEOMETRY. 
 
 93 
 
 a centre and any convenient radius draw a circular arc C^F" inter- 
 secting them. P>om tins arc set off on the prolongation of each 
 chord a distance equal to the chord itself, the chords on the left of 
 ^ being set oft" to the left of the arc, and vice versa; thus ef = pi, 
 
 .V Fig. 139 a 
 
 cd =^ pa^ ns ^= pg^ and so on. The cnvYQ fds thus determined 
 cuts 6^ F in a point o of the required tangent TT. 
 
 2. In Fig. 189 J, to iind the point of contact, TT being tan- 
 gent to the curve KL. Draw any 
 number of chords parallel to TT^ 
 through their extremities draw 
 parallel ordinates in opposite direc- 
 tions, and on each ordinate set off 
 from TT a distance equal to the 
 corresponding chord ; as, for ox- 
 ample, rs — mn = gh^ hd ^^ ef =^ 
 ac^ etc. ; the curve sdn^ passing 
 through the points just located, 
 cuts TT in the required point. 
 
 Such curves are known as 
 " curves of error," and with due care give very accurate results. 
 
 148. Problem 1. To draw a plane tangent to any single- 
 curved surface through any given point of the surface. ' 
 
 Analysis. Through the given point draw a rectilinear element, 
 and through the foot of that element draw a tangent to the base. 
 The plane of these two lines is the required tangent plane. 
 
 Construction. In Fig. 140, the given surface is a cone whose 
 vertex is O^ the plane of the base being perpendicular to T ; this is 
 represented, and the point P upon it assumed, as in Fig. 132. 
 Through P draw the element OPC^ and through its foot 6^ draw a 
 
94 
 
 DESCRIPTIVE GEOMETRY. 
 
 tangent to the base. The horizontal projection he of this tangent 
 will be tangent at c to the horizontal projection of the base, and its 
 vertical projection h'c' will coincide with the vertical projection of 
 the base. The traces of the tangent are M and G ; those of the 
 element are N and D ; therefore m! and n' are points in the verti- 
 cal trace, and d and g are points in the horizontal trace, of the re- 
 quired plane ; and these traces must meet at T in the ground line. 
 In Fig. 141, the surface is a cylinder, with its base in the ver- 
 tical plane. The tangent to the base is therefore the required ver- 
 
 ^ Fig. 142 
 
 tical trace, which cuts AB at T. The element PC pierces H in Z>, 
 thus determining dTt the horizontal trace. 
 
 In Fig. 142, the base of the cylinder is horizontal ; the liori- 
 zontal trace is therefore parallel to lic^ the horizontal projection of 
 the tangent to the base. The point cZ, found as before, fixes the 
 location of this trace, which, when produced, cuts AB in T. The 
 tangent KC pierces V in the point N\ and since the vei-tical trace 
 of the element PC \s> inaccessible, the direction of Tt' is determined 
 by drawing it through n'. 
 
DESCRIPTIVE GEOMETRY. 
 
 95 
 
 149. Problem 2. To draw a plane tangent to a cone through 
 a given point without the surface. 
 
 First Method. Analysis. Draw a line through the vertex of 
 the cone and the given point. Through the point in which this 
 line pierces the plane of the cone's base, draw a tangent to the 
 base. The plane of these two lines is the required tangent plane. 
 
 Construction. Let 0-XY^ Fig. 143, be the given cone, P the 
 given point. The line OP^ through the point and the vertex, 
 pierces H in D^ and the plane of the cone's base in S. The verti- 
 
 cal projection of the tangent is s'c' tangent to the vertical projec- 
 tion of the base ; its horizontal projection coincides with that of 
 tlie base itself, whose plane is perpendicular to H. This tangent 
 pierces H in 6^ ; and <7 and g are two points in the horizontal trace 
 of the required plane, which cuts AB in T, one point of the verti- 
 cal trace. Another point might be determined by finding the ver- 
 tical trace of OP ; but in this instance a third line, the element of 
 contact C?^', has been employed instead; it pierces T in ^, and 
 Tr't' is the required vertical trace. 
 
96 DESCRIPTIVE GEOMETRY. 
 
 When the cone becomes a cylinder, as in Fig. 144, the line 
 through the vertex becomes parallel to the elements. In the dia- 
 gram, the plane of the base is parallel to T, and OP pierces it at 
 8\ in this particular case the vertical projection s'c' of the tangent 
 to the base happens to be ]3arallel to AB, and since SO is therefore 
 parallel to the horizontal trace, the required plane will be parallel 
 to the ground line ; consequently it is sufficient to determine one 
 point in each trace. 
 
 When more than one tangent to the base can be drawn, there 
 will be more than one solution. 
 
 150. Second Method. Analysis. Pass through the given point 
 any plane cutting all the rectilinear elements of the given surface, 
 and in tins plane draw through the point a tangent to the curve of 
 intersection. Draw through the point of contact a ]-ectilinear ele- 
 ment ; this element, and the tangent to the curve, determine the 
 required plane. 
 
 Note. — This process may be applied to any single curved sur- 
 face ; but is of more particular advantage in cases analogous to the 
 one herewith illustrated. 
 
 Construction. In Fig. 145 the surface is a cone, with a circular 
 base situated in the horizontal plane : every section of it by a hori- 
 zontal plane will therefore be a circle whose centre lies upon the 
 line U^ drawn from the vertex to the centre of the base. Through 
 the given point P draw a horizontal plane LL ; it cuts U in 7?, 
 and also cuts tlie element OX in a point whose vertical projection 
 is s' ; therefore r's' is the radius of the circular section, which more- 
 over must be tangent to the extreme visible elements in the hori- 
 zontal projection. Draw through P a tangent to this section ; in 
 this case this tangent is parallel to the horizontal trace, which is 
 taugent to the base of the cone ; so that it is not necessary to make 
 use of the element of contact, OG. But, especially if the given 
 point P is but a small distance above H, it would be advisable to 
 use the vertical trace of that element if accessible, since the direc- 
 tion of Tt' would be thus more accurately determined. 
 
 151. Problem 3. To draw a plane tangent to a given cone 
 and parallel to a given right line. 
 
 Analysis. Through the vertex of the cone draw a parallel to the 
 
DESCRIPTIVE GEOMETRY. 97 
 
 given line ; and from the point in which it pierces the plane of the 
 cone's base, draw a tangent to the base. Tlie plane of this tangent 
 and the parallel, is the tangent plane required. 
 
 Construction. If in Fig. 143 we supj)ose the line OP to be 
 determined by the condition that it shall be parallel to a given line 
 J/jy, the construction is the same as in (149). 
 
 If the parallel line through the vertex pierces the plane of the 
 base in a point so situated that a tangent to the base cannot be 
 drawn through it, the problem is impossible ; if more than one tan- 
 gent can be drawn, there will be a corresponding number of solu- 
 tions. 
 
 If in this or the preceding problem tlie line through the vertex 
 be parallel to the plane of the base, the tangent to the base will be 
 parallel to that line. Should that line pierce the plane of tlie base 
 at a remote and inaccessible point, the tangent plane may be con- 
 structed as follows : Through any point of the parallel line pass a 
 plane cutting all the rectilinear elements ; and from the assumed 
 point draw a tangent to the curve of intersection. This tangent, 
 and the rectilinear element through the point of contact, will de- 
 termine the required plane. 
 
 152. Wlien the vertex is infinitely remote, a process analogous 
 to that of (150) may be employed; in any plane containing the 
 given line and cutting all the rectilinear elements of the cylinder, a 
 tangent to the curve of intersection may be drawn parallel to the 
 given line : the plane of this tangent and the element through the 
 l^oint of contact will be the required tangent plane. This is illus- 
 trated in Fig. 146, PC being the tangent and PE the element; 
 and obviously the intersection CE of the tangent plane with the 
 plane of the base will be tangent to the base at E^ the foot of the 
 element of contact. 
 
 The direct application of this process is not usually convenient ; 
 but from ii a tentative one is derived, based upon the consideration 
 that if any plane DOF\>Q constructed, containing a line (97^ paral- 
 lel to the elements, and another line DO parallel to the given line : 
 then the intersection 7)7^ with the plane of the base of the cylinder 
 will be parallel to CE^ which line, with the element through the 
 point of contact E, will determine the required plane. 
 
98 
 
 DESCRIPTIVE GEOMETRY. 
 
 153. The construction in accordance with the preceding argu- 
 ment is shown in Fig. 147, where JO^is tlie given line. Through 
 any point on any element of the cylinder, as OX, draw a parallel 
 to JfiV^; this line pierces the plane of the base at 6^, the element 
 pierces it at X, and XC is the line of intersection corresponding to 
 -Z>i^of Fig. 146. Parallel to X6' draw a tangent to the base; it 
 
 riG.146 
 
 pierces T in /^, H in G^ and E is the point of contact. The ele- 
 ment through ^pierces H in i> and V in R : joining d and ^, then, 
 we have the horizontal trace ; joining s^ and r\ the vertical trace is 
 determined ; and these traces meet at T in the ground line. 
 
 154. Special Cases of the Preceding Problems. In Fig. 
 14S a cone of revolution is given, with its axis parallel to AB : it is 
 required to draw a plane tangent to it, parallel to the given line 
 LM. The parallel through the vertex pierces W in Z>, V in iT, 
 ^nd the plane of the cone's base in S. In the profile, the circle of 
 the base is seen in its true form, and S> is projected at 6-, : the tan- 
 gent to the base through «, pierces H at ^,, whose distance from 
 FF determines the distance of g from AB in tlie horizontal projec- 
 tion. Then the points d and g fix the horizontal trace tT^ and 
 Tn't' is the vertical tr^ce. If necessary or convenient, the element 
 of contact, seen in the profile as o^c^^ and in the vertical projection 
 -as o'c\ may be used to locate points in the traces ; for example, the 
 point /'' in the vertical trace is found by producing that element. 
 
 In the profile. Fig. 149, a cylinder of revolution is shown, its 
 
DESCRIPTIVE GEOMETRY. 
 
 99 
 
 axis O being parallel to the ground line. This figure sufficiently 
 illustrates the fact that in this case, whether the plane is to be tan- 
 gent at a given point P on the surface, to pass through a given 
 point R without the surface, or to be parallel to a given line LM^ 
 the solution is at once effected and the result clearly exhibited by 
 drawing the profile. The projections on the principal planes are 
 not here given ; as in Fig. 74, they would be made up chiefly of a 
 
 
 ^ 
 
 ^ 
 
 [- 
 
 Xb---'-/^^'^^ 
 
 •^ 
 
 V 
 
 
 \X 
 
 
 
 ;■ 
 
 \ 
 
 ^ \ 
 
 
 
 '\ 
 
 \, 
 
 
 Fig. 149 
 
 
 > 
 
 V 
 
 confusing series of parallels : and all of this is equally true whether 
 the base of the cylinder is circular or of any other form. 
 
 155. PiANES Tangent to the Helical Convolute. In 
 Fig. 150 is shown so much of this particular convolute as is neces- 
 sary to illustrate the application to it of the preceding processes. 
 
 in representing it, the helical directrix should be accurately 
 drawn ; and this is facilitated by first drawing the front and top 
 views of the cylinder upon whose surface it lies. Now, the axis of ; 
 this cylinder being vertical and its base lying in H, it was shown 
 in (128) tliat the horizontal trace of the convolute will be the in- 
 volute of the circle of that base ; and this should next be carefully 
 cjonstructed, since it is the locus of the points in which H is pierced 
 fy the rectihnear elements of the surface. Then, om being equal 
 to the quadrant oli^ and gn being three times as great, it is clear 
 that OM is tangent to the helix at 0^ and GN tangent to it at G\ 
 and these are the extreme visible elements in the front view of the 
 
100 
 
 DESCRIPTIVE GEOMETRY. 
 
 portion shown, wliicli embraces three fourths of the circumference of 
 the cylinder, the rectilinear elements terminating, as in Fig. 135, 
 at their points of tangency to the directrix. 
 
 156. To assume a point on the surface, assume the horizontal 
 projection, asj9. Through^ draw the horizontal projection of an 
 element; it is tangent to the circle of the cylinder's base at c, ver- 
 tically projected at c' on the helix, and cuts the involute at <?, ver- 
 tically projected at e' in AB : j^' must lie on c'e' the vertical projec- 
 
 tion of the element. By reversing this construction the horizontal 
 projection may be found if ^' is assumed. 
 
 157. Problem 4. To draw a plane tangent to this convolute 
 and parallel to a given right line. 
 
 The tangents to the helix make equal angles with the plane of 
 the base, and it is apparent that this surface, though not warped, 
 has a cone directer; which, as shown in Fig. 151, is a cone of 
 revolution, whose elements make the same angle with the plane of 
 the base. If it be required to draw a plane tangent to the convo- 
 lute and parallel to the given line XY\ lirst draw a plane parallel 
 to this line and tangent to the cone directer (151): ss is its hori- 
 
DESCKIPTIVE GEOMETRY. 101 
 
 zoiital trace, and the horizontal trace tT oi the required plane is 
 parallel to ss and tangent to the involute. Next draw a tangent to 
 the circular base of the cylinder, perpendicular to tT\ this is the hori- 
 zontal projection of the element of contact, and locates the point of 
 tangency d^ which is vertically projected at d' in AB. The point 
 of contact ic with the circle is projected to u upon the helix, thus 
 determining d'u' the vertical projection of the element of contact, 
 whose vertical trace is B\ and Tr't' is the vertical trace of the 
 tangent plane. Should 7'' be inaccessible, draw through any point 
 L upon DU 2, parallel to tT\ it is a horizontal line of the plane, 
 and pierces V in K^ a point of the required vertical trace. 
 
 158. To draw ajplane tangent at any assumed pointy as Pi 
 the element CE through tliis point is one line of the plane ; the 
 horizontal trace is drawn through ^, perpendicular to ce^ and the 
 vertical trace is then found as above. 
 
 To dravj a tangent plane through a point without the surface. 
 
 Draw first a horizontal plane through the given point (150): 
 its intersection with the surface will be another involute, to which 
 a tangent is to be drawn through the given point. This will be 
 one line of the required plane ; the element through the point of 
 contact, which is found as in (157), is another, and by means of 
 these the traces may be determined. 
 
 159. In these illustrations, the lower nappe only of the convolute 
 has been represented and considered, in order to avoid confusion 
 in the diagrams. But it must not be f(>rgotten that the rectilinear 
 elements can be indefinitely extended both ways from the points 
 of contact with the helix ; so that a plane tangent to the surface at 
 any point is tangent to it all along a line lying on both nappes : 
 and it makes no diiference whether the upper or the lower one is 
 em])loyed in the process of construction. 
 
 160. Considering any point as P, Fig. 150, at a given dis- 
 tance CP from the point of contact C between the element and the 
 directrix : it is clear that in the generation of the surface the path 
 of this point will be a helix whose pitch is the same as that of the 
 directrix itself. This particular convolute, therefore, is one of the 
 numerous family of helicoids^ of which all the others are eithei' 
 warped or double-curved surfaces. And though it is usually pre- 
 
102 DESCRIPTIVE GEOMETRY. 
 
 seiited and represented in a manner so imperfect and ol)scure as to 
 conceal the fact, it possesses a certain practical interest, because it is 
 in fact the surface of a screw-thread, which, as will subsequently 
 be shown, can be cut in a lathe in the usual manner. 
 
 161. In general, a plane cannot be passed through a given 
 right line and tangent to a single-curved surface. The problem, 
 however, is possible in the following cases, viz. , if the given line 
 lies on the convex side of a cylinder, and is parallel to its rectilinear 
 elements ; if it passes through the vertex of a cone ; or if it is tan- 
 gent to a line of any single- curved surface. 
 
 PLANES TANGENT TO DOUBLE-CURVED SURFACES. 
 
 162. A plane tangent to any double-curved surface at a given point 
 
 can in general be determined (140) by drawing at that point a tan- 
 gent to each of two lines of the surface which ]3ass through that 
 point, the selection depending upon considerations of convenience. 
 Of the surfaces of this class which are used in the mechanic arts, 
 those of revolution constitute by far tlie larger portion, by reason of 
 the facility with which they can be formed in the lathe or on the 
 potter's wheel; they serve as well as any for illustrative purposes, 
 and to them we shall confine our attention. In dealing witli them 
 in the manner stated, the two curves which would naturally be se- 
 lected are the meridian line and the circumference of the transverse 
 section through the given point. 
 
 163. If a right cone be tangent to a surface of revolution 
 which has the same axis, it will be tangent all round the circumfer- 
 ence of a circle. Any plane tangent to this cone will be tangent 
 all along an element; therefore the plane will be tangent to the 
 surface at the point in which this element of contact cuts that 
 circle of contact. Such an auxiliary cone may sometimes be used 
 to advantage in determining a plane tangent to a double-curved 
 surface. 
 
 164. If the contour of a surface of revolution be such that the 
 direction of the normal to it can be readily determined, a plane tan- 
 gent to the surface at a given point can . be constructed with great 
 facility, since it is perpendicular to tlie normal at its extremity. 
 
DESCRIPTIVE GEOMETRY. 
 
 ws 
 
 165. Problem 1. To draw a plane tangent to a spliere at a 
 given point on the surface. 
 
 Construction. In Fig. 152, let C be the centre of tlie sphere, 
 and P the given point, assumed as in Fig. 138; then CP is the 
 radius of contact. The horizontal projecting plane of this radius 
 contains a horizontal diameter of the spliere ; revolving that jjlane 
 about this diameter until it is parallel to H, P falls dXp'\ and the 
 
 Fig. 153 
 
 horizontal trace appears as H" H" parallel to cp^ the distance cx'^ 
 being equal to ex' . Draw at^'' a tangent to the great circle; it 
 cuts H" H" in d" ^ the revolved position of a point d in the re- 
 quired horizontal trace, v/hich is perpendicular to cp (43). Draw 
 at P a tangent to the horizontal circle of the sphere through that 
 point ; it is a line of the tangent plane, and pierces V in ^ : the 
 line PD pierces V in N\ therefore the points r\ n' ^ determine the 
 vertical trace, which must be perpendicular to c'p' (43), and meet 
 the horizontal trace at T in AB. 
 
 I*^OTE. Regarding tlie sphere as a surface of revolution with a 
 vertical axis, it is seen that DPO is an element ot a right cone 
 tangent to the sphere around the circle described by P\ the base 
 
104 DESCKIPTIVE GEOMETRY. 
 
 of this cone in H is a circle with radius cd — x"d" \ and Tt is tan- 
 gent to this circle. 
 
 166. Pkoblem 2. To draw a plane tangent to any surface of 
 revolution at a given point on the surface. 
 
 Coiistructioii. In Fig. 153, the assumed point P lies upon the 
 circle whose vertical j^rojection is m'n'. Let c be the centre of cur- 
 vature of the contour 7ig% then en' is normal to the curve at n \ 
 its prolongation cuts the axis in (9, and o'n'm' is the vertical pro- 
 jection of a cone normal to the given surface. Tlierefore o'p' is 
 the vertical and op is the horizontal projection of the normal to the 
 surface at P. Through P di^avf, as in Fig. 67, a plane tTt' per- 
 pendicular to P(9; it is the tangent plane required. 
 
 167. Problem 3. 2^o draw through a given line a j)l(ine tan- 
 gent to a given sphere/ 
 
 First Method. Analysis. Eegarding the given line as the in- 
 tersection of two planes tangent to the sphere, each plane is perpen- 
 dicular to a radius at its extremity; therefore the plane of these 
 two radii is perpendicular to the given line. If then a plane be 
 passed through the centre of the sphere and perpenaicular to the 
 given line, and from the point in which it cuts the line a tangent 
 be drawn to the great circle cut from the sphere ; then the plane 
 determined bj that tangent and the given line is the one required. 
 
 Construction. Let (7, Fig. 154, be the centre of the given sphere, 
 MW the given line. 
 
 On the horizontal projecting plane of the line make a sup- 
 plementary projection; in this II' 11' is the horizontal plane, c^ the 
 centre of the sphere, m^n^ the given line, and LL the plane ])ei - 
 pendicular to the line, of which II is the horizontal trace. This 
 plane contains a horizontal diameter of the sphere, of which ge is 
 the horizontal projection ; and it cuts MN in the point A. Ee- 
 volve the plane about ge until it becomes horizontal ; its trace II 
 then appears as l"l" ^ and a falls at a" . Draw a" d" tangent to the 
 great circle of the sphere, and find o" the point of contact. Mak- 
 ing the counter-revolution, a" returns to ^, d" falls at d in II 
 (vertically projected at d' in AB), and o" goes to o on da., whence 
 it is vertically projected to d on d'a' . 
 
 The tangent line 7>^ .pierces H in Z^, the given line pierces it 
 
DESCRIPTIVE GEOMETRY. 
 
 105 
 
 in iV^; therefore tlie points d and n determine the horizontal trace 
 tT^ which must also be perpendicular to co : JfiV^ pierces V in B, and 
 r is a point in the vertical trace Tf of the required tangent plane ; 
 and tliis trace must also be perpendicular to c'o\ the vertical pro- 
 jection of the radius of contact. 
 
 168. Second Method. Analysis. If any point of the line be 
 taken as the vertex of a cone tangent to the sphere, a plane con- 
 
 1 ^ 
 
 Zyv^ 
 
 ,'/ 
 
 X' TF '/ \n' 
 
 ^ 
 
 ~^^\/ \ja" ^"^^ 
 
 
 m^ihg 
 
 taining the given line and tangent to this cone will also be tangent 
 to the sphere. The cone is tangent to the sphere all round the 
 circumference of a small circle ; the plane is tangent to the cone 
 all along an element; and the intersection of the circle and the 
 element is the point of contact between the sphere and the plane. 
 
 Construction. In Fig. 155, C' is the centre of the sphere, MN 
 the given line. The point Z>, in which the line is cut by a liori- 
 
106 
 
 DESCRIPTIVE GEOMETRY. 
 
 m/ ; n\ 'X' z\ 
 
 V 
 
 0^ 
 
 .Tig. 157 
 
DESCRIPTIVE GEOMETRY. WT 
 
 zoTital plane LL through C, is selected as the vertex of the auxili- 
 ary cone, merely for convenience ; its axis being thus made hori- 
 zontal, the plane of the circle of contact is vertical, and appears as 
 a right line nw in the horizontal projection. This plane cuts MJV 
 in a point whose horizontal projection is e and vertical projection 
 e' ; in the supplementary projection on this plane it appears as e^ : 
 through e^ draw a tangent to the circle of contact, find the point of 
 tangency <?,, and produce it to cut H' H' in g^. These two points 
 are projected back to o and g\ g is one point, and m is another, in 
 the horizontal trace of the required plane, which must also be per- 
 pendicular to CO. The vertical projection o' will be in a perpen- 
 dicular to AB tJirough o, at a distance o'z' above AB equal to o,s, in 
 the supplementary projection ; the given line pierces T in iV^, and 
 the element OD of the cone pierces it in S; therefore Ts'ii't' is 
 the vertical trace of the tangent plane, which nmst be perpendicular 
 to c'o\ the vertical projection of the radius of contact. In order to 
 avoid confusion, the vertical projection of the auxiliary cone is not 
 drawn, no use being made of it in the construction. 
 
 169. Third Method. Analysis. If any two points of the given 
 line be taken as the vertices of cones tangent to the sphere, the 
 planes of the circles of contact will intersect in a common chord, 
 whose extremities w^ill lie on all three surfaces. A plane contain- 
 ing the given line and either of these points will be tangent to the 
 sphere. 
 
 Construction. This is illustrated pictorially in Fig. 156, where 
 S is the sphere, MN the given line, and 7>, E., are the vertices of 
 the two cones. The construction in projection is given in Fig. 157, 
 (7 being the centre of the sphere, and vJ/iTthe given line. A hori- 
 zontal plane LL through C cuts the line in D^ and a plane FF 
 through C and parallel to V cutfS it in E. Take D and E as the 
 vertices of the two cones ; then the axis of the first being horizon- 
 tal, the plane of the circle of contact will be vertical ; kg is its 
 horizontal trace. The axis of the second will be parallel, and the 
 plane of the circle of contact perpendicular, to the vertical plane ; 
 Ic'g' is the vertical trace. Therefore Ug^ lc'g\ are the projections of 
 the intersection of these two planes. This line pierces H in 6^; 
 in order to find where it pierces the surface of the sphere, make a 
 
108 
 
 DESCRIPTIVE GEOMETRY. 
 
 supplementary projection on a plane perpendicular to the axis of 
 the first cone. In this projection the line of contact is seen in its 
 true form as the circle of wliich ^i^ is the centre, and the line k^g^ 
 cuts its circumference at 6>, , Avhich is projected back to o, and 
 thence vertically to o\ the distance o'b' being equal to o^z^. Then 
 DO is> Si line of the required plane; its vertical and horizontal 
 traces are xS'and I^; those of the given line are JV and J/; there- 
 fore mr 7^ is the horizontal and Tn's' is the vertical trace of the 
 tangent plane : and these traces are respectively perpendicular to the 
 projections of CO the radius of contact. 
 
 The vertical projection of the first cone and the horizontal pro- 
 jection of the second are omitted ; no use being made of either in 
 effecting the solution, their introduction would simply confuse the 
 diagram. A like result would have followed had the determina- 
 tion of the other tangent plane been represented, and the construc- 
 tion of one is sufficient to illustrate the principles of either method. 
 
 170. Problem 4. To find the jpoint of contact of any surf ace 
 of revolution with a plane perjpendicular to a given line. 
 
 Construction. In Fig. 158, the axis of the surface is vertical. 
 
 No ground line is drawn, but the axis may be regarded as lying in 
 a vertical plane, represented in the top view by the horizontal cen 
 
DESCRIPTIVE GEOMETRY. 
 
 109 
 
 tre line. Revolving the given line MN about a vertical axis until 
 it lies in this plane, it takes the position MN^. The plane LL is 
 drawn tangent to the contour and perpendicular to m'n^^ and 0\ is 
 the revolved position of the point of contact. In the counter- 
 revolution this point describes an arc of a horizontal circle, the angle 
 o^co being equal to the angle n^mn^ and is the true position of 
 the required point. 
 
 In Fig. 159, the axis is horizontal, and a vertical plane contain- 
 ing it is represented by the vertical centre line in the end view. 
 The given line MN and the surface together are revolved about 
 the axis until the line comes into that vertical plane ; and the re- 
 maining steps require no explanation. 
 
 In Fig. 160, the axis lies in the vertical plane, but is inclined, 
 the object accordingly appearing foreshortened in the top view. 
 The line to which the tangent plane is to be perpendicular is given 
 
 by the projections mn^ m'n' ; in the supplementary end view it ap- 
 pears as m,i^j, the distance n^x^ being equal to nx in the top view. 
 This line is revolved about the axis of the surface into the plane 
 FF as in Fig. 159, n^ going to n^, which is projected back to n" \ 
 
110 
 
 DESCRIPTIVE GEOMETRY. 
 
 the plane LL is then drawn tangent to the contour, and the point 
 of contact o" is projected to o^, revolved to <?,, and re-projected to 
 o' . In the top view, o is directly under o\ at a distance from the 
 centre line equal to the distance of o^ from FF in the end view. 
 
 KoTE. The determination of the outline in the top view, Fig. 
 160, depends upon this principle, viz., that the yisible contour of 
 any object is the envelope of all the lines upon its surface. In the 
 present case, all the transverse sections are circles, whose horizontal 
 projections are ellipses, and the contour is tangent to all these 
 ellipses, 
 
 171. Problem 5. To draw a plane making given angles with 
 the pi'incipal planes of projection. 
 
 Argument. In the profile. Fig. 161, (9 and P are the vertices 
 of two cones tangent to the same sphere, the axis of one being ver- 
 
 
 \ 
 
 o 
 
 i 
 
 P> 
 
 \ \ 
 
 
 
 M 
 
 Fig. 161 
 
 tical, that of the other perpendicular to V. All planes tangent to 
 the first are equally inclined to H ; all those tangent to the second 
 are equally inclined to Y : a plane through the line MN joining 
 their vertices, if tangent to one, will be tangent to both, as in Fig. 
 157. J/ will be a point in the horizontal trace, N 2, point in the 
 vertical trace ; and each trace will be tangent to the base of the 
 cone which lies in the correspondhig plane of projection. 
 
 In the application, the angles at the bases of the cones must be 
 made equal to the assigned angles which the required plane is to 
 make with H and V respectively. 
 
 Construction. In Fig. 162, the profile is first drawn, the cen- 
 
DESCRIPTIVE GEOMETRY. Ill 
 
 tre c^ of the sphere being for convenience placed in the ground 
 line. Tangent to the outline of the sphere, draw x^s^^ making the 
 angle x,s^g^ equal to the assigned angle with H ; and also z{r^ , the 
 angle z{t\G^ being equal to the assigned angle with V. Then draw 
 an indefinite perpendicular to AB through any convenient point (7, 
 and on it set off Cx above, equal to (?,a?, ; also Cz below, equal to 
 c,2^. About C describe above AB an arc with radius Cr' = c^r^, 
 and draw x'T tangent to it; also about C describe below AB an arc 
 with radius Os = c^s^, and draw sT tangent to it. These two tan- 
 gents will be the traces of the required plane, which must intersect 
 at T in the ground line. 
 
 JN'oTE. A different solution of this problem has already been 
 given in (94), but the one here presented as a neat application of 
 tangent planes is preferable, being more reliable as well as more 
 expeditious. 
 
li;;i DESCRIPTIVE GEOMETRY. 
 
 CHAPTEE Y. 
 
 OF INTERSECTIONS AND DEVELOPMENTS. 
 
 Intersection of Surfaces by Planes. Development of Single-curved Surfaces. 
 Tangents to Curves of Intersection, before and after Development. Prob- 
 lem of the Shortest Path. Intersections of Single-curved Surfaces. In- 
 finite Branches. Intersections of Double-curved Surfaces. Intersection 
 of a Cone with a Sphere. Development of the Oblique Cone. 
 
 172. If a line be drawn upon one surface, tlie point in wliicli it 
 pierces any other surface will lie upon the intersection of the two. 
 In order to find that point, an auxiliary surface is passed through 
 the line ; this intersects the second surface in another line, which in 
 turn cuts the first line in the point sought. 
 
 This is illustrated in finding the point in which a given right 
 line pierces a plane ; the auxiliary plane intersects the given plane 
 in another right line, and that cuts the first one in the required 
 point. It is clear that the line here spoken of as given might have 
 been cut from any given rnled surface by the auxiliary 23lane ; then 
 the point, located as above, would have been one point in the 
 intersection of that surface by the given plane ; and other points could 
 be found by means of other auxiliary planes. 
 
 Again, were any other surface substituted for the given plane, 
 the construction would be modified only in this, that the auxiliary 
 plane might intersect it in a curve instead of in a right line : but 
 this curve would still cut the rectilinear element of the first surface 
 in a point of the line of intersection of the two surfaces. 
 
 173. The point last mentioned is the one in which the first 
 surface is pierced by the curve cut from the second ; and evidently 
 would be, whether the line cut from the first by the auxiliary plane 
 were straight or otherwise. Which shows that the point in which 
 any plane curve pierces a given surface may be found by first de- 
 
DESCRIPTIVE GEOMETRY. 113 
 
 termining the intersection of its plane with that surface ; it will be 
 a line cutting the given curve in the required point. 
 
 But even in the case of a right line piercing a plane it is not 
 necessary that the auxiliary surface should be a plane : if, for ex- 
 ample, a cone or a cylinder be passed through the line, intersecting 
 the given plane in a circle, it will now be apparent that the circum- 
 ference will cut the given line in the point required. And in deal- 
 ing with a line of double curvature the simplest auxiliary surface 
 that can be used is cylindrical. Thus, in order to find where the 
 curve DEG^ Fig. 131, pierces the vertical plane, we make use of 
 the horizontal projecting cylinder, from which V cuts the element 
 M', and this intersects the curve in h'^ the point sought. 
 
 The same figure illustrates the process of finding the intersection 
 of two cylindrical surfaces ; one, whose base is ceh^ having vertical 
 elements, while the elements of the other, whose base is c'e'k' ^ are 
 perpendicular to Y. An auxiliary plane ee\ parallel to the rec- 
 tilinear elements of both cylinders, cuts from each a riglit line, and 
 these intersect in E\ and by other planes parallel to this the points 
 D and G are determined ; these points lie upon both surfaces, and 
 the curve cEk' is the required intersection. 
 
 174. From the preceding it will be perceived that the problems 
 of finding the points in which surfaces are pierced by lines, and the 
 lines in which surfaces intersect each other, are correlated and inter- 
 woven, involve the same principles, and require for their solution a 
 previous knowledge of the intersections of the surfaces under con- 
 sideration by certain others, which may be used as auxiliaries. In 
 all of them the ultimate determinations consist in the location of 
 points by the intersections of lines; and the auxiliary surfaces 
 should be so selected that these lines shall cut each other as nearly 
 as possible at right angles. 
 
 175. Tangents to Curves of Intersection. If a surface is cut by 
 a plane, the tangent to the line of intersection at any point will lie 
 in that plane, and also in the plane tangent to the surface at the 
 given point ; therefore it will be the intersection of those planes. 
 
 If two surfaces intersect, the tangent to the common line at any 
 point will be the intersection of two planes, one tangent to each sur- 
 face at that point. 
 
114 DESCRIPTIVE GEOMETRY. 
 
 Note. If, as sometimes happens, the two planes whose inter- 
 section should determine the tangent coincide, this method obvi- 
 ously fails to give any result. In this event, the direction of the 
 tangent can be determined only by methods depending upon the 
 mathematical properties of the curve, of which this branch of sci- 
 ence does not treat. 
 
 176. Development of Surfaces. A prism is capable of rolling, in 
 a hobbling and imperfect manner, upon a plane; turning about 
 each edge in succession as an axis, so that one face after another is 
 brouglit into coincidence with the plane. If the number of edges 
 be increased, the hobbling will diminish, and when the number be- 
 comes infinite, it will disappear entirely : the rolling is perfect, the 
 change from one axis to another going on continuously, since the 
 edges are now consecutive elements of a cylinder. In this way all 
 the elements of the cylinder, without change of relative position, 
 can be brought into the plane, the area rolled over being equal to 
 that of the surface wdiich has rolled over it. 
 
 A pyramid, treated in like manner, ultimately becomes a cone, 
 which possesses the same property : indeed it is a sufiiciently 
 famihar fact that either of these surfaces can be unrolled into a 
 plane, without extension, compression, or distortion of any kind. 
 
 177. This process is called the development of the surface. 
 And the two illustrations above given show clearly upon what its 
 possibility depends : the plane of development is not only tangent 
 to the surface, but contains two of its consecutive elements, and 
 therefore the elementary surface included between them. These 
 elements must be rectilinear, since each in turn is an axis of rota- 
 tion, and an axis is a right line ; no two consecutive elements of a 
 warped surface lie in the same plane, therefore all single-curved 
 surfaces, and no others, are capable of development. 
 
 It is evident that if any surface can roll upon a plane, the plane 
 is equally capable of rolling upon the surface ; and this develop- 
 Mient by rolling on a fixed plane is, in a limited sense, the converse 
 of the generation of the surface by a moving plane. 
 
 But in order to execute this process, it is not enough to know 
 that the surface is developable : it is necessary also to know before- 
 hand the developed forms of one or more lines of the surface which 
 
DESCRIPTIVE GEOMETRY. 
 
 no 
 
 intersect the rectilinear elements, in order to determine tlie relative 
 positions of these elements on the plane of development. 
 
 INTERSECTIONS OF SURFACES BY PLANES. 
 
 178. Problem 1. To find the intersection of a right circular 
 cylirider with a jplane. 
 
 Construction. In Fig. 163, the axis of the cylinder is vertical, 
 its base lies in the horizontal plane, and the given plane tTt' is per- 
 pendicular to V. 
 
 The points in which the elements through the points cc, 1, 2, 
 etc., of the base pierce the plane are seen directly at u\ 1', 2'. 
 
 etc. , in the front view ; and u'^'z' is the vertical, and xZy is the 
 horizontal, projection of the required intersection. 
 
 To draw the tangent at any point P. This must lie in the tangent 
 plane, which is vertical, and has the horizontal trace po tangent to 
 the circumference of the base ; the tangent also lies in the cutting 
 plane, therefore its vertical projection ^'o' coincides with Tt'. 
 
116 DESCRIPTIVE GEOMETRY. 
 
 To show the true form of the curye. Make the supplementary 
 projection S, looking perpendicularly against the given plane. In 
 this view, tt will aj)pear as tf^ perpendicular to Tt\ u'z' will be 
 seen in its tnie length as u^z^ parallel to Tt' ^ and tlie chords verti- 
 cally projected at V ^ 2\ etc., will also be seen in their true lengths 
 as 1,1,, 2,2,, etc. ; these lengths are equal to those of 11, 22, etc., in 
 tlie top view ; this curve is an ellipse whose major axis is u'z, and 
 whose minor axis is equal to 33, the diameter of the given cylinder. 
 
 To draw the tangent to this curve in its own plane. The point 
 "whose horizontal projection is p and vertical projection p' is found 
 in the supplementary pro jectibn at^, : the tangent at this point 
 was seen in the top view to cut the horizontal trace at o, and x^ 
 produced cuts that trace in r. In the supplementary j)rojection 
 2^Ui produced cuts the horizontal trace in /•, ; now set off r^o^ = 7'o, 
 and draw^,6>, ; it will be the required tangent at the point j9,. 
 
 179. To develop the lower part of the cylinder. Suppose the 
 cylinder, formed of thin sheet metal, to be cut through the element 
 os'u\ and unrolled. The vertical elements will remain vertical, and 
 since the base is a continuous curve perpendicular to them all, it will 
 develop into a right line x^x^^ as shown in D, whose length is equal 
 to the circumference. If in the top view this circumference is 
 divided into equal parts at 1, 2, etc., then the developed base will 
 be similarly divided at 1^, 2,, etc. ; at each point of subdivision 
 there will be a vertical ordinate, representing an element; and 
 since the lengths of these elements remain unchanged, we sliall have 
 I3I, = 1^1', 2323 = 2'2', etc., and u^S^^u^ will be the development 
 of the curve of intersection. 
 
 To draw the tangent to the developed curve. The tangent to the 
 intersection at P contains two consecutive points of the curve, and 
 will contain them after development, and will therefore be tangent 
 to the developed curve. When the plane of development becomes 
 tangent to the cylinder at P, it will contain the element PN, the 
 tangent PO, and the subtangent iV6^ ; and these will remain un- 
 changed in magnitude and in relative position. In the develop- 
 ment D this element falls at p^n^ *, therefore, setting off n^o^ = no, 
 we have p./), as the tangent to the developed curve. 
 
 In drawing the tangents at u^ and 2^ in this manner, it is 
 
DESCRIPTIVE GEOMETRY. 117 
 
 obvious tliat tlie subtangents will be infinite ; the tangents at those 
 j^oints are therefore parallel to x^x^. 
 
 180. The Problem of the Shortest Path. Let it be required to 
 find the shortest path on the surface of the cylinder, between the 
 points x^ x\ and y, k' . In the development these points fall re- 
 spectively at a?2, A^2, and the least distance between them is the right 
 line which joins them. This line cuts the various elements at points 
 wliose distance from the base wall remain the same when the de- 
 veloped sheet is re-formed into a cylinder. These points, there- 
 fore, are projected back to tlie original positions of the elements, 
 as h^ to h\ g^ to g\ x^ to a?', etc., thus forming in the vertical pro- 
 jection the curve x'g'k'^ which represents the required shortest 
 path. In the development D it is seen that the distances of the 
 points of this curve from the base of the cylinder, measured on 
 equidistant elements, increase at a uniform rate; therefore the 
 curve itself is a helix, of which y'k' is half the pitch. In the ver- 
 tical projection the outer elements of the cylinder are tangent to 
 tlie curve at x and h' \ g' is a point of contrary flexure, and the 
 tangent at that point is parallel to xji^. 
 
 It is also to be noted that the development of the curve of inter- 
 section is identical with the projection of a helix wliose half-pitch 
 is equal to x^y.^ , lying on a cylinder of w^iicli the diameter is ti}^ ; 
 the tangent at S, is parallel to u'z' . 
 
 181. Practical Applications. Since the ellipse is perfectly sym- 
 metrical about both axes, it may be turned end for end ; thus the two 
 portions of a cylinder cut by a plane making an angle of 45° with 
 the axis may be joined together as shown at E, making what is 
 known as a "square elbow." By using other angles, the pieces 
 may be put together at different inclinations, as shown at F, which 
 represents a ' ' three-section elbow. ' ' In order to lay out the 
 sheet for the middle piece, cut it by a plane "tnni perpendicular 
 to the axis; this section wall develop into a right line as in D, 
 and the ordinates are set off each way from this to determine the 
 contour. 
 
 These problems of development are of direct use to w^orkers in 
 sheet metal. In theory it makes no difference along what element 
 the surface is cut ; but in practice, the rule dictated by plain com- 
 
118 
 
 DESCRIPTIVE GEOMETRY. 
 
 mon sense is to cut it so as to make the shortest seain^ unless there 
 is some good reason for doing otherwise. 
 
 182. Problem 2. To find the intersection of an oblique cylinder 
 hy an oblique jplane. 
 
 Construction. In Fig. 164, the plane and the cylinder are each 
 
 Fig. 164 
 
 inclined to both H and T. Make a supplementary projection S, 
 looking in the direction tT\ the horizontal plane appears as IP 11^ 
 perpendicular to tT^ and the given plane as Tj}^ , both planes being 
 seen edgewise and at their true inclination to each other. The 
 centre of the base is here projected at o, , and any point P of 
 the axis at j?j, the altitude ^^a?, being equal to j)'x '^ then the new 
 projections of the elements are parallel to o,p^. The extreme visi- 
 ble elements are determined by drawing at k and y in the horizontal 
 projection, tangents to the base, perpendicular to II' PP' ; drawing, 
 through k^ and y^ , parallels to o^p, , these hnes are cut by T^t^ at 
 
DESCRIPTIVE GEOMETRY. 119 
 
 ^1 and ^, , which obviously are the highest and lowest points of 
 the curve : e^ is projected back to e on the horizontal projection 
 of the element through K^ and thence upward to the vertical pro- 
 jection of the same element, e' being as far from AB as e^ is from 
 H' II' . The positions of z and z are determined in the same man- 
 ner, and bj repeating the process a point may be found on any ele- 
 ment at pleasure : it is particularly desirable to locate with accuracy 
 those jDoints which lie upon the limiting elements, as f in the hori- 
 zontal and c in the vertical projection, since they are points of 
 tangency. 
 
 183. The above operation might be defined as consisting i-n the 
 use of a system of auxiliary planes, parallel to the cylinder and to the 
 horizontal trace of the given plane ; these cut horizontal lines from 
 the plane and elements from the cylinder, whose intersections are 
 points in the required curve. Thus in the supplementary projec- 
 tion S, dj^^ may be regarded as representing a plane perpendicular 
 to the paper ; its horizontal trace is (^m, and its intersection wnth 
 tTt' is a line horizontally projected as/"/, piercing T in iV, and ver- 
 tically projected in n'f parallel to AB. The horizontal trace cuts 
 the base in d and m, and the horizontal projections of the elements 
 through these points determine/* and Z, vertically projected 2Xf' and 
 V ; and these are points in the required curve. 
 
 184. To draw a tangent to the curye of intersection. Let L be 
 the point at ^vhich the tangent is to be drawn. The plane tangent 
 to the cylinder at this point will contain the element whose hori- 
 zontal projection is lni\ and its horizontal trace, tangent to the 
 base at m, cuts Tt in r^ vertically projected at r in AB : therefore 
 RL is the required tangent, which if produced must pierce T in a 
 point of the vertical trace of the given plaiie. This point may be 
 determined by producing rl to cut AB in 5, which is its horizontal 
 projection, and s' in Tt' is its vertical projection. 
 
 To show the curve and its tangent in their own plane. Make a 
 second supplementary projection S', looking perpendicularly towards 
 Tj^, ; the horizontal trace will appear as t^^ perpendicular to T;t^ , 
 and the different points of the curve will lie upon projecting lines 
 drawn through <?, , ^, , etc. , parallel to t^^. In order to determine 
 their relative positions, draw in the horizontal projection any line 
 
120 DESCRIPTIVE GEOMETRY. 
 
 of the plane perpendicular to Tt^ as cm ; this will be seen in S' as 
 a^u^ perpendicular to tf^ , and the distances of the points of the 
 curve from this line will be the same as their distances from fn^ in 
 the horizontal projection : thus, a^e^ = ae^ h^2^ — hz^ "^Ji = "^^'f^ 
 etc. 
 
 To draw the tangent at l^ : set off on tji^ the distance ti^r^ — iii\ 
 and draw rj,^ ; it is the tangent required. 
 
 185. From (172) it is apparent that the vertical and horizontal 
 projections miglit have been constructed by means of the auxiliary 
 planes, without using the supplementary projection S ; and indeed 
 any other system of planes might have been substituted for the one 
 here employed. In either case, a rigid test of the accuracy of the 
 construction would be to revolve the plane and cylinder together 
 around a vertical axis until, as in Fig. 165, the horizontal trace 
 becomes perpendicular to AB ; the vertical projections of all points 
 in the curve should then fall in one straight line, Tt' . By the use 
 of the projection S this test is applied at the outset ; the construc- 
 tion of the projections on H and T is facilitated ; and the true form 
 of the curve is found by making the projection S', far more readily 
 than it can be in any other way. 
 
 186. Ill order to develop the cylinder, cut it by a plane per- 
 pendicular to the rectilinear elements ; this right section will de- 
 velop into a right line. On this line set off the rectified distances 
 between the elements, and perpendicular to it draw the elements 
 themselves; on these perpendiculars lay off the true distances, as 
 measured on the elements, from the right section to any line of the 
 surface whose developed form may be required. 
 
 To draw a tangent to such a developed curve. Draw first a tan- 
 gent to the curve in its original position, measure the angle be- 
 tween it and the element passing through the point of contact, and 
 draw a line making the same angle with that element at the cor- 
 responding point on the developed sheet. 
 
 187. Problem 3. To find the intersection of an ohlique cone 
 hy an ohlique plane. 
 
 Construction. This differs from that above explained, merely 
 in the respect that the elements are convergent instead of parallsl. 
 The point in which any element pierces the plane, is seen directly 
 
DESCRIPTIVE GEOMETRY. 
 
 121 
 
 in tlie vertical projection, Fig. 166, since the plane itself is per- 
 pendicular to V ; and the horizontal projection must of course lie 
 on the horizontal projection of the element. The true form of the 
 
 curve is found by constructing the supplementary projection S', 
 precisely as in Fig. 164; and the diagrams being lettered as nearly 
 as may be to correspond, no detailed explanation is necessary. 
 
 The determination of the tangent at Z, in the horizontal as well 
 as in tlie suppleuientary projection, is also made in the same man- 
 ner as in the case of the cylinder. 
 
 The development of such a cone cannot now be explained ; be- 
 cause no method has as yet been described of drawing a line upon 
 the surface which will develop into a curve of known form : this 
 matter will be discussed subsequently. 
 
 188. Problem 4. To find ilie intersection of a cone of revolu- 
 tion hy a plane. 
 
 Construction. In Fig. 167, the axis of the cone being vertical 
 and tTt' perpendicular to V, the vertical projections u\ e\ z\ etc.. 
 
122 
 
 DESCRIPTIVE GEOMETRY. 
 
 of the points in whicli the plane cuts the rectilinear elements are 
 seen by inspection of the front view, and may be thence projected 
 to v^ e, 2, etc. , in the top view. But in this top view the project- 
 ing lines would cut the elements very acutely in the neighborhood 
 of J^jP, thus making the determinations unreliable. In that region 
 another process is preferable : the element JF^I^ for instance pierces 
 tTf at a point whose vertical projection is ^' ; a horizontal plane 
 through that point will cut the cone in a circle whose radius is m', 
 and its intersection with the given plane will be a right line per- 
 
 
 p' 
 
 \ 
 
 /(\ / 
 
 1 
 
 
 
 *' / \ 
 
 \ 
 
 'A. /' 
 
 B 
 
 
 
 N^7 ' 
 
 \ 
 
 
 ^ 
 
 =y^'^ 
 
 
 eT^^' 1 
 
 
 
 
 / \ 
 
 
 "'/ 1\ 
 
 <^ 
 
 ' \ / 
 
 \ 
 
 
 
 Til I 1 
 
 / 
 
 
 
 a 
 
 
 \ 
 
 / 
 
 
 k' b- d' 
 
 /' 
 
 9' 
 
 T 
 
 
 
 / ^Ti 
 
 =j£- 
 
 \ \ 
 
 
 
 
 ly z^^p 
 
 
 
 
 r 
 
 r 
 
 
 
 d 
 
 B 
 
 ^ 
 
 
 
 
 
 V 
 
 
 
 
 
 
 
 t 
 
 T 
 
 Fig. 167 
 
 d, a 
 
 pendicular to V, seen in its true length in the top view as a chord 
 nn in the circle cut from the cone ; and this process may be re- 
 peated to determine other points. 
 
 189. Since the given plane in this case cuts all the elements, 
 the intersection is an ellipse, the true length of whose major axis is 
 the horizontal projection is also an ellipse, whose major axis 
 In order to determine the minor axis, bisect u'z' at c', 
 
 u'z'\ 
 
 IS uz. 
 
 through which point draw a horizontal plane hj ; this cuts the cone 
 
DESCRIPTIVE GEOMETRY. 123 
 
 in a circle whose radius is Ih, and c' is horizontally projected as a 
 chord CG of the circle described about j? with this radius. It is 
 perfectly legitimate, and practically it is preferable, thus to de- 
 termine the axes, and to construct the curve by any of the well- 
 known methods of drawing the ellipse, not only in the horizontal 
 projection, but in the supplementary projection S, w^here the inter- 
 section is seen in its true size. 
 
 The same modes of finding points in the curve may be used 
 when the plane cuts a parabola or an hyperbola from the cone ; 
 the use of auxiliary transverse sections being more particularly ap- 
 plicable wlien the angle at the vertex is acute. 
 
 190. To draw a tangent at any point of the intersection, as N". 
 The element through this point pierces H at F^ and/J?, tangent to 
 the circle of the base, is the horizontal trace of the tangent plane ; 
 tliis cuts Tt in c», and on is tlie horizontal projection of the tangent. 
 A plane through the axis, parallel to V, cuts Tt in r\ in the sup- 
 plementary projection S, set off r^o^ = ro, and o,n^ will be the tan- 
 gent to tlie curve in its own plane, as in Fig. 163. 
 
 If the given plane be parallel to the plane of any two elements, 
 tlie section will be an hyperbola ; the plane tangent to the cone 
 along either of these two elements, will intersect the cutting plane 
 in a line parallel to the element itself, and this line will be an 
 asymptote to the curve. If the cutting plane be parallel to one 
 element onlj^, it will be parallel to the plane tangent along that ele- 
 ment, and therefore wdll not intersect it at all ; w^hich is as it should 
 be, since the curve of intersection is a parabola, which has no 
 asymptote. 
 
 191. To develop this cone. Since every point in the base is 
 equidistant from the vertex, the base itself will develop into an 
 arc of a circle whose radius is the slant height ; and the length of 
 tliis arc will be equal to the circumference of the base. Thus in 
 the diagram D, the arc gjc^, described mth radius j?2^2 — i^V 5 ^^ 
 equal in length to the semi-circumference gfk. This semi-circum- 
 ference is bisected at /"; and bisecting gj{^^ at/*,, we have/!,^, as 
 the position of the element FP on the developed sheet : and in 
 like manner the position of any other element may be found. 
 
 To find the developed form of the intersection : Lay oft" on- the 
 
124: DESCRIPTIVE GEOMETRY. 
 
 elements in the development, the true distances from the vertex to 
 the pomts in which these element pierce the plane tTt'. Tims, 
 p^u^ = p'u\ and p^s^ = p'z' : the element J^J^' pierces the plane at 
 JV, but p'n^ is a foreshortened view of a line whose true length is 
 p'i, found by passing a horizontal plane through iT; therefore on 
 p^/^ make j)^7i.^ = p'i^ and n^ will lie on the i-equired curve ; also 
 p^e^ = p'e" ^ and so on. 
 
 To draw a tangent to the developed curve, at any point as n^. 
 In tlie horizontal projection the tangent at n is no^ the hjpothe- 
 nuse of a right-angled triangle yko, wiiich triangle lies in the tan- 
 gent plane, and will in the development be seen in its true size and 
 form. Therefore, drawj^/^, perpendicular to f^p^ and equal to fo\ 
 then o^n^ is the required tangent. The principle is, that the tan- 
 gent makes the same angle with the element after development as 
 before ; hence the tangents at u„ and z^ are perpendicular respec- 
 tively to g^p^ and \p^. 
 
 192. To tind the shortest path on the surface, between any two 
 points as X and Y. These points fall at x^ and y^ in the develop- 
 ment ; the right line x^y^ is the developed path ; it cuts pjb^ at w^ ; 
 set o^p'w" — Jp^^i draw through w" a horizontal line cutting ^'Z»' 
 in w' ^ the vertical projection of a point in the required curve : the 
 hoi izontal projection is best found by setting oif pw equal to aw" ; 
 and in like manner, other points may be determined. The highest 
 point M is found by drawing p^d.^ perpendicular to x^y^ , which it 
 cuts at m^ : make the 2iYQ,fd equal to the 'axcf^d^ , and project d to 
 d' ; then the position of M on PD is ascertained as above. 
 
 Note. The problem of the shortest path, on any surfaces which 
 can be developed either by unrolling, as in the case of single-curved 
 surfaces, or by unfolding, like prisms or pyramids, is always solved 
 in this manner, provided that a right line can be drawn on the de- 
 veloped sheet from one point to the other, which, however, is not 
 always possible. 
 
 193. It is clear that, as in the case of the cylinder, the ellipse 
 may be turned end for end, and the upper part of the cone joined 
 to the lower as shown in the small diagram E ; in going from the 
 old position to the new, the vertex describes a semicircle about an 
 axis perpendicular to the elliptical section at its centre. 
 
DESCKirTlVE GEOMETRY. 
 
 1^5 
 
 194. Problem 5. To develop the helical convolute. 
 
 Analysis. If through the tangent to the directrix at any point 
 a plane be passed containing the centre of curvature at that point, 
 tlie helix can be rolled upon that plane portion by portion ; and, 
 the radius of curvature being constant, it will thus develop into a 
 circle having that radius. The rectilinear elements of the surface 
 being tangents to the helix, will appear in the development as tan- 
 gents to this circle. 
 
 Constrnction. Let it be required to develop so much of the 
 lower nappe of the convolute shown in Fig. 168, as lies above H, 
 
 — ?, 
 
 EiG. 168 
 
 between the point D and the element GN. 
 P 
 
 ture of the helix is 
 
 cos CD 
 
 The radius of curva- 
 , in which p is the radius of the cylinder 
 
 on which the curve lies, and go is the ohliquity^ or angle made by 
 the tangent with a plane perpendicular to the axis. This may be 
 readily found graphically thus. The vertical projection of the tan- 
 gent at O is o'm\ which cuts d'f'^ the outline of the cylinder, at c ; 
 draw at o' a horizontal line, and at c a perpendicular to o'm : these 
 intersect at A.', and o'k is the radius of curvature required. 
 
126 DESCRIPTIVE GEOMETRY. 
 
 Then in the diagram D, draw about any centre a?, a circle with 
 tliis radius, and on it set off the arc d^o^ g^ equal to g'n' ; this will 
 be the development of the helical arc DOG. Since the horizontal 
 trace dm is the involute of the original helix, its develoj^ment will 
 be the involute d.r^n^ of the arc dfi^g^^ to which g^n^ , equal to g'n\ 
 will be tangent. 
 
 195. The Problem of the Shortest Path. Let it be required to 
 find the shortest path- on the surface, between the points 2> in H, 
 and F on GJS^. Tliese points fall in the development at d^ and ^, , 
 but they cannot be joined by a right line on the surface, since the 
 latter has no existence within the circle. The least distance is 
 .found by drawing from^j a tangent to the circle, and finding the 
 point of tangency ?/, ; the required line is, then, made nj) of the 
 circular dire d^u^ , and the right line u^p^. 
 
 The projections of this path on the original surface may be de- 
 termined as follows. In the horizontal projection bisect the quad- 
 rant ge at a, and draw al tangent to the circle ; it rej)resents an ele- 
 ment of which ajy^ is the developed position, where a^ bisects the 
 arc g^e^ ; this cuts j^j-w, at y,. The vertical projection of this point 
 must lie on a'l' ; in order to fix its altitude, set off n'y'' = l^y^ , and 
 draw through y" a horizontal line cutting a'V in y' . Project y" on 
 AB at 2/2 ; then n'y^ will be equal to Zy, the horizontal projection of 
 l^y^ in its original position. In like manner, the location of TF on 
 ER may be found, and by drawing intermediate elements any de- 
 sired number of points may be determined. The restored position 
 of 'Wi is best found by setting off ou^ the same fraction of the quad- 
 rant oe that o^u^ is of the arc o.e^., and the helical arc DOUiorm?^ 
 the first portion of the required shortest path. 
 
 The tangent to the helix at U is the element UZ^ which in the 
 development is i^j^j , a prolongation oi p,v,\ and it will be noted 
 that the shortest paths from P to any points on the portion DMZ 
 of the horizontal trace of the convolute, are equal. 
 
 196. Problem 6. To find the intersectionof a plane with any 
 surface of revolution. 
 
 Construction. Every transverse section is a circle, which in 
 general is the simplest line that can be drawn upon such a surface ; 
 and these circles are made use of precisely as in the case of the 
 
DESCRIPTIVE GEOMETRY. 
 
 127 
 
 cone. Tims in Fig. 169, 71' in the front view represents a chord 
 in the circle whose radius is cd^ which chord is seen in its true 
 length as nn in the top ^-iew and as n^n^ in the supplementary view 
 
 S ; in' represents a chord in the circle whose radius is eg^ seen in 
 its true length as rmn and m^m^ , and so on. 
 
 To draw a tangent to the curve of intersection, at any point as L. 
 Draw the tangent to the horizontal projection at l^ by the method 
 of Fig. 139. This tangent pierces the plane through the axis par- 
 allel to T, at the point P ; of which the supplementary j)rojection 
 is j9j : and Z,^i is the supplementary projection of the tangent. 
 
 Otherwise : Draw a plane tangent to the surface at L by the 
 method of Fig. 153; its intersection with the given plane is the 
 
128 DESCRIPTIVE GEOMETRY. 
 
 required tangent line, whose supplementary projection may be found 
 as in Fio^s. 163 and 167. 
 
 197. In Fig. ITO is giiown tlie "stub end" of a connecting- 
 rod. It is rectangular in section, and is joined to the cylindrical 
 neck by a surface of revolution whose contour is the circular arc 
 w'z' ^ described about the centre K\ w^e have, then, to find the inter- 
 sections of this surface by the two planes tt^ ss, parallel to the axis. 
 
 A transverse section at c' is a circle whose radius is cp' ; this 
 circle is seen in the end view to be cut by the plane tt at d, which 
 is projected back to d\ a point in the required curve ; and in like 
 manner other points may be determined. 
 
 The plane Sii is seen in the side view to cut the outline at Xy 
 which determines the vertex x^' of the curv^e seen in the top view. 
 A circle through e is seen in the end view to cut the vertical centre 
 line in g, which, projected back to the contour at g\ fixes the loca- 
 tion of a transverse section from wdiich by the preceding ])rocess 
 Avould be found the point e in the side view, corresponding to <?" in 
 the top view. A transverse section at any point as o', between x' 
 and g\ is a circle in which 7in, in the end view^, is a chord ; this 
 chord is also seen in its true length as 7i''n'^ in the top view. The 
 same circle is seen in the end view to be cut by the plane U at /% 
 which projected back to r' determines another point in the curve 
 of intersection in the side view. Other points may be found in a 
 similar manner. 
 
 198. Tangents to the Curves of Intersection. By applying the 
 method of (175), a tangent may be drawn to either of these curves 
 at any point, with one exception. 
 
 It is seen in the end view that the circle through w is tangent 
 to the plane tt, at the point y ; and the intersection of this plane 
 with the surface, as seen in the side view, consists of two sym- 
 metrical branches, which intersect at y\ The tangent plane at this 
 point coincides with the cutting plane ; consequently the tangent 
 line cannot be determined by the usual process. But a tangent at 
 this point to the lower branch may be constructed as follows : 
 
 The given centre of curvature at w' is the point A"; produce 
 Jvw' to /, bisect KI at O, and about describe a semicircle on 7i'/ 
 as a diameter, cutting tlie projection of the axis at J^Z On 7i/ set 
 
DESCKIPTIVE GEOMETRY. 
 
 129 
 
-30 ^ DESCRIPTIVE GEOMETRY. 
 
 o^y' G =^ y'F\ also about y' as a centre describe a circular arc 
 with radius y'w' : from G draw a tangent to this arc, then ylj<) 
 perpendicular to that line, is the required tangent at y' . 
 
 This construction is based upon the consideration that if the 
 contour, w'z\ be the osculating circle of an hyperl)ola of which w' 
 is the vertex and y' the centre, then y'L thus determined will be 
 one of tlie asymptotes. Had that hyperbola been the actual contour, 
 the section of the surface by the plane tt^ as will subsequently be 
 shown, would have been, not a curve, but the right line y'L itself. 
 
 INTERSECTIONS OF SINGLE- CURVED SURFACES. 
 
 199. In Fig. 171, draw any line otI' through the vertex o of 
 the cone ; it pierces the plane of the base, M^ at the point n. Draw 
 in the plane M any line nc through the ])oint n^ cutting the base at 
 € and d\ it is clear that the plane one cuts from the cone the two 
 elements oc, od. Draw in the plane J/^ through n^ a line tangent 
 to the base at / ; then the plane onl ig tangent to the cone along the 
 element ol. And the like will be true of any other cone whose 
 base is in the same plane J/, if its vertex also lies in the same line 
 on. It is now proposed to apply this in the solution of the follow- 
 iac: problem. 
 
 200. ir*KOBLEM 1. To find the intersection of two cones whose 
 bases are in the same jpland. 
 
 Analysis. Pass a series of planes through both vertices. These 
 will cut elements from each cone ; and the intersections of those 
 cut from one cone with those cut from the other will be points in 
 the required curve. 
 
 Construction. Since the bases are in the same plane, the two 
 cones may be so placed that this plane shall coincide with H, as in 
 Fig. 172. Join the vertices, P and 0^ by a right line, and pro- 
 duce it to pierce H in ]^. Then as in Fig. 171, nc is the horizon- 
 tal trace of a plane which contains the line T^iT, and this plane cuts 
 from the cone X two elements whose vertical projections are c'o' 
 and d'o' . The trace nc also cuts the base of the cone l"in e, ver- 
 tically projected at e in AB ; and e'p\ the vertical projection of the 
 element cut from that cone, intersects cVand d'o' in the points h' ^ 
 g'^ which therefore lie on the vertical projection of the required 
 
DESCRIPTIVE GEOMETRY. 131 
 
 curve : in a similar maimer any desired number of points may be 
 determined. 
 
 201. In the construction of such curves, there are certain criti 
 cal and limiting points which it is always desirable to locate. For 
 instance, o'f is the extreme visible element on the left, in the ver- 
 tical projection of the cone X. And, just as in Fig. Vl\^ fn is the 
 horizontal trace of a plane containing that element and the line 
 0N\ tliis plane cuts from the cone Y an element whose vertical 
 projection i^'jp' intersects o'f in a point s' : and at this point o'f is 
 tangent to the vertical projection of the curve. The points of con- 
 tact on the right-hand element of X, and on the left-hand element 
 of y, are found in a similar manner. 
 
 Draw nl tangent to the base of X\ it is the horizontal trace of 
 a plane tangent to that cone along the element whose vertical pro- 
 jection is I'o' ; this plane cuts from Y an element whose vertical 
 projection mp' is tangent to the vertical projection of the curve at 
 h' ^ its intersection with I'o' . And another tangent may be deter- 
 mined by means of another tangent plane, whose horizontal trace is 
 nx. 
 
 202. Attention has thus far been purposely confined to the 
 vertical projection of the curve. The horizontal projection, of 
 course, will be determined by the intersections of the horizontal 
 projections of the elements ; but the effect of introducing these, in 
 a diagram upon so small a scale, and necessarily involving so 
 many other lines, would have been simply bewildering, for which 
 reason they have been omitted. In constructing that projection, 
 it will be found advantageous to pass auxiliary planes through the 
 extreme visible elements of the cones in the top view also, in order 
 to locate the points at which the curve appears tangent to those 
 elements. 
 
 It is also to be noted, that the two projections of these curves 
 arc in a sense independent of each other. Thus, the bases, and 
 the vertical projections of the cones, remaining as they are, the 
 horizontal projections^, o, of the vertices may lie upon any oblique 
 line drawn through n. AYhence it appears that an infinite number 
 of pairs of cones may be constructed having the same bases and 
 altitudes, whose curves of intersection will all have the same verti- 
 
132 DESCRIPTIVE GEOMETRY. 
 
 cal projection. In a similar manner it may "be sliown that differ- 
 ent pairs of cones may intersect in curves wliicli have the same 
 horizontal projection. 
 
 203. Fig, 172 represents a case of complete interpenetration ; 
 the cone A" enters the cone Y on the left, passes bodily through it, 
 and emerges on the right, thus forming two distinct curves of in- 
 tersection. This is indicated in the horizontal projection by the 
 jircumstance that every one of the horizontal traces, inchiding the 
 two which are tangent to the base of A"^, cuts the base of Y in two 
 points. Each of the auxiliary planes, then, cuts two elements from 
 
 Y, although it may contain only one element of X; l)ut the latter 
 will intersect both the former; thus, I/o' cuts 7)i'j/ at k\ and it 
 also cuts ap' at q\ wdiich is the vertical projection of the point of 
 tangency between the second element and the other curve. 
 
 204. Now suppose the cone l^to be revolved around a verti- 
 cal line through P, in the direction indicated by the arrow. The 
 jDoints a and m, and consequently the points q' and /•', will ap- 
 ]3roach each other, until when the base becomes tangent to l/i as 
 shown by the dotted line, the auxiliary ])lane will be taiigerit to 
 both cones. The two curves will then coalesce, forming one con- 
 tinuous line, which, as shown in Fig. 173, will crusts itself at the 
 intersection oi LO andZP, the two elem^ents of contact, not being 
 tangent to either of them. 
 
 The interpenetration is still complete ; but the exact limit has 
 now been reached, and if the revolution of Y be carried any 
 farther, this will no longer be the case. The plane tangent to X 
 along the element Z will then pass outside of the cone Y\ no 
 part of either surface will be entirely buried within the other, and 
 the two will intersect, as shown in Fig. 174, in a continuous curve 
 which does not cross itself. In this diagram the cones are of equal 
 altitude ; the line joining the vertices is therefore parallel to H, and 
 for convenience it is here made parallel to V also, so that the traces 
 of the auxiliary planes are parallel to AB. The auxiliary plane 
 tangent to Y along the nearest element 77^, cuts from X the two 
 elements JO^ LO, which are tangent to the curve at the jjoints C, 
 D, in which they intersect IP. Sijnilarly, an auxiliary plane tan- 
 gent to A^ along its most remote element, will cut from l^two ele- 
 
DESCRIPTIVE GEOMETRY. 
 
 133 
 
 mcnts, both of wliicli are tangent to the curve ; and other points 
 are found as previously explained. 
 
 205. If the cones as ^iven hare not a common base, a very obvi- 
 ous expedient is to provide one, by passing a plane which cuts all 
 the rectilinear elements of both. This, however, is not absolutely 
 necessary nor always even desirable. The base of Y^ Fig. 175, be- 
 ing in the horizontal plane, it is clearly always possible so to place 
 the two cones that the base of X shall lie, as liere shown, in a plane 
 tTt\ perpendicular to V. Drawing OP, produce it to pierce H in 
 iV^, and the other plane in M. Revolve tTt' into Y ; then the base 
 of X will be seen in its true form and in its correct position rela- 
 tively to the point J/", w^hich falls at in" . All the traces of the 
 auxiliary planes upon tTt\ then, will pass through m\ and all 
 their horizontal traces through n. Draw "rri"/", tangent to the 
 base at e' ; this is the trace of a plane tangent to X along the ele • 
 ment EO. Set off Tf = Tf\ and draw nf\ this is the horizontal 
 trace of the same plane, which is now perceived to cut from l^tlie 
 two elements CP^ DP : these cut EO in two points of the curves 
 sought. 
 
134 
 
 DESCRIPTIVE GEOMETRY. 
 
 The construction, in short, is effected precisely as in the pre- 
 ceding cases : for instance, in order to find a point on IIP the ex- 
 treme right-hand element of Y^ draw nrg the horizontal trace of 
 an anxiharj plane containing that element, set off Tg" — Tg^ and 
 
 draw g^'m" cutting the base of X at s" ; project s'' to s\ then s'o' 
 is the vertical projection of an element, cutting r'^' in z\ which is 
 the vertical projection of the required point. 
 
 206. Manipulation in Construction — It is a very common mis- 
 take to suppose tliat in making such constructions as those of Figs. 
 172 and 175, time may be saved by drawing at once both projec- 
 tions of a great number of elements, and then selecting those 
 which intersect in the required points. ■ But the effect of such a 
 maze of lines is bewildering, and the attempt is more than likely 
 
DESCRIPTIVE GEOMETRY. 
 
 135 
 
 to result in error, confusion, and absolute loss of time. A much 
 safer and on many accounts better course is to find first a point 
 situated for instance on tlie nearest element of one of the surfaces, 
 and then to follow the curve around point by point in one direc- 
 tion, sketching it in lightly as each additional point is located. In 
 determining a point, do not draw in the trace of the auxiliary 
 plane, but simply mark where it cuts the bases ; and do not draw 
 the w^iole projection of either element, because ordinarily a short 
 bit of one will sufiice, upon which is marked the point in which the 
 other cuts it. 
 
 207. If the cones are so situated as to have two auxiliary tan- 
 gent planes in common, as in Fig. 176, then it is clear that all the 
 
 Fig. 176 
 
 elements of both surfaces will be cut, and that the line of intersec- 
 tion will cross itself at two points on opposite sides of the cones. 
 The line may be of double curvature; but if the bases of both cones 
 are conic sections, it will in general be composed of two plane curves, 
 
 which tlierefore are themselves conic sections. 
 
 An exceptional case is illustrated in Fig. 177, where J9, o, are 
 
136 
 
 DESCRIPTIVE GEOMETRY. 
 
 the vertices of two similar cones of revolution of equal altitude, 
 whose axes are vertical and therefore parallel. 
 
 The extreme elements jpr^ os^ intersect in A ; an auxiliary plane 
 through po cuts from one cone tlie elements pa^ pb^ and fi'om the 
 other, the elements oe^ of: pa cuts oe in n^ but it is parallel to 
 of^ and oe is parallel to ph. Consequently these lower nappes of 
 the cones can intersect in only one line, the hyperbola kng\ the 
 upper nappes obviously intersect in the opposite branch of the 
 same curve : and it is quite evident that a similar state of tilings 
 may -exist with cones of other forms. 
 
 In Fig. 178, the angles at the vertices are the same as in Fig. 
 177, and the vertices are equidistant from g^ the intersection of the 
 
 Fig. 178 
 
 Fig. 179 
 
 axes, whose inclination is such that the extreme elements j^^/, oii-.^ 
 ai*e not parallel. In this case pa not only cuts oe in 71^ but it cuts 
 
DESCRIPTIVE GEOMETRY. 137 
 
 of in h^ and oe cuts ph in m ; thus tlie lower nappes intersect in 
 the Jiyperbola ling^ and also in the ellipse seen edgewise as the 
 right line cd : the upper nappes intersect in the opposite branch of 
 the same hyperbola. 
 
 In Fig. 179 the same cones are show^n, the vertices equidistant 
 from i the intersection of the axes, but so inclined that jpy and ow 
 are parallel. In these circumstances the surfaces intersect in the 
 parabola Idg^ and also in the ellipse cd ; the upper nappes do not 
 meet at all. The same cones, it is clear, can be so placed as to in- 
 tersect each other in two ellipses, as do those shown in Fig. 176 : 
 in which case also the opposite nappes do not intersect. 
 
 208. In the case of any two given cones, the question whether 
 there will be any line of intersection wdth infinite branches may 
 be decided thus: Draw through the vertex of either a series of 
 lines parallel to the elements of the other; then in general this 
 third cone will either have only the vertex in common with the 
 first, or be tangent to it along one element, or cut it in two ele- 
 ments. 
 
 In the first case, no element of either of the given cones will 
 be parallel to any element of the other, and the intersection will 
 consist of one or two closed curves; which, w^hether of double 
 curvature or not, belong by analogy to the class of ellvptic inter- 
 sections. 
 
 In the second case, one element of each cone will be parallel to 
 one element of the other ; and there will be one line of intersection 
 consisting of an infinite branch with no asymptote: which there- 
 fore belongs to the class of parabolic intersections. 
 
 In the third case, two elements of each cone will be parallel to 
 elements of the other ; the surfaces will intersect in two infinite 
 branches with asymptotes, which belong by analogy to the class of 
 hyperholic intersections: and the asymptotes are determined by 
 drawing ])lanes tangent to the cones along the parallel elements; 
 these will intersect in the required lines. 
 
 In the last two cases there will be one closed curve of intersec- 
 tion, besides the infinite ones. But should the two cones have all 
 their elements respectively parallel, as in Fig. 177, there will be an 
 intersection composed of two infinite branches, and no other. The 
 
138 DESCllIPTlYE GEOMETRY. 
 
 cones in tliis case have necessarily two common tangent planes, and 
 the elements of contact are the ones to which the asymptotes will 
 be parallel. If the intersection is of single curvature, the asymp- 
 totes will be determined by the intersections of the plane of the 
 curve with the two tangent planes ; but if not, the asymptote can- 
 not be graphically determined : because the planes tangent to the 
 two cones along the parallel elements now coincide, giving no line 
 of intersection. 
 
 209. If one of the cones becomes a cylinder, the intersection may 
 be found in substantially the same manner. Draw a line, parallel 
 to the elements of the cylinder, through the vertex of the cone ; 
 then auxiliary planes through this line will cut from both surfaces 
 rectilinear elements, whose intersections will be i^oints in the re- 
 quired curve. Draw such planes tangent to each surface ; then, 
 as before, if both planes tangent to either cut the other, there will 
 be two separate curves ; if one plane tangent to each cut the other, 
 there will be one continuous curve ; if tliere be one common tan- 
 gent plane the curve wdll cross itself once ; and if two, it will cross 
 itself twice, and will consist of two conic sections when the base of 
 each surface is a conic section. 
 
 Infinite Intersections Since all the elements of the cylinder 
 
 are parallel to the auxiliary line through the vertex, they cannot 
 be parallel to any element of the cone, unless that line itself lies in 
 the conical surface ; if it does, there may be curves of intersection 
 with infinite branches. Draw a test plane tangent to the cone 
 along this auxiliary line ; it will either pass outside the cylinder, or 
 be tangent to it, or cut it in two elements. 
 
 In the first case, the intersection will be a single closed curve. 
 
 In the second case, it will in general consist of one infinite 
 branch with a single asymptote. 
 
 In the third case it will in general consist of two infinite 
 branches with two common asymptotes. 
 
 210. That these things are so, is very clearly shown by so 
 placing the surfaces that the auxiliary line is vertical. Thus in Fig. 
 180, which is a horizontal projection, let o represent this line; let 
 M be the horizontal trace of the cone, N that of the cylinder, TT 
 that of the test plane, and ox^ oy^ oz, those of auxiliary planes. 
 
DESCRIPTIVE GEOMETRY. 
 
 139 
 
 Each auxiliary plane contains the auxiliary line, which is parallel to 
 the elements of the cylinder ; but it also cuts from the cone another 
 element which is not parallel to them, and since M is in the hori- 
 zontal plane, while the vertex is above it, it is apparent that in this 
 
 Fig. 180 
 
 case every element of the cylinder must pierce the cone in a point 
 of the lower nappe. Had the cylinder been placed on the opposite 
 side of TT^ the intersection would have lain on the upper nappe. 
 Fig. 181 illustrates the second case, the test plane being tangent 
 
 Fig. 181 
 
 to N\ here it is evident that as the auxiliary plane approaches 
 coincidence with TT^ the elements cut from the cylinder approach 
 the line of contact a^ while the one .cut from the cone becoming 
 more nearly parallel to them, meets them at points more and more 
 remote, receding to infinity at the limit. Consequently the curve 
 of intersection lies wholly on the loM-er nappe, and the element a 
 of the cylinder is an asymptote to both sides of the single iniinite 
 branch, and lies between them, but nearer to one than to the other. 
 An exception to this occurs when, as in Fig. 182, the two sur- 
 faces are tangent along an element. Let oy be the trace of any 
 
140 
 
 DESCllIPTLVE GEOMETRY. 
 
 auxiliary plane, cutting Jf in r and JV in d: then ot^ is tlie projec- 
 tion ox that part of an element of the cone included between the 
 vertex and the base, and od the projection of that part included 
 between the vertex and the line of intersection. The length of the 
 former being always finite, that of the latter must also be finite ; 
 therefore the surfaces intersect in a single closed curve. 
 
 211. The third case is illustrated in Fig. 183; by comparison 
 with Fig. 181, it is obvious that the part of the cylinder mcluded 
 between TT'and the tangent auxihary plane 02, will intersect the 
 cone in an infinite branch on the lower nappe, to which the two 
 
 Fig. 183 
 
 Fig. 184; 
 
 elements a and h of the cylinder will be asymptotes. Also that the 
 portion on the opposite side of the test plane will give another 
 infinite branch on the upper nappe, liaving as asymptotes the same 
 elements a imd b. 
 
 There will be an exception to this also, when the two surfaces 
 intersect in the auxiliary line, as in Fig. 184. They will also inter- 
 sect in one continuous curve, w^iich, crossing the common element 
 at the vertex, extends both ways to infinity, and is asymptotic in 
 each direction to the other element h cut from the cylinder by the 
 test plane, 
 
 212. In the preceding arguments relating to tlie nature and the 
 number of the curves of intersection, it has been assumed that, as 
 is usual in practical cases, the cones and cylinders have closed 
 mirves as bases, and are externally convex throughout. Almost 
 endless variations miglit result from constructing one or both of the 
 
DESCRIPTIVE GEOMETRY. 141 
 
 surfaces with spiral, sinuous, or infinite curves for bases; these 
 liowever it is not proposed to consider, since they involve no prm- 
 ciple which has not already been explained. 
 
 213. Intersection of Two Cylinders. The same general method 
 is still applicable : Draw a plane parallel to the elements of each 
 cylinder ; then auxiliary planes parallel to this plane will cut from 
 both surfaces rectilinear elements, and their intersections will be 
 points in the required curve. By drawing two such planes tangent 
 to each surface, the questions as to whether there will be one or two 
 curves, one or two crossings, etc. , are decided exactly as in (208) ; 
 and w4ien there are two common tangent planes, the intersection 
 will consist of two conic sections, if the bases themselves are conic 
 sections. 
 
 As to infinite intersections, it is perfectly obvious that, if the 
 bases are closed curves, there can be none, unless the elements of 
 one cylinder are parallel to those of the other ; w^hen they will 
 intersect in either two or four elements, if at all. But any infinite 
 curve can be made the common base, and therefore the line of 
 intersection, of two cylinders. 
 
 214. Other Methods of Operationc It appears then from the 
 foregoing, that if two cones, two cylinders, or a cone and a cylinder, 
 intersect each other, it is edwajs possible to employ a system of aux- 
 iliary planes which cut rectilinear elements from each surface. But 
 it must not be inferred that this is the only or always even the most 
 eligible expedient. In probably the greater number of practical 
 cases, these surfaces have circular bases ; and when this is so, planes 
 which cut circles from each, or right lines from one and circles 
 from the other, can often be employed to great advantage. 
 
 Thus in Fig. 185, the horizontal plane i'f cuts from the cone 
 a circle, which in the horizontal projection is seen to pierce the 
 cylinder at e and n, which are vertically projected to e, n' ; and 
 any number of points may be found in like manner. 
 
 Draw ox^ oy^ tangent to the base of the cylinder; describe 
 through r and <?, the points of contact, a circle about o, cutting ow 
 in 6?, whose vertical projection is d' '^ then/'', c\ must lie on a 
 horizontal plane thrpugh d'^ and at those points the curve is tangent 
 to o'x\ o'y' . Draw through a^ the centre of the base of the cyl- 
 
142 
 
 DESCRIPTIVE GEOMETRY. 
 
 inder, the right line ov ; it cuts tlie base in s and Ic, whose vertical 
 projections, found in the same manner as r' and c\ are the highest 
 and lowest points of the intersection. 
 
 FiGf. 185 
 
 215. Only the front half of the cone is represented in Fig. 185 ; 
 and in Fig. 186 is given the development of that half, showing the 
 form of the hole which must be cut in tlie sheet for the insertion 
 of the cylinder. The base develops into the arc of a circle whose 
 radius o^z^^ is equal to o'z\ the slant height of the cone, and ^he 
 
DESCRIPTIVE GEOMETRY. 143 
 
 length of tlie arc z^ii^w^ is equal to that of the seniicircumference 
 zuii\ Make the arc it{i\ = arc tcv, set off the arcs ^',a?, , v,y^, equal 
 to the arc vx, and draw o.y,, o^v ,, o,x^. Describe about 6>, an arc 
 with radius o, J, , equal to o'd\ thus locating the points of tangency 
 r^ , c\ ; describe about o^ an arc with radius o^fi = o'/'', cutting o^v^ 
 in m, , set off the arcs m^n^ , m,e^ , each equal to me, then e^ , n^ , 
 lie on the developed curve : and other points may be found in a 
 similar manner. The vertices, s^ and A;, , are determined by setting 
 off on o^v^ the true distances of the points S and ^from the vertex 
 of the cone, which are respectively equal to o^t\ o'V . 
 
 • 216. The construction of Fig. 187 may be explained thus: 
 The point in which any line on either the horizontal or the incliae(? 
 cylinder pierces the vertical one, is seen directly in the top view, 
 Thus, the nearest element lik of the horizontal cylinder cuts the cir- 
 cumference at ^, which is projected vertically to h' in the front 
 view : any point c" on the circumference in the end view repre- 
 sents an element, seen as c'd' in the front view, and as cd in the top 
 view, where the distance xc from the centre line ie equal to x"g" ; 
 and d' is vertically over d. Similarly, the nearest element of the 
 inchned cylinder is represented by m'o' in the front view and by 
 "iiio in tlie top view, and the altitude of the point in which it 
 pierces the upright cylinder is determined by projecting o up to o' . 
 A supplementary view looking in the direction of the arrow shows 
 the base of the inclined cylinder in its true form, and any point n^ 
 upon its circumference represents an element of which the front 
 view is nr' and the top view is nr, located by making wn = w^n^ ; 
 and t' is vertically over r : any number of points in the curves may 
 he found in the same manner. In this way the determination of 
 the intersections might be made clear to one totally unfamiliar with 
 the stage machinery of Descriptive Geometry : though it is very 
 evident that the operations are actually equivalent to the use of a 
 sci-ies of horizontal planes in the first case and a series of vertical 
 ones in the second. 
 
 217. Just such cases as these last are the ones most often met 
 with in practice ; and accordingly, they are the ones most seldom 
 illustrated in theoretical treatises on the principles involved in them. 
 In the appHcation to sheet-metal work, the development is of im^ 
 
144 
 
 DESCRIPTIVE GEOMETRY. 
 
 portance; and that of tlie iipriglit cylinder is given in Fig. 188, 
 Supposing it to be cut vertically along tlie most remote element u, 
 and unrolled to right and left, the surface will form a sheet of a 
 breadth equal to the height of the cylinder, and a length equal to 
 its circumference. The intersection with the horizontal cylmder 
 
 Fig. 188 
 
 will develop into a curve symmetrical about a horizontal line whose 
 distance from the lower edge is equal to that of the axis from AB 
 in Fig. 187. Rectify the arcs ul^ Ik^ in the top view, in the de- 
 velopment set off the distances ul^ Ik^ equal to them, and subdivide 
 the latter into parts respectively equal to the partial arcs M, dfy 
 
DESCllTPTIVE GEOMETRY. 
 
 145 
 
 etc. ; at the points of subdivision erect vertical ordinates equal to 
 the distances of d' ^ f\ etc. , above the centre line of the horizontal 
 cylinder in the front view ; the required curve passes througli the 
 extremities of these ordinates. The intersection witli the inclined 
 cylinder will develop into a curve symmetrical about 22^ the posi- 
 tion of tlie right-hand element of the upright cylinder on the un- 
 rolled slieet; the altitudes of the vertices, <^', J, are taken directly 
 from the front view : on zz mark also the points 1, 2, 3, at altitudes 
 equal to the distances of s\ 0% /, above AB, and at these points 
 draw horizontal ordinates 1^, 2(9, 3^% respectively equal in length 
 to the rectified arcs sz^ oz^ rz^ in the top view : any desired mimber 
 of points may be found in a similar manner. 
 
 INTERSECTIONS OF DOCBLE-CURVED SURFACES. 
 
 218. Problem 1. To find the intersection of two surfaces of 
 revolutioru whose axes are in the same plane. 
 
 Analysis. If the axes intersect, take the point of intersection 
 as the common centre of a series of auxiliary spheres. Each spliere 
 will cut a circle from each of the given surfaces (137) ; the circum- 
 ferences of these circles will cut each ather in two points, which lie 
 upon the required curve. If the axes are parallel, the spheres be- 
 come planes perpendicular to the axes. 
 
 Construction. In the side view at the left, Fig. 189, the plane 
 of the axes, which is j^arallel to the paper, contains the visible coii- 
 
 Fia. 189 
 
 tours of the given surfaces and of the spheres whose common 
 
146 DESCRIPTIVE GEOMETRY. 
 
 centre is w ; tlie intersections of tlie former, at /' and s, give at once 
 two points of the required curve. A spliere tangent to one surface 
 around the circle oy^ cuts the otlier in the circle ^a?, and oy cuts^a? 
 in n ; another and larger sphere cuts the first surface in circles 
 through d and g^ and the second in a circle through c^ and the in- 
 tersections of these ' circles at e and h also lie upon the required 
 curve snr : any desired number of points may be determined in like 
 manner. 
 
 In the end view, the points s, r, are projected directly to s^, /\, 
 on tlie vertical centre line. The point 7i in the side view representis 
 a chord in the circle oy, whos6 circumference being seen in its true 
 form and size in the end view, the points n^ , n^., are found by pro- 
 jecting n across to that circumference; and ^,, /^,, are located in a 
 similar manner. 
 
 Note. This method is equally applicable to the case of single- 
 curved surfaces of revolution whose axes intei'sect. Its application 
 in the extreme case where the intersection is infinitely remote and 
 the spheres become planes, has already been illustrated in finding 
 the intersection of a cone with a cylinder. Fig. 185. 
 
 219. If the axes lie in different planes the determination of the 
 intersection of the surfaces is in general much more laborious. In 
 most cases, it would probably be advisable to use a system of auxiliary 
 planes perpendicular to one of the axes, thus cutting circles from 
 the surface to which that axis belongs ; but it still remains to deter- 
 mine the form of the line cut from the other surface by each in- 
 dividual plane. 
 
 220. Problem 2. To find the intersection of any ohlique cone 
 with a sphere. 
 
 Analysis. Pass a series of auxiliary planes through the vertex ; 
 each will cut a circle from the spliere and two elements from the 
 cone ; and the intersections of these elements with the circumference 
 of that circle will be points in the required curve. 
 
 Construction. In Fig. 190, 6'' is the centre of the sphere, i? 
 the vertex of the cone, whose base for the sake of convenience is 
 placed in H ; and the auxiliary planes are vertical. Let jpp, drawn 
 at pleasure through o, be the horizontal trace of one of these planes, 
 cutting the base of the cone in d and r, and the contour of the 
 
DESCRIPTIVE GEOMETRY. 
 
 147 
 
 sphere in/* and K\ tlieuyA istlie horizontal projection of the ckcle 
 cut from the sphere, and its middle point e is that of its centre. 
 Revolve this plane about ^ into H; o goes to o'\ oo" being equal 
 to the altitude of the cone ; and do'\ ro'\ are the revolved posi- 
 tions of the elements cut from the cone. Also, e goes to e'\ the 
 distance ee" bemg equal to that of C from H ; about e" with radius 
 
 ef describe an arc cutting do" ^ ro , in g 
 
 u 
 
 which will be the 
 
 -N m 
 
 revolved positions of the points in which the elements pierce 
 the sphere. In the counter-revolution, g" goes to g on pp, whence 
 it is vertically projected to g' on d'o\ tims determining a point G 
 on tiie required curve ; in like manner the projections of the point 
 whose revolved position is u" may be found, the construction being 
 omitted to avoid confusion in the diagram. 
 
 Otherwise: Revolve the auxiliary plane about the horizontal 
 
148 DESCRIPTIVE GEOMETRY. 
 
 projecting line of O nntil it is parallel to T ; this line is o'x in tlie 
 vertical projection, and setting off x'i\ = or^ and xd^ = od^ the two 
 elements appear as o'r^ , o'd^\ on the horizontal through c' set off 
 ye^ = oe^ then e^ is the revolved position of the centre about which 
 an arc with radius ^is to be described, cutting the elements in u^ 
 and (J,. 
 
 221. By repeating the above process, any number of points 
 may be found and the curve fully determined. The points at which 
 the projections of this curve are tangent to the extreme visible ele- 
 ments of the cone, as for instance n' on o'm ^ are determined as in 
 previous cases, by passing auxiliary planes through those elements. 
 But the points at which the vertical projection is tangent to the 
 contour of the sphere cannot be located by any direct means, since 
 there is no way of ascertaining which element of the cone will 
 pierce the sphere in the great circle cut from it by the plane II 
 parallel to V. These points may, however, be determined indi- 
 rectly, thus : the horizontal projection of the curve cuts II at s and 
 w ; these are the horizontal projections of those points of penetra- 
 tion, and the vertical projections s' and w' must lie on the contour 
 of tlie sphere. 
 
 222. In this instance, the vertex of the cone lies within the 
 body of the sphere ; wdien it lies outside the surface there may be 
 two curves of intersection, but the construction is the same for both. 
 If any plane tangent to the cone passes outside the sphere, tlie in- 
 terpenetration will be partial, and the surfaces will intersect in one 
 continuous curve ; if the cone and sphere have one common tan- 
 gent plane, this curve will cross itself once ; if they have two it 
 will cross itself twice. In order to ascertain wdietlier either con- 
 dition exists, draw a line from the centre of the sphere to the vertex 
 of the cone, and make it the axis of a cone of revolution with the 
 same vertex and tangent to the sphere ; if this test cone is tangent 
 to" the given one along one element, there will be one connnon tan- 
 gent plane ; and if the cones are tangent along two elements, there 
 will be two of them. If the cones cut each other in two elements, 
 there will be one closed curve of intersection ; if in four elements, 
 the given cone will intersect the sphere in two distinct curves. It 
 is here assumed, as in (212), that the transverse section of the 
 
DESCKIPTIVE GEOMETRY. 149 
 
 given cone by a plane perpendicular to the axis of tlie test cone, is 
 a closed curve and externally convex throughout. 
 
 The intersection of a sphere with a cylinder, obviously, is to be 
 deter mined by means of a system of planes cutting elements from 
 the latter and circles from the former, the process of construction 
 being substantially the same as in the case of the cone. 
 
 223. PiiOBLEM 3. To develop any ohlique cone. 
 
 Analysis. Intersect the cone by a sphere whose centre is at 
 the vertex. The curve of intersection vrill develop into an arc of 
 a circle whose radius is equal to that of the sphere. On this circle 
 lay oil' distances equal to the rectiiied arcs of the curve of intersec- 
 tion intercepted between rectilinear elements of the cone ; and on 
 the radii drawn through the points thus determined, set off the true 
 lengths of the corresponding elements. 
 
 Construction. The intersection with the sphere, in Fig. 191, is 
 constnicted as in Fig. 190. Since it is a double curved line, its 
 true length is not seen in either projection ; but it can be ascer- 
 tained by developing the horizontal projecting cylinder, as shown in 
 Dj. Here the elements of the cylinder appear in their true length 
 and at their true distances from each other ; thus ^,a?, is equal to 
 e'ic^ (j^y^ to (j'y^ and xij^ to the arc eg in the horizontal projection ; 
 the positions of other elements being determined in like manner, 
 the base develops into a -right line w^w^., and the double curve line 
 into a single curved one nfijiy_ . In the development of the cone, 
 shown in D^ , an indefinite circular arc is described about any centre 
 6>2 with a radius equal to that of the sphere ; on this, set oif an arc 
 e^(j^ equal to the arc e^g^ in D^, then g^in^ = ^,7?z, , and so on : the 
 points thus determined fix the positions of the elements of the cone 
 on tlie developed sheet, and on the radii drawn through them the 
 true lengths of these elements are set off, — as o.{Jl^ =^ OD^ o^f^ = 
 0I'\ etc. In a similar manner the developed form of any other 
 curve on the surface may be determined. 
 
 To draw a tangent to the developed base at any point, as f^. The 
 line o.f^ is the developed position of the element OF^ and the tan- 
 gent to the base of the cone at F is FQ^ in the horizontal plane. 
 Find the true angle included between OF and FQ^ and make the 
 angle o^f^q^ equal to it; theny!,^, is the required tangent. And the 
 
150 
 
 DESCRIPTIVE GEOMETRY. 
 
 tangent to the development of any otlier curve on the surface may 
 be drawn in the same way. 
 
 Fig. 191 
 
 224. Conditions of Symmetry. Tlie cone represented in Fig. 
 191, having a circular base, is symmetrically divided by tlie verti- 
 cal plane through tlie axis, which also cuts from it the longest ele- 
 ment OD and the shortest one 0C\ and the points E^ N^ in which 
 these elements pierce the sphere, are respectively the highest and 
 the lowest points of the curve of intersection. Again, any two ele- 
 ments OF^ OK^ equidistant from OD^ will be of equal length, 
 make equal angles with the plane of the base, and therefore pierce 
 the sphere in points of equal altitude ; moreover, since they are 
 equally foreshortened in the horizontal projection, og is equal to ol, 
 and eg equal to el- 
 
DESOKIPTIVE GEOMETRY. 151 
 
 From these considerations it follows that the horizontal projec- 
 tion of the curve of intersection is symmetrical with reference to 
 od ; that the development of the projecting cylinder is symmetrical 
 about e^x^ ; and that the development of the cone is symmetrical 
 about o^d^ . And it is evident that the same will hold true in re- 
 gard to any cone which is symmetrically divided by a plane through 
 the vertex perpendicular to the plane of the base. 
 
 225. Practical Suggestions. The above deductions are of im- 
 portance in the practical applications of this problem. For, a 
 curve which is symmetrical mth respect to a given line can always 
 be constructed more easily and more accurately than one which is 
 not : and the process of re-forming the original cone from the de- 
 veloped sheet is likewise much facilitated if the latter be symmetri- 
 cal. 
 
 Such work as this should in practice always be laid out upon as 
 large a scale as may be ; when this is done, the processes of rectifying 
 the arcs of the horizontal projection of the curve of intersection, in 
 order to develop the projecting cylinder, and of transferring the 
 length of the developed curve to the circular arc in developing the 
 cone, may be executed with sufficient accuracy by stepping them 
 of[ with the spacing dividers, the points of which are set so close 
 together that the difference between the chord and the arc shall be 
 practically inappreciable. 
 
152 DESCRIPTIVE GEOMETRY. 
 
 CHAPTER YI. 
 
 Of Warped Surfaces. 
 
 The Hyperbolic Paraboloid ; Its Vertex, Axis, Principal Diametric Planes and 
 Gorge Lines. The Conoid. — The Hyperboloid of Revolution.— The Ellipti- 
 cal Hyperboloid, and its Analogy to tlie Hyperbolic Paraboloid. The 
 Helicoid ; of Uniform and of Varying Pitch. The Cylindroid. — The Cow's 
 Horn. — Warped Surfaces of General Forms. — Planes Tangent to Warped 
 Surfaces. Warped Surfaces Tangent to Each Other. Interse3tions of 
 Warped Surfaces. 
 
 THE HYPERBOIJC PARABOLOID. 
 
 226. The Hyperbolic Paraboloid is a warped surface, with a 
 plane directer, and two rectilinear directrices wliicli lie in different 
 planes. It takes its name from the fact that its curyed sections by 
 planes are either hyperbolas or parabolas. 
 
 Any plane parallel to the plane directer will cut each directrix 
 in a point; and the right line joining these two points will be an, 
 element of the surface. 
 
 If a series of such parallel planes be drawn, they will divide the 
 two directrices proportionally. And conversely : If any two right 
 lines not in the same plane be divided into proportional parts, the 
 right lines joining the corresponding points of division will lie in 
 parallel planes, and be elements of a hyperbolic paraboloid. 
 
 If through any point in space two lines be drawn, respectively 
 parallel to any two of these elements, the plane of those two lines 
 will be parallel to the plane directer, and may be taken for it. 
 
 If in this j^lane directer any right line be drawn, the element 
 parallel to that line may be found thus : Through any point of 
 either directrix draw a line parallel to the given line ; that direc- 
 trix and this parallel will determine a plane cutting the other direc- 
 trix in a point : through that point draw a parallel to the given 
 line, and it will be the element required. 
 
DESCRIPTIVE GEOMETRY. 
 
 153 
 
 227. The Hyperbolic Paraboloid is Doubly Buled; that is to 
 say, it lias two sets of rectilinear elements, and consequently two 
 plane directers. 
 
 In Fig. 192, let XX be the plane directer, A O smd jBD the 
 directrices, CD and AB two elements, the latter lying in XX. 
 
 Fig. 193 
 
 Through BD draw a plane T^T^ parallel to A O, cutting XX in 
 
 TFTF: a plane parallel to this through AC cuts XX in AZ parallel 
 
 to WW, Draw Z>^ perpendicular to WW, also 6^^ perpendicular 
 
 to AZ', then ^^will be parallel and equal to CD. 
 
 Through any point ^ on ^ 6^ draw £[F parallel to the plane 
 
 A 7^ T^Tf 
 XX, and cutting BD in F', then -^ = -— (226), and ^i^ will 
 
 be another element of the surface. 
 
154 DESCKIPTIYE GEOMETRY. 
 
 Draw ^6^ perpendicular to J. Z and i^/ perpendicular to WW, 
 then GI will be parallel and equal to £^2^. We have also 
 
 EG - GH' ^^^ FD ~ IK ' 
 
 therefore j^fj^ = y^, consequently JTIT and GI cut AB in the 
 
 same point B. 
 
 Now draw any plane parallel to YY, cutting AB in Z; its 
 intersection with JC^ will be parallel to AZ, and will cut GI in 
 
 Tvr zxz^- ir • • ^^ ^^ ^^' rpi- 1 
 JV, HK m jM, giving -y^ = ttt/ = ~Ttp' ^^^^^ plane also cuts^ 
 
 the parallelogram GF in iV^O parallel and equal to GE^ and the 
 parallelogram HD in MP parallel and equal to IIC. Therefore 
 
 LN NO ^., , x^7. . . 1 T 
 
 we have ■y^J^J^ = jifl* '^ which proves that LOF is a riglit line, 
 
 intersecting the three elements AB, FF, CD. Also, the elements- 
 
 CD, EF, AB, are proportionally divided by the planes parallel 
 
 ,. CP EG AL 
 to YY, so that -p-^ = -Q^ = z^- 
 
 If then AB and CD be taken as directrices, and YY as a 
 plane directer, the resulting surface will be identical with that 
 having A C and BD as directrices and JlX^ for a plane directer. 
 The rectilinear elements CD, EF, etc., which are parallel to XX., 
 are called elements of the first generation ; those of the other set, as- 
 AC, LP, etc., which are parallel to YY, are called elements of 
 tlie second generation. And from the preceding it appears tliat 
 every element of either generation intersects all those of the other. 
 
 228. Vertex and Axis. It is clear that a plane through R par- 
 allel to YY, in Fig. 192, will determine an element FT of the 
 second generation, parallel to DK, which is perpendicular to WW. 
 In a similar manner (226) an element of the first generation may 
 be found which shall be parallel to any line drawn in the plane XX 
 and perpendicular to WW. Thus in Fig. 193, having found RT 
 as above, draw AF perpendicular to TFTF, pass through C a plane 
 parallel to XX, cutting J^I^in DQ, and draw CE, perpendicular 
 to DQ and consequently equal and j)arallel to AF. Then EFy 
 
DESCRIPTIVE GEOMETRY. 
 
 155 
 
 equal and parallel to A C\ cuts BD in iT, and N'M parallel to AF 
 is the required element. 
 
 The elements BT^ MN^ therefore, lie in a plane, which is per- 
 pendicular to TFTF, the intersection of the two plane directers, and 
 cuts it at /. The point in which these elements cut each other 
 is called tlie vertex of the surface ; and the line TJTI^ drawn through 
 parallel to TFTF, is called the axis. The plane determined by 
 RT and MN^ again, is tangent to the surface at the vertex 
 (142). _ ■ 
 
 Obviously, the angle NOR between these two elements is 
 equal to the angle NIR between the two plane directers : if this 
 angle is a right angle, the surface is called a right, or rectangular, 
 hyperbolic paraboloid ; if not, the surface is called oblique. 
 
 229. In the absence of a model, the pictorial representation in 
 Fig. 194 may aid in forming a conception of this surface. NS in 
 the horizontal plane, and MT in the vertical plane, are divided 
 into the same number of equal parts, and the right lines joining the 
 
 Fig. 194 
 
 corresponding points of division are elements of one generation; 
 those of the other generation are determined by like treatment of 
 >&l!fin V and NT\u H. 
 
 Draw Tt' parallel to J/iS', then tTt' is the plane directer of the 
 first generation ; in like manner Ss' parallel to MT determines s8s\ 
 
156 
 
 DESCIlirTIVE GEOMETRY. 
 
 the plane directer of tlie second generation. The vertical traces of 
 these two planes intersect at G on the vertical plane, their hori- 
 zontal traces at iV^on the horizontal plane, thus determining WW^ 
 the intersection of the plane directers. 
 
 Since N8 is an element of one generation, and NT one of the 
 other, the horizontal plane is tangent to the surface at N (142) ; 
 and for a like reason the vertical plane is tangent to it at M. This 
 is most clearly shown in Fig. 195, which represents the same sur- 
 face in projection on the profile plane. 
 
 230. The Principal Diametric Planes. Every hyperbolic parabo- 
 loid, whether rectangular or oblique, is symmetrically divided by 
 two planes perpendicular to each other, passing through the axis, 
 and bisecting the angles formed by the two elements which deter- 
 mine the tangent plane at the vertex. 
 
 Fig. 196 represents in profile the system of Fig. 193, for con- 
 
 FiG. 198 
 
 venience so placed that the plane RN coincides with the paper, and 
 that the plane directers are equally inclined to the horizontal plane 
 HH, wliich contains their intersection TFTF, represented by the 
 
DESCRIPTIVE GEOMETRY. 157 
 
 point /: thus the axis is represented by the point (9, and the ele- 
 ments of the two generations by the Hnes respectively parallel to 
 XX and YT. 
 
 Set off on OB the points a, h, and on 0^ the points c, d, all 
 equidistant from O ; through a and h draw parallels to XX, and 
 through c and d, parallels to YY: the intersections of these paral- 
 lels determine the rhombus egfh. Through e and f draw the plane 
 ZZ, and through g and k the plane i^P, both perpendicular to tlie 
 paper; these planes are perpendicular to eacli other, intersect in the 
 axis 0, and bisect the angles formed by Oli and OJV^. 
 
 Now, the sides of the rhombus represent four elements of the 
 surface; and since a, h, c, d, lie in the plane of the paper, the 
 intersections e, y, g, h, do not, because those elements are inclined 
 to that plane. Suppose e to lie at any distance behind it; then 
 because elj, hh, hd, d/, etc., are all equal, the point /* must also lie 
 at tlie same distance beliind it ; and the points g, A, at exactly the 
 same distance in front of it. 
 
 Set off hk on IcO equal to cs on JV^O, and complete the rhom- 
 bus mpnq, then by the same reasoning it appears that m and n are 
 at the same distance behind the plane of the paper, and p, q, at an 
 equal distance in front of it. Produce the sides of the lirst rhombus 
 to cut those of the otlier, then r, I, y, w^ are at equal distances 
 beliind the plane, while lo, t, v, a?, are at equal distances in front 
 of it. Then rl, sk, ut, etc. , are chords of the surface parallel to 
 the planes PP and PX; and the plane ZZ bisects them all. So 
 also all chords parallel to ZZ and 7?X, as for example tv, ly, mn, 
 etc., are bisected by PP. Consequently, as stated, the surface is 
 symmetrical with respect to both planes; which are called the 
 principal diametric planes. 
 
 231. Tlie Gor^e Lines. There may be an indefinite number of 
 hyperbolic paraboloids having the same plane directors XX, YY, 
 and the profile, Fig. 196, represents them all. JS'ow, the point e 
 may be at any distance beyond the plane of the paper ; but if that 
 distance be assigned, the inclination of the directrices to that plane 
 is thereby fixed, and the surface fully determined. 
 
 Let Ok be twice Oh ; then since el — ce, I will be twice as far 
 beyond the plane as e, and since Im — Ik, m will be twice as far 
 
158 DESCRIPTIVE GEOMETRY. 
 
 beyond it as Z, or four times as far as e ; wliile the distance of ra 
 from the axis is but twice that of e. 
 
 Had Oh been three times Oh^ m would have been three times 
 as far as e from the axis, but nine times as far beyond the plane of 
 the paper ; and so on. 
 
 Consequently LL cuts the paraboloid in a curve, symmetrical 
 with respect to the axis of the surface, having its vertex at 0^ ex- 
 tending to infinity beyond the plane i?^^, and having its abscissas 
 proportional to the squares of the ordinates ; that is to say, in a 
 parabola. And in a similar manner it may be shown that PP 
 cuts the surface in another parabola having its vertex at 0, and ex- 
 tending in front of the plane PI^. 
 
 232. Since the plane directers in Fig. 196 are equally inclined 
 to the horizontal plane, PP is horizontal and LL is vertical. Fig. 
 197 is a projection upon the latter plane; in which the planes PP 
 and PN are seen edgewise. Set off 01 , the assumed distance of 
 e from PN^ and Og equal to it ; also set off 6^4 and Op equal to 
 four times 01. Taking the lengths of the- chords ef^ mn^ from 
 Fig. 196, draw the projections of the rectilinear elements through 
 9") ^j ^j/*? i^? ^^? andj9, n. Thus the parabola mOn is shown in 
 its own plane ; also, it is seen that ge^ j^m, are tangent to this curve 
 at e and m, (the subtangents l)eing bisected at the vertex) : which 
 is as it should be, because eg., <?A, being elements of different gener- 
 ations, deteiTuine a plane tangent to the surface at their intersec- 
 tion e (142). The other parabola is shown in its true form in Fig. 
 198, which is a projection on the plane PP:, the construction is 
 obvious, the lengths of the chords gh^ jpq^ being transferred from 
 Fig. 196. 
 
 Tliese parabolas are upon different scales, since the chords ef^ 
 gh^ equidistant from the vertex, are respectively equal to the diago- 
 nals of a rhombus; this is because this particular surface is an 
 oblique one : liad it been rectangular, the rhombus would evidently 
 have been a square, its diagonals equal, and the parabolas identical. 
 
 233. If in Fig. 197 a plane parallel to PNhe drawn through 
 any point as ^, it will cut any pair of elements abov^e 6, Sispm^ qm, 
 thus determining a chord of the curve of intersection, seen as P£! 
 in Fig. 198. By drawing other elements, other points of tliis 
 
DESCRIPTIVE GEOMETRY. 159 
 
 <iurve are easily found ; and it is clear that e must be its vertex, 
 ])ecause the elements (je,, he^ determine a plane tangent to the sur- 
 face at that point, and that plane intersects the cutting plane in a 
 line perpendicular to LL, which is the tangent to the curve at the 
 point under consideration. The same plane will cut the lower 
 part of the surface in a similar and opposite curve, as shown in 
 Fig. 199, which is a projection on the profile plane BW. In this 
 figure (7, C^ are the curves cut from the surface by the plane 
 tin-ough e^ and 2>, Z>, are tliose cut from it by a parallel plane 
 through ^, at the same distance from EN but on the opposite side. 
 It can be proved tliat these curves are, in fact, hyperbolas, of 
 which the centres lie in the axis of the surface, and the asymptotes 
 are parallel to the elements Oli^ 0N\ but without discussing their 
 mathematical peculiarities, it is evident from what precedes that 
 they are convex toward the axis, and that their vertices lie upon 
 the two parabolas shown in Figs. 197 and 198: which suffices to 
 show that those parabolas are true gorge lines, as defined in (129). 
 
 234. The Plane a Limiting Form. Referring to Fig. 194, sup- 
 pose the elements to be perfectly elastic lines, fixed at each end in 
 the two planes H and V. Then if V be revolved about the ground 
 line in the direction of the arrow, these elements will be length- 
 ened, the curvature of the surface becoming less and less, until at 
 the limit, when J!/^ falls in H beyond the ground line, the parabo- 
 loid will have been extended into a plane. 
 
 If on the other hand V be revolved in the opposite direction, 
 the curvature of the surface will become greater and greater, the 
 elements contracting, until Jf falls in H in front of the ground line, 
 and the paraboloid will have collapsed into a plane. 
 
 235. The Warped Quadrilateral. Four right lines, connecting 
 any four points not in the same plane, constitute what is sometimes 
 <?alled a loarjped qtiadr Hater al^ and may define a portion of a 
 hyperbolic paraboloid : not necessarily, however, since there are 
 other warped surfaces which are doubly ruled, and such a quadri- 
 lateral can evidently be drawn upon any one of them. 
 
 But supposing that they do ; it is to be remarked that the same 
 four points may be joined two and two 'by right lines in three dif- 
 ferent ways, thus forming three different quadrilaterals, as shown 
 
160 
 
 DESCKTPTIVE GEOMETRY. 
 
 ill Figs. 200, 201, and 202, and determining as many diverse 
 hyperbolic paraboloids. 
 
 Fig. 201 
 
 Tig. 202 
 
 Fig. 200 
 
 236. Representation of the Surface. In Fig. 203, draw ^Z^^ 
 MN^ any two right lines of definite length not lying in the same 
 plane, and divide each into the same number of equal parts ; the 
 lines joining the corresponding points of division will be elements 
 of a hyperbolic paraboloid (226). 
 
 i>' 
 
 
 
 r 
 
 
 
 '^" 
 
 — 
 
 
 ^ 
 
 :^ 
 
 -/ 
 
 
 
 ^'/ 
 
 o 
 
 
 ^ 
 
 rr- 
 
 / 
 
 / 
 
 
 _. 
 
 -4' 
 
 tA 
 
 7^ 
 
 
 h' 
 
 
 
 Fig. 203 
 
 Fig. 204 
 
 To assume a point upon this surface. Assume the horizontal 
 projection, as 6>, Fig. 203 ; the point must then lie upon a vertical 
 line through o\ of which h't' is the vertical projection. Through 
 this vertical line pass any plane, as II ; it cuts the elements at Gy 
 E^ etc., thus determining a curVe of intersection, whose vertical 
 projection g'k'It cuts h't' in o\ the vertical projection of the as- 
 sumed point : and in a similar manner the horizontal projection 
 may be found if the vertical projection is assumed. In either case 
 
DESCRIPTIVE GEOMETRY. 161 
 
 tlieiv^ maj be two determinations, since a right line may pierce the 
 surface in two points ; but no more. 
 
 The* same process, obviously, is applicable to any otlier warped 
 eurface. 
 
 To draw an element through the point thus found. In Fig. 204, 
 the quadrilateral P 31 and the point O, in order to avoid confusion 
 in the diagram, are cojDied from Fig. 203., Through draw OB 
 parallel to MJV, and O/S parallel to Pi>; lind as in Fig. 83 the 
 points X and Y, in which the plane of these two lines cuts DM 
 and PJV: then XO Y is an element of the surface. 
 
 To draw a plane tangent to the surface at this point. Draw 
 through O a plane parallel to Z>i¥ and PiY, and find as above the 
 points in which it cuts PD and MN-^ then WOZ drawn through 
 those points is an element of the other generation, and the plane 
 determined by XY oiid WZ is tangent to the surface at O (142) : 
 and its traces if required may be found in the usual manner. 
 
 237. Through any two given right lines not' in the same plane, 
 any number of hyperbolic paraboloids may be passed ; for taking 
 those two lines as directrices, there may be an infinite number of 
 plane directers. This may be seen from another point of view, by 
 consider ing that the ratio between the lengths of PP and 31 X^ in 
 Fig. 203, is entirely arbitrary ; and that any change in this ratio 
 must modify the form of the surface, since one of the plane direct- 
 ers is always parallel to DM and PX^ and the other to PD and 
 3IX, It follows from this that if any two right lines not in the 
 same plane be drawn of indefinite length, and equal spaces be set off 
 upon each of them by any two scales of equal parts, whatever the 
 ratio between the units of those scales : the right lines joining the 
 successive points of division will be elements of a hyperbolic para- 
 boloid. 
 
 238. Practical Applications. The hyperbolic paraboloid is 
 sometimes met with, forming the basis if not the actual surface of 
 a practical structure, without revealing its true nature to the casual 
 observer. Thus in Fig. 205, A is a front view and B is a top view 
 of the "pilot," or "cow-catcher," of an American locomotive, 
 drawn of course in skeleton ; tins being divided by the central ver- 
 tical plane LL^ C is a view from one side, and D a view from the 
 
16'^ 
 
 DESCRIPTIVE GEOMETRY. 
 
 otiier side, of that plane. Considering first the part on tlie left of 
 LL in view A, and regarding the warped quadrilateral aoeb as lying 
 upon a surface of tliis kind; then by subdividing ao., he^ as above 
 explained, we determine a series of elements parallel to LL^ as 
 
 r L 
 
 Fig. 205 
 
 a 
 
 A 
 
 
 L 
 
 
 
 
 
 
 
 e 
 
 C 
 
 
 
 
 
 h 
 
 r 
 
 
 
 
 
 L 
 
 shown in view C and in the lower half of view B. Upon the other 
 side of LL there is a warped quadrilateral scrd^ symmetrical to the 
 first ; in this case, by means of corresponding subdivisions of cr and 
 scl^ a series of horizontal elements is determined, as shown in view 
 D and in the upper half of view B. 
 
 The pilot, then, is composed of two symmetrically placed hy- 
 perbolic paraboloids, of which the horizontal plane and the vertical 
 plane LL are the common plane directers ; and the bars may lie 
 either in vertical planes or in horizontal planes : both arrangements 
 have been used, the preference being given to the former. 
 
 239. This surface has also been employed, in designing the bow 
 of a boat, as shown in Fig. 206, where the water-lines are shown at 
 A, while B is the sheer plan and C the body plan. The warped 
 quadrilateral dhco being treated as in Fig. 200, the lines xx^ etc., as 
 well as the frames nn, etc. , are right lines, as shown not only in B 
 but in the left-liand part of C and the lower half of A. But the 
 true water-lines, 1, 2, 3 (or horizontal sections, as shown in the 
 right-hand half of C), are not straight, but what is technically called 
 liollow, — that is, outwardly concave, as seen in the upper half of A. 
 This circumstance is obviously due to tlie fact tliat dh in this case is 
 
DESCRIPTIVE GEOMETRY. 
 
 Xt)3 
 
 not horizontal, but droops as it recedes from do^ the vertical line of 
 the stem ; which is most clearly shown in the sheer plan B. And 
 
 ' h 
 
 y 
 
 so 
 
 i> 
 
 - 
 
 p 
 
 m 
 
 n 
 
 
 
 
 y 
 
 
 
 
 \- —4 
 
 X 
 
 
 
 
 J\ 
 
 p m n 
 
 Fig, 206 
 
 it is equally clear that if dh had been horizontal, all the water-lines 
 would have been straie^lit instead of concave. 
 
 THE CONOID. 
 
 240. The Conoid is a warped surface, with a plane directer, 
 and two linear directrices, one of which is a right line, and the 
 other a curve. Thus the elements of the surface, instead of con- 
 verging to a single point as in the cone, pass through all the points 
 of the rectilinear directrix. If that directrix is perpendicular to the 
 plane directer, it is called the Line of Striction, and the surface is 
 called a Right Conoid. 
 
 The definition being perfectly general, the curved directrix may 
 be of single or of double curvature. But the term conoid is fre- 
 quently used in a limited sense, as including only those surfaces in 
 which this directrix is a closed curve, lying in a plane perpendicular 
 to the plane directer. 
 
 241. In Fig. 207 is represented the most simple of the conoids, 
 which bears to others the same relation that the cone of revolution 
 has to all other cones ; and in all probability gave the name to the 
 class. This is a right conoid, of which the plane directer is V, and 
 
164 
 
 DESCRIPTIVE G p]OM ETRY. 
 
 the curved directrix a circle lying in H ; the line of striction inter- 
 sects at O the axis of this circle, which is evidently a line of sym- 
 metry, and may be called the axis of the surface. It is apparent 
 that this surface is symmetrically divided by two planes through the 
 axis, one of which is parallel, and the other perpendicular, to the 
 plane directer; also that it is divided by the line of striction into 
 
 '\ 
 
 Ji 
 
 
 7i, 
 
 
 
 
 
 
 
 '\ 
 
 n \ 
 
 
 
 
 
 
 
 
 
 
 
 \\fc' \^ 
 
 / 
 
 
 
 
 0, 
 
 
 
 
 d'// 7 ' 
 
 
 
 
 
 
 
 
 n< ^ 
 
 . / '' 
 
 e' Z' 
 
 XV ^' 
 
 
 Ci 
 
 r 
 
 
 V. 
 
 ^. / 
 
 /w 
 
 
 
 
 / / I 
 
 \ 
 
 \ A 
 
 
 f 1 I 
 
 I d g 
 
 Ic 
 
 / c 
 
 /■ 
 
 
 Pig. 207* 
 
 w\ p 
 
 a'i 
 
 ' z/ / 
 
 u ' 
 
 • \ \ 
 
 ./> 
 
 / 
 
 T 
 
 T ^**"-^^-»i 
 
 ^^■7/ 
 
 ^ ^m 
 
 7/ 
 
 7 
 
 
 two nappes, like a cone, which are similar and symmetrically placed. 
 
 The intersection of this conoid by a plane perpendicular to the 
 axis is an ellipse. 
 
 Thus, let ZZ be such a i3lane, lying between the directrix and 
 the line of striction : then it is seen that in the horizontal projec- 
 tion the ordinates go, rx, are less than, but directly proportional 
 to, the corresponding ordinates do, yx, of the circle ; therefore the 
 curve is an ellipse whose major axis mn is equal to the diameter of 
 the circle. If the elements be extended, -and cut by a horizontal 
 plane, as II, more remote than the directrix from the line of stric- 
 
DESCRIPTIVE GEOMETRY. 165 
 
 tion, the ordinates will be greater than those of the circle, and the 
 minor axis of the elliptical section shown in a dotted line will be 
 equal to the diameter of the directrix. 
 
 242. Planes Tangent all along Rectilinear Elements. It is verj 
 *easy to draw the plane tangent at any point, as A for instance, 
 
 since it must contain the element through that point, and also the 
 tangent at the same point to the elliptical section by a horizontal 
 plane; the horizontal trace of the required plane being drawn 
 through ^, the foot of the element, and parallel to that taiigent, the 
 vertical trace is drawn parallel to az' . And it is obvious that in 
 general such a plane will not be tangent along the element, because 
 the tangent to the circle at z is not parallel to the tangent to the 
 ellipse at a. 
 
 But the plane TT^ parallel to V, contains the vertical element 
 EH^ and also a tangent at one of the vertices to every elliptical 
 section : it is therefore tangent all along the element. 
 
 So, again, the plane sSs\ perpendicular to T and containing 
 CP^ is tangent to the surface all along that element ; and two other 
 planes possessing this peculiarity can be drawn on the opposite 
 sides. 
 
 243. In Fig. 208, Y is the plane directer, the curved directrix 
 is the horizontal circle CC^ and the rectilinear directrix DD is hori- 
 zontal but inclined to T, and intersects at O the axis of the circle. 
 The upper nappe of the surface is not represented, but the elements 
 are continued below CC to the horizontal plane. 
 
 It is obvious on inspection that this surface is not symmetrical 
 like the preceding one ; the two nappes will not be similar, and the 
 directrix is the only circular section. 
 
 In regard to the horizontal trace, we have, since the elements 
 are proportionally divided by parallel planes, the values 
 
 fg sv _0G _ 
 
 an de 
 
 gk ~ vw Gu 
 
 m em ' 
 
 therefore that trace is an ellipse, and the same is true of a section 
 by any horizontal plane as LL. 
 
 244. Since there must be a vertical element at R and another 
 at S^ it follows that rs will be a common diameter of aU these 
 
166 
 
 DESCRIPTIVE GEOMETRY. 
 
 ellipses, as well as of tlie circle. Draw gn^ a diameter of the cir- 
 cle perpendicular to rs^ and through g and n draw the projections 
 of two elements, determining the points /»• and z in the horizontal 
 trace. Then regarding rs as a diameter of the ellipse, kz will be 
 
 the conjugate diameter ; and in like manner, in any other horizon- 
 tal section, the extremities of the diameter conjugate to rs will lie 
 xv^onfgk and anz. The tangents to these ellipses at the extremi- 
 ties of these conjugate diameters are parallel to rs^ consequently 
 the planes determined by DD and the eleinents FK^ AZ, are tan- 
 
DESCRIPTIVE GEOMETRY. 167 
 
 gent to the surface all along those elements ; the true angles y5, &?, 
 which these planes make with H, are seen in the supplementary 
 projection. 
 
 Draw (?<?, ex^ two conjugate diameters of the directrix, respec- 
 tively parallel and perpendicular to V. The elements through the 
 extremities of the first determine the points u^ y^ of the horizontal 
 trace ; then reasoning as before, m^ the extremity of the diameter 
 conjugate to ^ty^ must lie upon the element through e : so also does 
 j9, the extremity of a diameter conjugate to hi^ in the section by 
 the plane LL. From which it appears that, in this case also, two 
 planes parallel to V can be drawn, each tangent all along an ele- 
 ment. 
 
 245. Limiting Forms of the Conoid. These two examples illus- 
 trate sufficiently for our j^urpose the peculiarities of this surface, 
 which, however, is susceptible of variations without number, since 
 the straight directrix may have any position, and the curved one 
 any form ; in consequence of this, it happens that, in special cases, 
 warped surfaces of other classes are also in fact conoids. 
 
 The less the inclination of the straight directrix to the plane 
 director, the more acute will be the angles between it and the ele- 
 ments ; and at the limit, when that directrix is parallel to the plane 
 director, the elements will become parallel to it and to each other. 
 In that event, if the other directrix be of double curvature, or lie 
 in a plane which cuts the rectilinear directrix, the conoid will be- 
 come a cylinder ; if it lie in a plane parallel to the rectilinear direc- 
 trix, the surface will degenerate into a plane. 
 
 THE HYPERBOLOID OF REVOLUTION. 
 
 246. This surface is generated by a right line, Avhich revolves 
 about an axis lying in a different plane. In Fig. ^09, let the axis 
 be vertical, the point o being its horizontal, and w'o'y its vertical 
 projection ; and let the generatrix MN be parallel to Y : then each 
 point of MN will describe a horizontal circle, of which the true 
 radius is seen in the horizontal projection, and the exact altitude in 
 the vertical projection. 
 
 Thus, tlie point N describes a circle whose radius is on ; the 
 vertical projection of this circle is determined by setting off on each 
 
168 
 
 DESCRIPTIVE GEOMETRY. 
 
 side of the axis, on tlie horizontal through n\ the distance 
 y'h' = on : similarly, the vertical projection of the circle described 
 by the point Ehfe'f\ where u'f ~ oe, and so on. The circle of 
 least diameter is that described by the point O, since oc is the com- 
 mon perpendicular of the axis and the generatrix. Tiiis circle, 
 
 TiQ.2.09 FiQ. 210 
 
 whose vertical projection is d'c'd' ^ is called the gorge circle, and its 
 plane is called the gorge plane. Since points of JfiT which are equi- 
 distant from C are also equally distant from the axis, and therefore 
 describe equal circles, it follows that the portion d'V of the contour 
 above d'd' is precisely like the portion d'h' below that line ; and 
 that the surface is symmetrically divided by the plane of the gorge. 
 247. Practical Suggestions. This surface has practical applica- 
 tions in mechanism ; and it is proper here to point out most em- 
 phatically, tliat in constructing the contour it is not only useless, 
 but worse than useless, actually to draw the vertical projections 
 d'd', ff\eiG.^ thus covering the paper with superfluous lines to be 
 subsequently erased : no good draughtsman will do it, nor w^ill any 
 
DESCRIPTIVE GEOMETRY. 1C9 
 
 good instructor permit pupils to adopt such slovenly methods. It 
 is not even necessary to draw the horizontal projections of the radii ; 
 having marked the projections of any point of J/iV, as r, r'-^ mark 
 on the axis the point x on the horizontal through r\ then taking 
 the distance or in the compasses, a very short arc only is drawn as 
 at g\ g\ on each side, with x' as a centre ; the precise points on 
 these arcs being finally marked on the horizontal through z' . 
 
 248. This Surface is Doubly Ruled. The portion here repre- 
 sented is limited by two planes equidistant from the gorge plane; 
 which may be called its bases. Projecting m to the lower base at 
 s\ and n to the upper base at^', it is seen that Tucn is the horizontal 
 projection of a second line whose vertical projection is s'g'2>' - This 
 line makes the same angle as MN^ Avith the plane of the base, but 
 elopes in tlie opposite direction ; moreover, it is also a generatrix of 
 the surface, because each of its points, in revolving around the axis, 
 describes the same circle as a corresj)onding point of MN\ for in- 
 stance, Z and R^ on the same side of the gorge and equally distant 
 from it, are also equidistant from the axis, and describe the same 
 horizontal circle. This facilitates the construction of the contour : 
 thus a' is vertically over e\ consequently those two points are equi- 
 distant from {?', and the circles drawn through them will have the 
 same radius oe ; the radius of the circle through V ^ vertically over 
 r% is equal to or^ and so on. 
 
 249. Through any point of the surface, then, two elements can 
 be drawn ; and it is apparent that either of them if produced will 
 intersect all those of the other generation except that one which is 
 parallel to it, for the simple reason that they do lie upon the same 
 surface ; and it may be said to intersect that one at an infinite dis- 
 tance. From this it follows that if any three elements of either 
 generation be taken as directrices, any element of the other may be 
 taken as a generatrix, whose motion will produce the same surface. 
 Considered in this light, the surface is one with three rectilineat 
 directrices ; regarded as a surface of revolution, it is one with a 
 cone directer and two circular directrices, — the former being a right 
 circular cone whose ano-le at the vertex is s'c'n' in FIh". 209 : the 
 base of which is II in the horizontal projection : whan thus situ' 
 ated this cone is obviously asymptotic to the surface. 
 
170 DESCRIPTIVE GEOMETRY. 
 
 If a plane be drawn cutting tliis cone directer in an ellipse, a 
 parabola, or an hyperbola, a plane j^arallel thereto will cut this 
 hyperboloid in a curve of the same kind. 
 
 250. To assume a point on the surface. In Fig. 210, the contour 
 having been found as above, assume the vertical projection c ; then 
 a horizontal through c' determines the radius y'x' of the circle upon 
 whose circumference c must lie. Conversely, if the horizontal pro- 
 jection c be assumed; a circle through c, about centre <?, will be cut 
 by the meridian plane LL parallel to V, in the point a?; which pro- 
 jected to the contour at x' will determine the altitude of the vertical 
 projection c' . 
 
 Otherwise^ if the horizontal projection c be assumed : Draw 
 through c two tangents to the gorge circle ; these are the horizontal 
 projections of the two elements which pass through the assumed 
 point. One of these pierces the lower base in D and the gorge 
 plane in i^; the other pierces the lower base in E and the gorge 
 plane in U\ and their vertical projections (^y, eic\ intersect in c\ 
 the vertical projection of the assumed point. 
 
 To draw a plane tangent to the surface at any point. Draw through 
 the point an element of each generation. Thus in Fig. 209, the 
 two elements whose horizontal projection is ran determine the 
 plane tangent to the surface at the point C on the gorge circle. 
 And in Fig. 210 the \?lements DF^ EU^ determine the plane tan- 
 gent at the point C\ in this figure the lower base coincides with H, 
 consequently tt^ drawn through d and e^ is the horizontal trace of 
 the plane. This trace is perpendicular to the radius oc^ which fact 
 enables us to draw it with precision, even when d and e are very 
 close together ; and indeed it enables us to dispense altogether with 
 one of the r.Iements through C^ since, the direction being known, 
 either one of the points, d or e^ suffices to locate the trace. 
 
 251. It is to be observed that, having determined the circular 
 paths of any two points in the generatrix, as for instance Jf and N 
 in F^.g. 209, the projections of any number of elements might have 
 been drawn ; and instead of constructing the contour line h'd'V by 
 points as above, we could then have drawn it as the enveloj^e of the 
 vertical projections of those elements (170), each of wliich would be 
 tangent to it at a point in the meridian plane parallel to V : thus in 
 
DESCKIPTIVE GEOMETRY. 171 
 
 Fig. 21C, tlie element EG lias but one point, K^ in common with 
 the meridian curve, and no point either of the element or the sur- 
 face lies to the left of that curve ; consequently h' is a point of tan- 
 gency in the vertical projection. 
 
 252. The Meridf.a.'» Curve is an Hyperbola. In Fig. 209, the in- 
 tercepts e'f'^ r'g\ etc., diminish as they recede from the gorge. 
 In fact, each intercept, as r^g' for instance, is the difference between 
 the hypothenuse or and the base cr of a right-angled triangle 
 whose altitude oe remains constant. Let the hypothenuse = h, the 
 base — b, the altitude = a; then a' z= h' — b' =: (h -|- b)(h — b), 
 
 JS^ow the value of this fraction diminishes as the 
 
 h-f- b 
 
 denominator increases, and reduces to zero when h + b becomes in- 
 finite : consequently c'n\ c'jp\ are asymj)totes to the meridian curve 
 l:'d'l\ of which d' is the vertex and the horizontal line through c' 
 is the axis. 
 
 In Fig. 210, MN^ ^y^x^ ^re two positions of the same genera- 
 trix, both parallel to Y, but on opposite sides of the surface ; thus, 
 on\ o'm^\ are asymptotes to the contour. Draw any element DF 
 of the other generation, cutting 3f N at B and M^N'^ at S. This 
 element pierces the meridian plane LL at P, and d's' is tan- 
 gent at p' to the contour (251.). Now, rp = ps, consequently 
 r'p' = p's' ; that is to say, the intercept between the asymptotes is 
 bisected at the point of contact : which is a property peculiar to the 
 hyperbola. 
 
 Whence it appears that the contour lines Tc'd'V ^ h'd'l\ in Fig. 
 ^09, are the opposite branches of an hyperbola of which the centre 
 IS o and the major axis is d'd' \ and that the surface may be gen- 
 erated by revolving this hyperbola about its conjugate axis. This 
 surface being unbroken, is distinguished as the hyperboloid of revo- 
 lution of one nappe 'y the same hyperbola by revolving about th? 
 major axis will generate an hyperboloid of two separated nappes, — 
 which however is obviously a double-curved surface. 
 
 225a. In Fig 210a, the planes TT^ LL^ parallel to Fand to 
 the axis, and tangent to the circle of the gorge, each cut from the 
 surface two rectiHnear elements, whose vertical projections coincide 
 and are asymptotes to the meridian hyperbola. Draw two other 
 
172 
 
 DESCRIPTIVE GEOMETRY. 
 
 planes YY, WW, parallel to the others and equidistant from the 
 axis but nearer to it ; the sections will evidently be similar curves, 
 the projections of those on the right-hand side of the axis coinciding 
 in ev'h'. Any rectilinear element of the surface, as IiS, pierces 
 these planes in the points 0, 0, 3f, JV; and shice oc = 7mi, it fol- 
 
 lows that o'c' = m'nf ; consequently e^'N is also an hyperbola 
 whose asymptotes are the same as those of the meridian outline. 
 
 A plane parallel to these but farther from the axis, as XX, will 
 cut the surface in an hyperbola whose vertical projection g'f'k' will 
 have the same asymptotes, but its major axis will be vertical instead 
 of horizontal. 
 
 253. In Fig. 209, the first position of the generatrix, MN\ 
 
DESCRIPTIVE GEOMETRY. 173 
 
 was parallel to Y ; but in applications of this surface as an auxiliary, 
 the given generatrix may not be so conveniently situated. To illus- 
 trate : suppose, in Fig. 210, tlie vertical axis and the line EG to be 
 given, from which d?.ta the surface is to be constructed. Draw ou 
 perpendicular to eg ; this is the radius of the gorge circle, the alti- 
 tude of whose plane is found by j)i*ojecting i^ to u' on eg' ; this, 
 done, MN is drawn parallel to T, whence the asymptotes are deter- 
 mined and the contour constmcted as in Fig. 209. 
 
 If we regard e'g' in Fig. 210 as the vertical trace of a plane 
 perpendicular to T, and tangent to the hyperboloid at K\ then that 
 plane must contain the companion generatrix passing through K^ 
 whose horizontal projection is A'J; also, it will cut the meridian 
 plane LL in a right line which intersects the axis of the surface at 
 Z, and it is ajDj^arent that KZ is the true tangent at K to the hyper- 
 bolic outline. These considerations will be of use in employing 
 this surface as an aid to the solution of the following problem. 
 
 254. Problem. To pass through a given right line a plane 
 tangent to any given surface of revolution. 
 
 Analysis. Revolve the given line about the axis of the given 
 surface, thus generating a warped hyperboloid of revolution ; the 
 required plane will be tangent to this hyperboloid at some point of 
 the given line (142). It will also be perpendicular to the meridian 
 plane through the point of contact, on either surface ; but the two 
 surfaces having a common axis, these meridian planes will coincide, 
 and will cut fi-om the common tangent plane a right line, tangent 
 to both meridian curves. This tangent and the given line deter- 
 mine the required plane. 
 
 Construction. In Fig. 211, let CN be the given line, through 
 which it is required to pass a plane, tangent to the surface of revo- 
 tion with a vertical axis, whose contour is shown in the vertical 
 projection. Find the radius and altitude of the gorge circle, and 
 construct the hyperboloid, as explained in (253). Draw, mechani- 
 cally, s's' tangent to the given contour and also to the hyperbola 
 li'g'V ; this is the vertical trace of the required plane when revolved 
 aboQt the axis of the surface until it is perpendicular to V. The 
 plane in this position contains an element C^D^ (the revolved po- 
 sition of the given line), which pierces the meridian plane LL par- 
 
174 
 
 DESCRIPTIVE GEOMETRY. 
 
 allel to V, at 0^ the revolved position of tlie point of contact with 
 the hjperboloid. The revolved position of the point of contact 
 
 X' 
 
 FiG. 211 
 with the given surface is vertically projected at j^/ , found as in 
 (147), and horizontally at j^?, on LL, 
 
 255. In the counter-revohition, O, returns in a horizontal 
 circle to O on CD\ o is at once located by setting off do = d^o^ ; 
 then, drawing the radius oe^ the horizontal trace of tlie required 
 plane is tcT perpendicular to oe^ and the vertical trace is Tt\ 
 passing through n ^ tlie vertical trace of the given line. The point 
 of contact P may be located in a similar manner by setting off on 
 oe the distance ep — ejp^ ; the vertical projection jp' lies upon a hori- 
 zontal line through j^i'. 
 
 Note. The common tangent line PO^ mentioned in the argu- 
 ment, is not here made use of, since the directions of the traces 
 •can be determined without it, and in this instance more conven- 
 iently ; but in other cases the use of that line might be essential. 
 
 256. Limiting Forms of tliis Surface. If in Fig. 200 the radius 
 0C be gradually diminished, other things remaining unchanged, the 
 
DESCRIPTIVE GEOMETRY. 175 
 
 gorge circle will become less and less, tlie surface assuming a closer 
 resemblance to a cone, which is the limiting form reached when 
 the connnon perpendicular becomes zero and the generatrix inter- 
 sects tlie axis. 
 
 If, oc i-emaining unchanged, we suppose MN to make a greater 
 angle with the horizontal plane, the curvature of the surface will 
 diminish, and it will more and more resemble a cylinder, which is 
 the limiting form reached when the generatrix is parallel to the 
 axis : the diameters of tlie gorge and the base circles then becoming 
 equal. 
 
 If on the other hand the generatrix becomes horizontal, it must 
 always lie in a plane perpendicular to the axis, which therefore is a 
 third limiting form of this hyperboloid. The plane generated in 
 this manner, however, cannot abstractly be considertjd to exist 
 within the circumference described by the point C\ wliicli is in 
 fact a true edge of regression. 
 
 THE ELLIITICAL HYPERBOLOID. 
 
 257. Let the smaller circle E^ in Fig. 212, be the horizontal 
 projection of the gorge, the larger one, F^ that of the upper and 
 lower bases equidistant from it, and m,n that of the generatrix, 
 of an hyperboloid of revolution ; whose vertical projection is con- 
 structed as above explained. If the ordinates of both these circles 
 which are perpendicular to LL be reduced in the same proportion, 
 the circles will be transformed into two ellipses, / and J\ which 
 are similar to each other, since the ratio of the major to the minor 
 axis is the same in both. Now, if these ellipses be substituted for 
 the original circular gorge and bases, a right line moving in contact 
 witli them all will generate a new surface, called an elliptical hyper= 
 boloid. 
 
 If we constnict as in Fig. 209 the cone directer of the origina.. 
 liyperboloid of revolution, and treat its base in like manner, that 
 <3one will be transformed into the elliptical cone directer of the new 
 surface. The sections of this cone and of the hyperboloid itself by 
 parallel planed will be either ellipses, parabolas, or hyperbolas, 
 according to the positions of the cutting planes. 
 
176 
 
 DESCKIPTITE GEOMETRY. 
 
 258, The chord Tee of the elhpse «/, tangent to the ellipse / at 
 the extremity of the minor axis, is the horizontal projection of an 
 
 Fig. 212 
 
 element of this surface : this element is by construction parallel 
 and equal to the element MN of the hyperboloid of revolution ; 
 and will have the same vertical projection, since mh^ ne^ are per- 
 pendicular to T. 
 
 Let qf be the horizontal projection of any other element of the 
 hyperboloid of revolution ; it is tangent at o to the circle E^ and 
 cuts LL in p. From g and f draw ordinates per23endicular to LL^ 
 cutting the ellipse Jin c and d ; and from o draw another, cutting 
 the ellipse I'm r\ then since by construction gc^ fd, or^ are equal 
 fractions of these ordinates, the chord cd will pass through 7' and^, 
 and by a known property of the curve it will be tangeir': at r to the 
 smaller ellipse. 
 
 Consequently cd is the liorizontal projection of an element o£ 
 
DESCRIPTIVE GEOMETRY. 177 
 
 tlie elliptical liyperboloid, just as ^/is of an element of tlie circular 
 one ; and since gc^ fd^ are perpendicular to V, these two elements 
 will have the same vertical projection c'd' : which is also the ver- 
 'tical projection of another element, wdiose horizontal projection i& 
 drawn through ^, tangent to the gorge ellipse on the opposite side. 
 
 From the preceding it follows that the elliptical hyperboloid is 
 doubly ruled, and that wdien placed, as here shown, witli the major 
 axes of its bases parallel to V, its vertical projection will be identi- 
 cal with that of the circular one from which it is derived in the 
 manner above set forth. 
 
 Evidently, the ordinates of the circles might have been in- 
 creased, instead of being diminished, in any desired proportion, 
 without affecting the argument : in which case the minor instead of 
 the major axis would have been parallel to T. 
 
 In the profile, it is evident that A/, ^^, will be equal to the 
 minor axis of the larger ellipse, and ab to that of tiie smaller one. 
 The contour in this view may logically be determined thu§ : Draw 
 xyz the contour of the original circular hyperboloid ; then ordinate^ 
 from this contour perpendicular to the axis are to be reduced in 
 the same proportion that was adopted in constructing the ellipses / 
 and J. Treating in the same way the asymptotes st^ su, to the 
 curve Xi/2, we shall have two right lines, ww, vv ; which are the 
 two elements represented by wv in the horizontal projection. From 
 this construction it is seen that sw, sv, are apymptotes to the con- 
 tour /^ta^; which can be shown to be an hyperbola by reasoning 
 similar to that of (252). 
 
 260. To assume a point upon this surface. If the horizontal 
 projection be assumed, the horizontal projection of an element 
 through the point w^ill be a tangent to the gorge ellipse.; and the 
 vertical projection of the point must lie upon the vertical projec- 
 tion of this element : it can therefore be determined directly. 
 
 But if the vertical projection be assumed, as at S^ ; it is then 
 necessary to draw the section of the surface by a plane JTX passing 
 through the point and perpendicular to the axis. It follows from 
 what precedes that this section will be an ellipse of which the major 
 axis is Gllm the vertical projection, and the minor axis is DQ in 
 the profile. This ellipse will be seen in its true form in the hori- 
 
178 DESCKIITIVE GLUMETllY. 
 
 zontal projection, and tlie horizontal projection of the assumed 
 point must Re upon tlie curve. 
 
 To draw a plane tangent to this surface at a given point. The 
 surface being doubly ruled, the tangent plane is determined by 
 drawing through the point an element of eacli generation. 
 
 261. If, in Fig. 212, the ordinates perpendicular to LL in the 
 horizontal projection liad been increased instead of diminished in 
 any given ratio, the circles would have been transformed into 
 elHpses with their minor axes parallel to V. But, the preceding 
 argument still holding true, the vertical projection of the resulting 
 elliptical hyperboloid would liave been identical with that of the 
 original circular one : accordingly, the profile in the figure may be 
 regarded as the vertical projection of an hyperboloid of revolution, 
 in which the diameters of the gorge and the bases are respectively 
 equal to the minor axes of the ellipses 1 and «/; as indicated in the 
 diagram B, drawn below the profile. 
 
 It follows, then, not only that from a given circular hyperboloid 
 an infinite number of elliptical ones may be derived ; but that the 
 former is merely a special case, in which the axes of the three ellip- 
 tical directrices become equal. 
 
 262. Limiting Forms of tlie Elliptical Hyperboloid. By rea- 
 soning analogous to that of (256), it will be apparent that when the 
 generatrix intersects the axis, the surface will become an elliptical 
 cone ; when it is parallel to the axis, the surface will become an 
 elliptical cylinder ; and if it is tangent to the gorge ellipse, the sur- 
 face will become a plane with ati elliptical perforation. 
 
 263. Tlie Hyperbolic Paraboloid a Special Case. A limiting form 
 may be reached, however, in a different manner. In Fig. 213, 
 let c be the axis and rarn the generatrix of an hyperboloid of revo- 
 lution ; the latter retaining its present position, suppose the axis to 
 recede, as indicated by the successive positions <?, , c^ : it is clear that 
 the difference between the radii cr and cm will become less and 
 less, until when the axis is infinitely remote, those radii will be 
 equal, the two circumferences will become one right line, and the 
 surface will become a plane ; which therefore is a Ihniting form in 
 this case. 
 
 In Fig.- 214., the gorge and base ellipses having been derived 
 
DESCRirTlVE GECMETRT. 
 
 179 
 
 from tlie circles shown, as in Fig. 212, let ^ and e be their corre- 
 sponding foci : then from the mode of derivation it follows that the 
 semi-parameters os^ er\ have the same ratio as the radii ca^ cb. 
 Kow, the points <^, 5, and^* remaining fixed, suppose again the axis 
 
 The curves, as has been seen, will always 
 
 to recede to 
 
 '15 ^a) ^^^' 
 
 be similar ellipses; and their semi-parameters approach equality as 
 the axis recedes, becoming equal when its distance is infinite : but 
 at this limit, the conjugate focus o' being also infinitely remote, the 
 gorge ellipse becomes a parabola; and from the preceding argu- 
 ment it follows that tlie outer ellipse will also become a parabola 
 liamng tlie same parameter. In these circumstances, the elliptical 
 hyperboloid becomes a hyperbolic paraboloid; which in view of 
 this mode of derivation might more consistently be called a para- 
 bolic hyperboloid. 
 
 264. It will now^ be clear that the gorge plane PP^ and the 
 plane /.//, in Fig. 212, correspond in position to the principal di- 
 ametric planes in Figs. 196-199; and supposing the axis of 'the 
 surface to recede to the left, the gorge ellipse /will at the limit 
 become the gorge parabola of Fig. 198, and the hyperboloidal con- 
 
 * As c recedes, os will increase, while he and er will diminish ; until, when ac 
 becomes infinite, we shall have he — ao, and er = os = 2ao, 
 
180 
 
 DESCRIPTIVE GEOMETRY. 
 
 tonr HOp' will become the other gorge parabola, shown in Fig. 
 197; O^ the connnon vertex of the ellipse and the hyperbola iu 
 Fig. 212, will remain the connnon vertex of the two gorge parab- 
 olas : it will also be the vertex of the resulting hyperbolic parabo- 
 loid, — whose axis, however, has no relation to the axis of the original 
 surface, since it is merely the common axis of the two gorge lines, 
 being the intersection of tlie planes PP^ LL. 
 
 265. The ellipse «/, in Fig. 212, represents the intersection of 
 the elliptical hyperboloid by either of two planes, parallel to PP 
 and equidistant from it. And we observe that any chord cd of 
 this ellipse which is tangent to the smaller one /, is bisected at r^ 
 the point of contact ; which from the mode of derivation must 
 hold true whatever the relative magnitudes of the corresponding 
 axes, — and these again are independent of the distances from the 
 gorge plane. 
 
 At the limit both ellipses become parabolas ; whence the fol- 
 lowing problem may arise. Given, in Fig. 215, the parabola 
 
 Fig. 215 
 
 HOK^ of which O is the vertex and OR the axis : Required, to 
 construct another parabola with the same axis, and a given vertex 
 A.^ any chord of which, tangent to the given one, shall be bisected 
 at the point of contact. 
 
 Any parabola having the same axis will satisfy the condition 
 for the chord tangent at 0'^ it remains then to find one which will 
 satisfy it in regard to some other chord. Set off OB = OA^ erect 
 the ordinate BC\ draw J. 6^ tangent to TlOIy^ produce it, and make 
 CD = CA ; then Z> is a point in the required curve PAL Draw 
 
DESCRIPTIVE GEOMETRY. 
 
 181 
 
 tlie ordinates DE^ OG^ then by construction AE= 2AB = 4:A0, 
 and since the curve is a parabola, DE= 2 GO] but by construc- 
 tion DE= 2BC, consequently GO — BC. That is to say, the two 
 parabolas are precisely alike; which agrees with the conclusion 
 otherwise reached in (263). 
 
 2.66, In Fig. 216, tlie two parabolas Tiok^ dai, in the horizontal 
 projection, are the same as those of the preceding diagram; the 
 
 "K^ 
 
 ^^ 
 
 m 
 
 
 
 
 
 
 
 ^^v 
 
 ^.S' 
 
 
 
 
 
 > 
 
 ^ 
 
 V 
 
 
 ^ 
 
 \ 
 
 
 r. 
 
 
 
 \ 
 
 
 "yC 
 
 % 
 
 z 
 
 i 
 
 
 ^ 
 
 
 
 
 yn 
 
 ja 
 
 
 vertical projection of the former is in the line PP, while s'd', a'i\ 
 parallel to PP and equally distant from it, are the vertical projec- 
 tions of the latter. Kow, whatever the actual magnitudes of x's' 
 and x'a\ it is to be observed that s'i'^ a'd\ will intersect at y' on 
 PP^ giving y'o' = o'x\ and will therefore be tangent to a parabola 
 of which o' is the vertex ; and in like manner it may be sliown that 
 m^n'^ w'z\ and in short the vertical projections of all tangents to 
 liok^ will be tangent to the same parabola a'o's' ; which lies in the 
 vertical plane LL : the surface thus determined is, then, 2. hyper- 
 bolic paraboloid. 
 
183 DESCRIPTIVE GEOMETRY. 
 
 Bj similar reasoning it may be sliown tliat sections hy planes 
 parallel to ZL, will be parabolas precisely like a'(/s' ; that is to say, 
 all sections of tins surface by planes parallel to either of the princi- 
 pal diametric planes, will be similar and equal to the gorge parab- 
 ola which lies in that plane. 
 
 267. If we suppose the parameter of the gorge hok to be 
 indefinitely reduced, other things remaining unchanged, that pa- 
 rabola will ultimately become c, right line, and the surface Mill 
 become a plane coinciding with ZZ^ its elements being then tan- 
 gents to ao's. Similarly, if the -horizontal gorge remain un- 
 changed, while the parameter of a'o's' is reduced, the elements will 
 "ultimately become tangent to hok, and the surface a plane coincid- 
 ing with Z*P. 
 
 268. This derivation of the hyperbolic jjaraboloid, though 
 showing it to be doubly niled, gives thus far no evidence of the 
 existence of plane directers. But if it be established tliat the ele- 
 ments of each generation divide those of the other proportionally, 
 it must follow that those of either set lie in a series of parallel 
 planes. 
 
 If this relation is true as to the elements in space, it must also 
 be true of their projections, in which all j)arts are equally fore- 
 shortened ; in other words the tangents to the gorge parabolas in 
 Fig. 216 must divide each other proportionally. That they do so, 
 is a fact made use of in the familiar construction shown in 
 Fig. 217 ; the sides J^C, CD, of the isosceles triangle ZJCIJ being 
 divided into tlie same number of equal parts, the points of division 
 are numbered in opposite directions, and the corresponding num- 
 bers joined by right lines : the envelope of these lines is the parab- 
 ola EAD. The following demonstration of this depends upon two 
 properties of the parabola, viz., that the abscissas are proportional 
 to the squares of the ordinates, and that the subtangent is bisected 
 at the vertex. 
 
 269. In Fig. 218, let A be the middle point of BC\ which 
 bisects the vertical angle of the isosceles triangle EDC. On the 
 equal sides set off DF — Oil of any length at pleasure, draw ZII 
 cutting ^^'in G, and on JjO set oif AM — AG. Perpendicuki 
 
DESCRIPTIVE GEOMETRY. 
 
 183 
 
 to BCdmw FO, ML cutting FH'm Z, ^P cutting FH'm iTand 
 I) Cm P, and ///cutting BCin iV^and CD in / 
 
 Kow since BCis bisected at A^ CD is tangent at /> to a pa- 
 rabola of which A is the vertex ; and since MG is bisected at A, 
 FJIis tangent at Z to a parabola of which A is the vertex. Also 
 these parabolas have the common axis DC', but in order to prove 
 
 F D 
 
 Fig. 2L7 
 
 Fio. 21S 
 
 that they are one and the same, it is necessary to show that 
 AM\ AB :: MD : BD\ 
 
 But AM^ AG, AB = AC, and ML : BD :: AK \ AP-, 
 consequently the above proportion may he written AG : AG :: 
 
 AK' : AF^\ or, in fractional form, -j-^ = ~TW^' 
 
 In order to demonstrate this, we proceed as follows: Draw 
 througli F 'd parallel to DC, cutting BD in F' , and ML produced, 
 in F''-, draw also NF', NF", Then since FF' ^ CN, NF' ie 
 parallel to CD\ and since FF" = NG, NF" is parallel to /7Z 
 Also, since FH, FL, are bisected by J./*, we have ZTA^ = iV/ 
 = KP, = F'D, = LF" ; therefore NF' passes through K, and 
 NF" passes through P. 
 
184 DESCIIIPTIVE GEOMETRY. 
 
 Then from similar triangles GAK, NAP, we liave ~, = ^~ I 
 
 and '' '' " JVAK,OAP," " ^=^\l^,\ 
 
 AG _ AK' 
 •*• AC-AP^' ^•^•■^• 
 
 269a. These hjperboloids, viz., the circular, elliptic, and para- 
 bolic, are the only doubly-ruled warped surfaces, and also the only 
 ones having three rectilinear directrices. Each has two gorge lines, 
 lying in planes perpendicular to each other, and having a common 
 vertex ; and from the preceding it is apparent that if sections be 
 made by planes parallel to and equidistant from either gorge, their 
 projections on the plane of that gorge must coincide, be similar to 
 the gorge curve and similarly placed, and satisfy the condition that 
 any chord of the outer curve, tangent to the inner, shall be bisected 
 at the point of contact. It has been shown that this condition can 
 be satisiied by the conic sections ; and it is not known that it can 
 be satisiied by any other curves. 
 
 THE HELICOID. 
 
 270. \Ye shall use the term helix in the restricted sense in which 
 it is commonly employed ; as designating the path of a point which, 
 w^hile revolving uniformly around an axis, also moves uniformly in 
 a direction parallel thereto. 
 
 This curve, then, lies upon the surface of a cylinder of revolu- 
 tion, cuts all its rectilinear elements at the same angle, and becomes 
 a right line when tlie cylinder is developed into a plane. 
 
 This being understood, we shall use the term Helicoid to desig- 
 nate any surface generated by a right line whose motion is deter- 
 mined by helical directrices lying upon concentric cylinders. 
 
 The right line, thus controlled, must necessarily have a motion 
 of revolution ; and it may either intersect the axis, like the genera- 
 trix of a circular cone, or remain always at a fixed distance from 
 it, like that of an hyperboloid of revolution. 
 
 271. Helicoidsof Uniform Pitch and of Tarying Pitch. If all 
 the helical directrices have the same pitch,- every point of the gen- 
 eratrix will travel, in a direction parallel to the axis, at the same 
 
DESCRIPTIVE GEOMETRY. 
 
 185 
 
 rate ; and that line will either lie in, or make a constant angle 
 with, a plane perpendicular to the axis : the surface is then said to 
 be of uniform pitch. 
 
 But the directrices may be of different pitches ; in which case, 
 the rate of arial advance being different for the various points of 
 the generatrix, that line will continually change its inclination to 
 the transverse plane ; and the surface is then called a helicoid of 
 varying pitch. 
 
 272. Right and Ohlique Helicoids. The hehcoids of uniform 
 pitch may be subdivided into two classes. In Figs. 219 and 220, 
 let the generatrix DE vqyo\yq uniformly around the vertical axis, 
 
 Fig. ^9 
 
 Fig. 220 
 
 while at the same time all its points move uniformly in a direction 
 parallel to the axis; then the point C'will describe tha helix MCN 
 lying on the inner cylinder, and the points D and E will describe 
 helices of the same pitch lying on the outer cylinder ; of which 
 r's't'^ xy'u\ are the vertical projections. In both cases, the gen- 
 eratrix remaining at the fixed distance oc from the axis, will always 
 lie in a plane tangent to the inner cylinder, touch that cylinder in 
 a single point, and cut the vertical element through that point at a 
 constant angle. 
 
 In Fig. 219, the generatrix is jperjpendicular to that element^ 
 
186 DESCRIPTIVE GEOMETRY. 
 
 and therefore parallel to the transverse plane ; that plane, then, h 
 the plane directer of the surface, which is called a right helicoid. 
 The radius oc being arbitrary, may be reduced to zero ; in which 
 case the generatrix intersects the axis, and the helicoid becomes a 
 special form of the right conoid, the axis being the line of striction : 
 this is a most familiar modification in practice, since it is the sur- 
 face of the common square-threaded screw. 
 
 In Fig. 220, the generatrix is inclined to the vertical element 
 of the inner cylinder through the point of contact. The surface is 
 then called an oblique helicoid, and has a cone directer instead of a 
 plane directer : if ^c be reduced to zero in this case, the surface is 
 that of the ordinary triangular or Y-threaded screw. 
 
 273. Representation of the Helicoid. Since each of the points 
 2>, 6^, E\ in the two preceding figures, describes a helix of the 
 same pitch, it is easy to draw as many elements as may be desired. 
 But the mere projection of a number of elements of indefinite 
 length does not ordinarily convey an adequate idea of the form of 
 any surface. This is better done by representing a limited portion, 
 as in Fig. 221 : the conditions here are similar to those in Fig. 220, 
 and the surface is at once recognized as resembling that of the 
 groove in an auger or a twist drill. This figure illustrates the 
 principle already mentioned, that the visible contour is the envelope 
 of all lines of a surface ; it is, accordingly, tangent to the helical 
 paths of the points Jf, AT, etc. , which are used for this determina- 
 tion in preference to the rectilinear elements : the latter would of 
 course have served the same purpose, but practically would have 
 been more confusing and very little if any easier to construct. 
 
 274. To assume a point on the surface. Supposing the axis to 
 be vertical, as shown ; then if the horizontal projection be assumed,, 
 an element can at once be drawn through the point. If the verti- 
 cal projection be assumed, the point must lie on a perpendicular to 
 V through the point ; any plane through this line will cut the ele- 
 ments in points determining a curve, whose liorizontal projection 
 "will cut that of the line, in the required horizontal projection of 
 the point. 
 
 275. Peculiarities of Meridian Sections. If in Fig. 221 the sur- 
 face be cut by a plane through the axis, perpendicular to T, the 
 
DESCRIPTIVE GEOMETRY. 
 
 187 
 
 section will be the curve gfji^ , shown in the profile. This curve 
 is easily constructed by drawing elements of the surface, and finding 
 the poiuts in which they pierce the cutting plane : for example, 
 the element XY pierces that plane at P ; the altitude of jy, is the 
 samt as that of p' ^ and its distance jp^o^ from the axis is equal to po. 
 
 This curve is convex toward the axis, and tangent at its vertex <?, 
 to the vertical element of the inner cylinder. This helicoid, then, 
 surrounds a cylindrical well, within which it cannot extend ; and 
 since all the meridian sections are obviously alike, the surface is 
 tangent to the cylinder all along the helix KCL, which is a true 
 
 In Fig. 222, the generatrix is perpendicular to the element of 
 the inner cylinder through the point of contact, as in Fig. 219-' 
 the diameters of the two cylinders, as well as the helical pitch, are 
 
188 
 
 DESCRIPTIVE GEOMETRY. 
 
 the same as in Fig. 221. This change in the position of the gen- 
 *^ratrix lias caused the helices described by D and E to approach 
 
 each other, so that the breadth g'h! of the groove is less than be- 
 fore. 
 
 Still the form of the meridian section is not greatly changed ; 
 and it is evident upon consideration that if the generatrix be 
 lengthened, thus enlarging the outer cylinder, the curve gcji^ will 
 continue to expand as it recedes from the vertex c, , though less 
 and less rapidly; the curve having two horizontal asymptotes 
 through Z, , ^j . 
 
 276. In Fig. 223, T>CE\% again inclined, but in the opposite 
 
DESCRIPTIVE GEOMETRY. 
 
 189 
 
 direction ; and to such an extent that D now lies upon the lower 
 and E upon the higher of the two helices on the outer cylinder. 
 
 The result is that the meridian section is a looped curve, crossing 
 itself at a point u^ between the two cylinders. Thus this helicoid 
 
190 DESCRIPTIVE GEOMETRY. 
 
 encloses not only the cylindrical wall, to which it is tangent along 
 
 Fig. 224 a 
 
 the gorge helix KCL^ but also a serpentine void whose transverse 
 section is of the form of the loop xi^w^c^z^ . ' 
 
 In these circumstances, a groove of the form li{a^g^ can be cut 
 
DESCRIPTIVE GEOMETRY. 191 
 
 in the outer cylinder, bounded by helicoidal surfaces of which DE 
 is tlic generatrix : but the actual formation of the surface below 
 the point ?/, is impracticable. 
 
 277. In Fig. 223, the angle d'c'Ii'^ between i>^ and the verti- 
 cal element of the inner cylinder through C^ is nevertheless greater 
 than that between this element and the tangent to the helix KCL. 
 ]S^o\v in Fig. 224, these angles are equal ; in other words, the gen- 
 eratri x DE is tangent at G to the helix : the surface, although an 
 oblique helicoid, is now developable, being in fact the helical con- 
 volute previously discussed. 
 
 In the meridian section, the loop has now disappeared, u^ hav- 
 ing retreated to c, , at which point the curve g^cji^ forms a cusjpy 
 the tangent to this curve at this point, moreover, is horizontal in- 
 stead of vertical, and the helix KCL is no longer a gorge line but an 
 edge of regression ; along which the helicoid intersects the cylinder, 
 to which it is not tangent, as has been erroneously stated. 
 
 Finally, in Fig. 225 the angle dldh' is less than the angle be- 
 tween the tangent to KCL and the vertical element through C, 
 We now have again as the meridian section a curve whose tangent 
 at its vertex <?, is an element of the inner cylinder ; and as before 
 the helix Is^CL is a gorge line, along which the helicoid is tangent 
 to that cylinder. 
 
 278. Peculiarities of TransTerse Sections. The transverse sec- 
 tion of any helicoid may be determined in the ordinary manner, by 
 finding the points in which any number of elements pierce the cut- 
 ting plane. But though perfectly correct in theory, this method 
 is in this case, as in many others, practically more laborious and far 
 less reliable than that of constructing the curve by the aid of other 
 known properties. 
 
 Tims, let it be required to draw the transverse section by a 
 plane through (7, in Fig. 224. The axis being vertical, this plane 
 is horizontal, and tlie required section TcT' is no other than the 
 horizontal trace of the surface ; and this (128) is the involute of the 
 circular base of the cylinder upon which the helix KCL is traced. 
 Drawing tangents at 5, Z, «, etc. , set off upon them hp — arc cb^ 
 Im = arc cl, and so on ; the curve passing through the points thus 
 located is the required section : the distance rs^ set off upon the 
 
192 
 
 DESCRIPTIVE GEOMETRY. 
 
 most remote tangent, is in this case equal, of conrse, to the semi- 
 circumference of the base. 
 
 N. B. Tliis curve forms a cusp at C^ as did the meridian section ; 
 and so will the section of this particular surface by any other plane 
 
 Fig, 225 
 
 through the same point, with the exception of those planes which 
 pass through the generatrix DE. 
 
 279. The transverse section of the surface shown in Fig. 223 
 will not be a true involute, because the generatrix is not tangent to 
 the helix KCL ; but it may be constructed in a manner analogous 
 to that above described. In Fig. 223a, let c'l'r' be a half turn of 
 the helix ; draw through c' a horizontal line c 8 ^ and through r' 
 the vertical projection of an element of the helicoid; then o'^' is 
 evidently the distance to be set off as rs on the most remote tan- 
 
DESCRIPTIVE GEOMETRY. 193 
 
 gent. The motion of tlie generatrix consists of a uniform rotation, 
 combined with a unifoi-m axial advance; consequently, dividing 
 c's' in Fig. 223<2 into equal parts by the points 1, 2, 3, and the 
 8emi- circumference clr into the same number of equal parts, we 
 draw tangents at the points of subdivision 5, Z, a, and set off upon 
 them the distances hi = c'l, 12 — c'2, a^ — c'Z : the curve passing 
 tlirough the points 1, 2, 3, s, is the section required. As many 
 intermediate points as may be deemed necessary can be located in 
 like manner; and the resulting curve in this case will be looped, 
 crossing itself at u, the distance uo being equal to u^o^ in the me- 
 ridian section. 
 
 280. The formation of this loop in Fig. 223 is obviouslj due 
 to the fact that the distances hi, Z2, etc., set off on the tangents, 
 are greater than the arcs ch, d, etc. In Fig. 225, on the other 
 hand, these distances, determined in the same manner, are less 
 than the corresponding arcs. The result is that the curve TcT has 
 a wave instead of a loop : and ^t is observed that the meridian 
 section also possesses this distinguishing feature. 
 
 In thus passing from the loop of Fig. 223 to the cusp of Fig. 
 224 and the wave of Fig. 225, the angle d'c'h^ has been pro- 
 gressively diminished. If on the other hand that argle be in- 
 creased, the loop will become larger, the curvature at c growing 
 less and less, until when the angle reaches the limit of 90°, as in 
 Fig. 222, there will be no curvature at all, and the transverse sec- 
 tion will be simply the generatrix D^ itself. If this limit be 
 passed and the angle d'c'h' made obtuse, as in Fig. 221, the section 
 TcT, still tangent at to that of the inner cylinder, will curve in 
 the opposite direction. 
 
 281. Intersections of the Oblique Helicoid with Itself. The 
 plane PP, in Fig. 224«^, is tangent to the inner cylinder, upon 
 which lies the helical directrix of the developable helicoid repre- 
 sented by this figure. This plane cuts the surface in the genera 
 ivix UPS, and also in the curve WPT, as shown in the side view. 
 After one revolution, ^-^S' will be found at rs, after another at r's\ 
 and so on ; and in these new positions will cut the curve PT in 
 the points 5, s\ s'\ etc. The point s describes a helix xx, lying on 
 a cylinder of which aa is an element ; the point 6'' traces the helix 
 
194 DESCRIPTIVE GEOMETRY. 
 
 yy^ on a cylinder of wliicli JA) is an element ; tlie jDoint s" in like 
 manner traces a helix, not shown, on a cylinder of which cc is an 
 element — and so on indefinitely ; and these helices, all having the 
 same pitch as the original directrix on the inner cylinder, are evi- 
 dently the lines of intersection between the successive convolutions 
 of the surface. 
 
 And it is obvious that as shown in the figure a screw may be 
 cut, the crest of the thread being the helix xx^ and its root being 
 the directrix : or a larger one of the same pitch, the thread having 
 yy for its crest and xx for its root ; these screws will be Hivgle- 
 threaded^ and their surfaces are portions of the same helicoid. 
 
 Fig. 224, on the other hand, represents a doidjle-threaded 
 screw; as seen in the side view, two helices, k'c'l\ f'q't\ are 
 traced on the inner cylinder, to wdiich the two generatrices, d'c'e\ 
 f''it\ are respectively tangent : these lines thus generate two dis- 
 tinct helicoids, which intersect each other in the helices d'h'u\ 
 v'g'e\ forming the crests of the threads. 
 
 282. That the surface of a warped helicoid will also intersect 
 itself in a series of helices is shown in Fig. 225 ; a plane through 
 DE^ parallel to the axis, cuts the surface in the curve [fc'FJ^ 
 which, however, is not tangent to the generatrix, but crosses it at 
 the point C. After one revolution, this generatrix will have the 
 position rs^ cutting the curve at s^ which point will describe a helix 
 corresponding to xx in the preceding figure ; and successive ones 
 may be determined in the manner above explained. 
 
 In the case of the right helicoid, however, the plane tangent to 
 the inner cylinder cuts the surface in the generatrix only ; and 
 since this is parallel to its position after any given number of com- 
 plete revolutions, this surface does not intersect itself like the 
 others, no matter how far extended. 
 
 283. Special Case of the Right Helicoid. In Fig. 226, the gen 
 eratrix intersects the axis at right angles, and the rotation being 
 in the direction indicated by the arrow in the end view, the point 
 d describes the helix def^ while g describes the helix ghh ; any other 
 points of the generatrix, as x, s, will describe lielices of different 
 obliquities, since they lie upon cylinders af different diameters, but 
 because they all have the same pitch, these helices will in the side 
 
DESCRIPTIVE GEOMETRY. 
 
 195 
 
 view intersect in the points s, s\ etc. , lying on the projection of 
 the axis. This figure represents an ideal single- threaded screw, 
 
 Fig. 226 
 
 Fig. 228 
 
 which may be conceived as being formed by uniformly twisting a 
 thin rectangular strip of flexible metal about its longitudinal centre 
 line. Suppose two such strips to be fastened together at right 
 
196 DESCRIPTIVE GEOMETRY. 
 
 angles, presenting when viewed endwise the form of the cross, dg^ 
 mn, Fig. 227 ; let each be iiniformlj twisted in the same direction 
 and at the same rate as in the preceding figure : then 711 will 
 describe the helix mlq^ while n describes the helix nj[)r^ and the 
 result will be the formation of the surface of an ideal double- 
 threaded screw. The practical application is shown in Fig. 228, 
 in the construction of a single-threaded screw, cut upon a cylinder 
 whose radius is od^ the depth of the groove being limited by the 
 central ' ' core, ' ' who^e radius is ox ; upon the surface of this inner 
 cylinder lie the hehces corresponding to xs^ ss, in Fig. 226. 
 
 284. Special Case of the Oblique Helicoid. In Fig. 229, the 
 generatrix DCJ^ cuts the axis acutely at C; the helix described by 
 the point C therefore coincides with the axis itself. Let c'm' be 
 one half the pitch, then at the end of a half revolution the genera- 
 trix will have the position e'-mlf ^ which intersects the first position 
 in E\ the path of this point, then, is the first of the helices in 
 which the generated surface intersects itself. Set up 7}i r' equal to 
 the pitch, then at the end of a revolution and a half the generatrix 
 will have the position r's' ^ which cuts d'e' produced in 8\ whose 
 path s'xm' is the second of the series of helices, as explained in 
 (281). 
 
 The visible contour, as in all other cases, is tangent to the pro- 
 jections of all lines of the surface which it intersects (except those 
 whose rectilinear tangents at these points of intersection are per- 
 pendicular to the plane of projection); thus, it is tangent to the 
 helix d'c'e\ and at V and ^', to the path of the point K: again, 
 uu^ ww^ are the horizontal projections of two elements, and the 
 contour is tangent at u' and w' to their vertical projections. It is 
 also obviously tangent to the axis at a\ the middle point of c'm' ; 
 this therefore is the vertex of the curve, which is symmetrical with 
 respect to the horizontal line through a' ^ e' . The contour line 
 through o\ when produced, as o'q'^ will also be tangent to the 
 helix mix's' ; it is evidently asymptotic to d'c's'^ and closely resem- 
 bles the hyperbola having the same vertex and asymptotes ; but the 
 latter, as shown by the broken line o'jp' ^ lies within the curve under 
 consideration. 
 
DESCKlillVE GEOMEIKY. 
 
 1^7 
 
198 DESCRIPTIVE GEOMETRY. 
 
 285. The horizontal trace of this helicoid is an Archimedean 
 spiral. If c'd' be supposed to rotate without advaneing, it will 
 after half a revolution occupy the position c'y' ; Imt it does mean- 
 time actually advance to the position me' parallel to cy\ and when 
 produced pierces H at 6^ : and since c'm' = 2o'c', we will have 
 y'g' — 'Ic'y' ^— d'y' ^ — de. Since the revolution and the axial 
 advance c:re both uniform, the trace is, as stated, an equable spiral, 
 of which y'g\ or eg^ is the radial expansion in a half revolution. 
 This should be carefully constructed, by dividing eg and the semi- 
 circumference dwe into proportional parts, and setting oft' the dis- 
 tances cl, c2, etc., on the corresponding radiants: by this means, 
 the horizontal traces of any given elements can be more accurately 
 located than in any other way. 
 
 If the surface be limited by the second helix of intersection. 
 m'x's\ a section by a transverse plane through y**' will be bounded 
 by two synmietrical curves, of which one is the portion dzg of this 
 trace. If the helicoid be limited by the iirst helix d'ce'f\ the 
 section by the same plane, or by H itself, will be the shaded loop 
 of the spiral : the meridian section by a plane parallel to V in this 
 case consisting merely of a series of isosceles triangles whose bases 
 coincide with the axis, as shown on a reduced scale in the small 
 detached diagram. 
 
 286. Practical Application. This helicoid, as previously stated, 
 forms the surface of the Y-threaded screw. It is quite obvious 
 that such a screw^ may be formed by taking, in Fig. 229, the helix 
 m'x's for the crest, and the helix d'c'e' for the root, of the thread ; 
 both sides of which will then be formed of portions of the same 
 helicoid. But it is not essential that this should be so ; and in the 
 majority of practical cases the opposite sides of the thread are por- 
 tions of two different helicoids. Thus in Fig. 230, the screw may 
 be regarded as being formed by winding a bar of flexible metal, 
 having the triangular section e7no^ around the central core; the 
 pitch trin being equal to eo. Prolonging me^ quo, to cut the axis 
 in c and d^ it will be seen that if cd be 1^ times the pitch, it will 
 correspond to c'r' of the preceding figure, md being then simply 
 another position of mc. This, however, is not the case ; we have 
 here two independent lines each generating the helicoidal surface 
 
DESCRIPTIVE GEOMETRY. 
 
 199 
 
 of one side of the thread. These lines are usually, as in this illus- 
 tration, equally inclined to the axis ; which, nevertheless, is by no 
 means essential, since the section emo of the flexible bar might 
 have been a right-angled, or a scalene, instead of an isosceles 
 
 Fig. 231 
 
 triangle; and such non-symmetrical screw-threads are sometimes 
 used in practice. 
 
 If the pitch be doubled, it is clear that another thread may be 
 formed in the intervening space ; if it be trebled, two threads can 
 be added ; and so on indefinitely. 
 
 For the sake of uniformity the helices have been drawn right- 
 
200 DESCRIPTIVE GEOMETRY. 
 
 handed throughout ; it need hardly be stated that they might have 
 been made left-handed without in any way affecting the argument : 
 and in practice they often are, in the construction of ordinary 
 screws as well as of screw-propellers. 
 
 287. Helicoids of Yarying Pitch. If in Fig. 231 the line DC, 
 which is perpendicular to die vertical axis, were to rise in a half 
 revolution to the positiony/',- rotating and advancing uniformly, 
 it would generate a right helicoid, the point D tracing the helix 
 DGF. Now let the line rotate uniformly, in contact with this 
 helix and with the axis, the point C at the same time advancing 
 uniformly, but less rapidly than before, so that in a half revolution 
 it reaches the altitude ef instead of l\ By this motion a different 
 surface w^ill be generated, which is a helicoid of varying pitch : 
 since e'f is longer than c'd', it is evident that the point Z> does 
 not describe the helix DGF^ but since (7 remains in contact with 
 the axis, there is a sliding of the generatrix upon the guide helix 
 during the motion supposed. 
 
 Now draw the inner cylinder, with any radius oc at pleasure ; 
 the generatrix in its first position pierces this cylinder at 0^ and 
 after a half revolution, pierces it at P. 
 
 Set off the arc dg^ say one-third of the semi- circumference, and 
 araw gc^ the horizontal projection of the generatrix in an inter- 
 mediate position ; project g to g' on the outer helix ; the altitude 
 h'g' will be one-third of s'f : set up c'ti = -J c'e\ then g'h' is the 
 vertical projection of this line, which pierces the inner cylinder at 
 
 JV, 
 
 It is obvious that the altitude m'n' will be one-third of r'j?^ ; 
 and since the same argument applies to any other position of the 
 generatrix, it follows that the inner cylinder cuts the helicoid in a 
 true helix ONP, and will do so whatever its radhis may be, the 
 pitch, evidently, varying with the radius : and the same applies to 
 exterior cylinders as well. 
 
 288. In Fig. 232, the generatrix DC \w its first position lies in 
 H ; let DBF be the guide helix, and FF the position of the gen- 
 eratrix after a half revolution ; let mc, nc be the horizontal pro- 
 jections of intermediate positions; if these be revolved about 
 the axis until parallel to V, their outer extremities will appear as 
 
DESCRIPTIVE GEOMETRY. 201 
 
 m\ 7i\ dividing s'f in the same projjortion in wliicli their inner 
 extremities divide c'e. These lines will therefore if prolonged 
 pierce H in the same point ; as also will k'l \ the position of the 
 generatrix after a revolution and a half — from which it follows that 
 all the elements of this surface pierce H at the same distance from 
 the axis. In other words the horizontal trace of the surface con- 
 sists of the circle whose radius is eo, and also of an indefinite right 
 line coincidino; with Z^6\ the element which lies in the horizontal 
 plane. 
 
 289. The transverse section by any other plane as PP will be 
 a spiral, wux, of which the points u and w are obtained directly, 
 since the elements g7i\ Vh\ in the vertical projection pierce this 
 plane at u' ^ w' . To find other points, draw intermediate elements, 
 as through m and n in the horizontal projection ; these when re- 
 volved until parallel to V will pass through in" ^ n'\ points dividing 
 h'h" into parts proportional to the divisions of g'V^ and will be cut 
 by PP at points whose distances from the axis are to be set off on 
 cm^ en : and in like manner any number of points may be deter- 
 mined, and the curve extended in either direction. This curve is 
 peculiar in possessing a circular as well as a rectilinear asymptote ; 
 if continued in the direction uw^ it will pass through the pole c, 
 and then again expanding at a decreasing rate, it will after an in- 
 finite number of turns be tangent internally to the circle oqt : if 
 continued in the opposite direction, it will be tangent at infinity to 
 a line parallel to ed, and lying at a distance from it equal to the 
 circumference of the circular asymptote. 
 
 290. In Fig. 233, the generatrix PC in its first position also 
 lies in H, but at a distance cic from the axis ; DKF is a helix traced 
 on the cylinder whose radius is ud. Now let DC revolve uniformly 
 about the axis, in contact with this helix, the point C at the same- 
 time moving uniformly along the element of contact with the cyl- 
 inder whose radius is cu^ so that after a half revolution the line 
 occupies the position FE. By reasoning similar to that used in 
 connection with Fig. 231, it can be shown that the surface thus 
 generated will intersect any cylinder having the same axis in a true 
 helix; this form of the helicoid is, then, the general one, of whicl^ 
 that above discussed is a special case. 
 
202 
 
 DESCRIPTIVE GEOMETRY. 
 
 Fig. 233 
 
 1 ^ 
 
 p 
 
 / >~X / 'A 
 
 ""'' 
 
 
 L 
 
 
 
 , 
 
 
 
 \ 
 
 \ "1 
 
 ~c " y 
 
 
 
 
 / \ / /\. V 
 
 \ 
 
 
 \ // 1 A 1 ; 
 
 
 ( If 
 
 \ 
 
 1 w \ / J <^ 
 
 '■■\ 
 
 /// 
 
 M 
 
 'j^ yfCj 
 
 i^- 
 
 TV 
 
 ]■ ( 
 
 
 ^.Qi 
 
 -==>_. 
 
 r 
 
 ^1 
 
 
 
 
 
 Fig. 234 
 
DESCRIPTIVE GEOMETRY. 203 
 
 Drawing mj?, nr^ tangent to tlie circle whose radius is uc^ and 
 revolving the elements of which these are the horizontal projections 
 nntil thej are parallel to T, it mav be shown as in Fig. 231 that they 
 will then pierce H in the same point O : showing that the horizontal 
 trace of this surface consists of the circle whose radius is uo^ and 
 also of a right line coinciding with DC^ the original position of the 
 generatrix. 
 
 291. Practical Application. AVhen the generatrix intersects 
 the axis, the helicoid of varying pitch is often practically employed, 
 forming the acting surfaces of screw-propellers with what is techni- 
 cally called ' ' radially increasing pitch ' ' ; which are swept up, of 
 course, by that portion only of the generatrix which lies on one side 
 of the axis. 
 
 To aid in gaining a clear idea of the nature of this surface, 
 there is represented in Fig. 234 a portion of it generated by a line 
 of definite length JS/Z; the pitch at that distance from the axis 
 being LP^ PE^ while the pitch at the axis itself is MC^ CD. The 
 generatrix ML being perpendicular to the axis, will after successive 
 revolutions take the positions CEP^ DFR^ etc. ; the point L thus 
 tracing a ^w<a^6- /-helical curve LIEK^ Iji^g on a surface of revolu- 
 tion whose meridian line is LEFG\ since, as shown in (288), 
 7?Z>, PC^ etc., when produced w^ill meet in the same 23oint on 
 the prolongation of LM^ the outline of this surface of revolution is 
 tlie waved branch of a conchoid, of w^hich the pole is and the 
 directrix is NN the axis of the helicoid. . In order to throw the 
 surface into stronger relief, a concentric cylinder is introduced, 
 which, as before shown, cuts it in a true helix, of which portions 
 are visible. 
 
 292. From this it is clearly seen that the surface is divided into 
 two symmetrical parts by the median plane LL^ which contains 
 the element perpendicular to the axis. In the immediate neighbor- 
 hood of this plane the surface resembles the right helicoid, while at 
 sensible distances from it there is a greater similarity to the oblique 
 helicoid. And it is specially to be noted, that the generatrices in- 
 cline in opposite directions on the two sides of this plane ; so that 
 in the construction of a propeller it does not suffice to giye merely the 
 pitches at the rim and the hub respectively ; since, while the form 
 
204 DESCRIPTIVE GEOMETRY. 
 
 of the surface would thus be definitely fixed, the particular pprt of 
 it to be used would not l)e located : it is necessary therefore to give 
 in addition the precise inclination to the axis of some specified rec- 
 tilinear element of the proposed blade. 
 
 THE CYLINDEOID. 
 
 293. The Cjlindroid is a warped surface with a plane directer, 
 and is derived from the cylinder in a manner which will be readily 
 understood by the aid of the pictorial representation, Fig. 235. On 
 the left is shown the half of a circular cylinder, with its axis in II 
 and its elements parallel to V ; and this semi-cylinder is cut obliquely 
 by two vertical planes, forming the sections erd^ msn : the elements 
 cut the outlines of these sections in the corresponding points 1 1 , 
 2 2, etc. 
 
 Now, the section erd remaining fixed, let the other section be 
 moved upward by translation in its own plane through any given 
 distance; the diameter mon will then, as shown on the right, be 
 vertically over and parallel to its orginal position, here represented 
 by m,6»,7i, : moreover, the relative positions of the points 1, 2, 3, 
 upon the arc ns^ will be the same as before. 
 
 ]S"ext, joinmg these points with the corresponding points 1, 2, 
 3, upon the arc di\ the new lines 11, 2 2, etc., will be the rectilinear 
 elements of the surface under consideration. By construction these 
 elements are parallel to T, which therefore is the plane directer in 
 the case here illustrated. 
 
 The method of representing this surface in projection is too 
 obvious from the above to require further explanation ; nor does 
 the surface itself possess any remarkable features, with the excep- 
 tion that the limiting tangent planes, as LL for instance, are tan- 
 gent all along the elements which they contain. 
 
 294. Practical Application. The cylindroid may be used to 
 form the roof of a transverse gallery connecting two parallel arched 
 passages on different levels. 
 
 The floor of such a gallery, if constructed on the same principle, 
 will also be a warped surface ; it is evident that in the right-hand 
 figure, mn^ de^ are the directrices, and nd^ me^ are two elements, 
 
DESCRIPTIVE GEOMETRY 
 
 205 
 
206 DESCRIPTIVE GEOMETRY. 
 
 of a hyperbolic paraboloid, wliick lias H for one plane directer and 
 '*'' for the other. 
 
 Ill the preceding illustration, the circular cylinder was selected 
 merely for convenience ; it is clear that a similar process may be 
 employed, whether the roof of the original arch be circular, ellipti- 
 cal, or of any other section. 
 
 THE cow's HORN. 
 
 295. This is a warped surface liaying three directrices, viz., a 
 right line, and two circles in parallel planes : a plane perpendicular 
 to the latter contains the centres of both circles and also the recti- 
 linear generatrix. 
 
 In Fig. 236, the circles onrn^ hsd, are of equal diameters and 
 lie in vertical planes, to which the rectilinear directrix 2)0 is per- 
 pendicular ; also, the centres, c and e, are on opposite sides of po 
 and equidistant from it. Under these conditions tlie surface is 
 syjiimetrically divided by the vertical plane through ^>>6> ; and has a 
 practical application in the construction of the warped arch. 
 
 Any plane through po, evidently, will cut the planes of the 
 circles in parallel lines, as ok, pi, thus determining an element Ik 
 of the surface. The element mb cuts po in a? ; gk cuts it in w, 
 farther from o ; Ik would cut it in a point still more remote, and 
 so on, until, when the cutting plane becomes vertical, pr and os 
 being equal under the assumed conditions, rs is parallel to po. 
 The elements beyond rs will then intersect op on the opposite side 
 of the vertical plane ; dn, evidently, cutting it at a point y, mak- 
 ingpy equal to ox. 
 
 in Fig. 237, none of the above special conditions are imposed ; 
 the two circles are of different diameters, the directrix po is not 
 perpendicular to their planes, and the centres c and e lie on the 
 same side of po, but at unequal distances from it ; moreover these 
 distances are so chosen that the points p and o do not divide tlie 
 radii en, ed, in the same proportion. This figure, then, represents 
 a general case of the surface, of which the warped arch is only a 
 special form. All the elements now cut the rectilinear dii-ectrix in 
 ironfc of the vertical plane, and at finite distances, since no one of 
 them is parallel to it. 
 
DESCRIPTIVE GEOMETRY. 20? 
 
 296. The obvious nse of sucli surfaces is in forming the roofs 
 of arched passages ; which naturally leads to the selection of circles, 
 ellipses, or other symmetrical curves, lying in parallel planes, for 
 the curved directrices. The construction of the roof requires the 
 use of only one-half of each of these curves, which accordingly is 
 all that is i^hown in the figures : if we suppose the other half to be 
 added, It is evident that the plane containing the centres of these 
 curves and the rectilinear directrix, will divide the complete sur- 
 face symmetrically, and in general no other plane will do so. The 
 above pictorial representations show not only the nature of the sur- 
 face, but the method of determining its elements, more clearly than 
 would its projections, which can readily be drawn without further 
 explanation. Since no plane can be tangent to such a surface along 
 an element, the visible contour will in all cases be a curve, though 
 sometimes a very fiat one. 
 
 Substantially the same method would be employed were the 
 curved directrices in planes not parallel to each other, not similar 
 to each other, or even were they of double curvature ; and indeed 
 it may be said that the Cow's Horn is only a special variety of a 
 general class of warped surfaces, having one rectilinear directrix and 
 two curved ones of any kind whatever. 
 
 WARPED SURFACES GENERAL FORMS. 
 
 297. In addition to the preceding, warped surfaces having no 
 specific names are sometimes met with in practical operations. 
 These must necessarily be determined either by two curved direc- 
 trices and a plane director, or by three curvilinear directrices ; and 
 in representing them, three problems may arise. If there be a 
 plane directer, it may be required to draw an element either 
 parallel to a given line therein, or through a given point on one of 
 the directrices ; if there be none, it may be required to draw an 
 element through a given point upon either directrix. We will 
 consider these problems in the order given. 
 
 298. I. In Fig. 238, let CD, EF, be the directrices of a 
 warped surface ; it is required to draw an element parallel to the 
 line jollying in the plane directer tTt' . 
 
 Analysis. Pass through either directrix a cylinder whose ele^ 
 
208 
 
 DESCKIPTIVE GEOMETRY. 
 
 ments are parallel to the given line. The other directrix will 
 pierce this cylinder in one or more points, through either of which 
 an element of the surface may be drawn parallel to the given line. 
 
 Construction. Through any points G^ 11^ K^ etc. , on (7i>, draw 
 parallels to MN\ these lines pierce the vertical projecting cylinder 
 of EF in the points 7?, 0^ Z, etc. , thus determining a curve verti- 
 cally projected in €'f\ and horizontally projected in e^f^. This 
 curve is the intersection of the two cylinders, and cuts EF in the 
 point Z7, through which is drawn the required element ZZZ, paral- 
 lel to M]S\ 
 
 299. II. In Fig. 239, let OD, EF, be the directrices, tTt' the 
 
 Fig. 238 
 
 plane directer ; it is required to draw an element of the warped 
 surface, through the point O on CD, 
 
 Analysis. Pass through the given point a plane parallel to the 
 plane directer; it will cut the other directrix in a point of the 
 required element. 
 
 Construction. Assume in tTt' any point P and also any line 
 JO^, and join P by right lines to any points B, /S, etc. , upon MJV, 
 Through O draw parallels to PP, PS, etc. ; this series of lines 
 determines a plane parallel to tTt'. This parallel plane cuts the 
 horizontal projecting cylinder of EF in a curve whose projections? 
 are ef, e^f^ ; and this curve cuts EF in X, thus determining OX^ 
 the element required. 
 
DESCRIPTIVE GEOMETRY. 
 
 209 
 
 300. III. In Fig. 240, MN, CD, EF, are the directrices of a 
 warped surface; it is required to draw a rectilinear element 
 through the point O on MN. 
 
 Fig. 239 
 
 Analysis. Pass through either of the other directrices a cone 
 of which the given point is the vertex. The third directrix will 
 pierce this cone in one or more points, through either of w^hich 
 and the given point an element of the surface may be drawn. 
 
 /A 
 
 FiQ. 240 
 
 Construction. Through any points G, R, K, etc. , on CD, draw 
 lines from O ; these are elements of the cone, and pierce the hori- 
 zontal projecting cylinder of EF in the points H, jS, Z, etc : thus 
 
210 DESCRIPTIVE GEOMETRY. 
 
 determining a curve, of wliicli the horizontal projection is ef and 
 the vertical is e„f^. This curve is the intersection of the cone 
 with the projecting cylinder, and cuts EF in X, a point of the re- 
 quired element OX. 
 
 PLANES TANGENT TO WARPED SURFACES. 
 
 303.. A plane tangent to any warped surface at a given point 
 may be constnicted by the general rule of (140), viz. : Draw 
 through the point any two intersecting lines of the surface, and at 
 the point a tangent to each line ; these tangents determine the re- 
 quired plane. 
 
 The rectilinear element through the given point is one line of 
 the tangent plane, in all cases ; and if the surface be doubly ruled, 
 the plane is at once determined by drawing through the point an 
 element of each generation (142) ; as has already been illustrated 
 in the cases of the hyperbolic paraboloid (236), the hyperboloid of 
 revolution (250), and the elliptical hyperboloid (260). 
 
 If there be only one set of rectilinear elements, that curve of 
 the surface should be selected to which the tangent can most read- 
 ily be drawn : the helicoid affords a good illustration, since the 
 tangent to the helix is easily determined. 
 
 302. Problem 1. To draw a j^lane tangent to an oblique heli- 
 coid at a given point. 
 
 Construction. In Fig. 241, let 2>(7, parallel to V, be the gen- 
 eratrix, and DFE the helix traced by D : to draw a plane tangent 
 to the surface at the point P upon this helix. 
 
 Since the generatrix cuts the axis, set up c'g' = p'y\ then g'p' 
 is the vertical projection of the element through P, whose traces 
 are and S. Since tho given helix pierces H at Z>, makej?/', per- 
 pendicular to cp^ equal to the arcj9(i; then rpm is the horizontal, 
 and r'p'w! is the vertical, projection of the tangent at P to the 
 helix. It is the horizontal trace of this tangent ; therefore tovT is 
 the horizontal, and Ts't' is the vertical, trace of the required tan- 
 gent plane. 
 
 Note. The horizontal trace of this surface is the Archimedean 
 spiral hodl.^ constructed as in Fig. 229 ; and it is to be particularly 
 observed that tT is not tangent to this trace, because the plane is 
 
DESCRIPTIVE GEOMETRY. 
 
 211 
 
 tangent to the surface only at the point P^ and not along an ele- 
 ment. The horizontal trace of a plane tangent to the helicoid at 
 
 Fig. 241a 
 
 2>, or any other point in H, of course would be tangent to this 
 spiral. 
 
^18 DESCRIPTIVE GEOMETRY. 
 
 303. Problem 2. To find the jj^int of tangency hetween a 
 given ohliqii^e helicoid and a jplane containing a given element 
 thereof. 
 
 Analysis. Since the plane is not parallel to the other elements, 
 it will cut each of them in a point ; tlie curve passing through the 
 points thus found will cut the given element in tlie required point. 
 
 Construction. In Fig. 242, let DC parallel to V, be the gen- 
 eratrix of the helicoid, whose pitch is also given. Let GO be the 
 given element of the surface thus determined, and tTt' the given 
 plane. 
 
 Having drawn the horizontal trace of the surface, hodl^ as 
 before, draw <?m, c?i, etc., the horizontal projections of elements 
 on each side of GO. The points in which these elements pierce 
 tTt' can be most readily found by means of a supplementary pro- 
 jection on a plane zu'z'\^ perpendicular to tT. In that projection, 
 the intersection of this plane with the given plane appears as tlie 
 line 2,s/, the axis as c,<?, perpendicular to IIH., and the points 
 ^7^, 71, /*, are projected at 7n^ , n^ , 'i\ , respectively. The rate of 
 axial advance being known, the supplementary projections of the 
 elements through m,, t?,, etc., are readily determined; they cut s.s/ 
 at ^«?l , ^, , y, , whence they are projected back to %o on cm, e on en., 
 etc., thus determining the curve wf^ which cuts co at^, the hori- 
 zontal projection of the point sought. 
 
 304. Problem 3. To draw a jplane tangent to an oblique heli- 
 coid^ and perjpendicular to a given right line. 
 
 Analysis. Construct the cone directer of the surface, and draw 
 through its vei'tex an auxiliary plane perpendicular to the given 
 line. If the problem be possible, this plane will in general cut 
 from the ronfe two elements; these are respectively parallel to 
 those elements of the helicoid, through either of which a plane 
 may be drawn parallel to the auxiliary plane and tangent at some 
 point to the surface. The point of contact is found as in the pre- 
 ceding problem. 
 
 Construction. In Fig. 241a, 6^ is the vertex and DYXh the 
 base of the cone directer, LL is the given line, sSs is the auxiliary 
 plane perpendicular to it, and cp^ en, are the horizontal projections 
 of the two elements : had the plane been tangent to the cone, it is 
 
DESCRIPTIVE GEOMETRY. 
 
 213 
 
214 DESCRIPTIVE GEOMETRY. 
 
 evident that onlj tlie elements of the helicoid parallel to the line of 
 contact, could have planes drawn through them which would satisfy 
 the assigned condition. 
 
 Note. The cone directer here drawn, is that of the helicoid 
 represented in Figs. 241 and 242 ; in each of which also the tan- 
 gent plane is parallel to s/Ss' of this diagram, and contains the same 
 element parallel to CP ; which facilitates a comparison of the con- 
 structions. 
 
 305. The same general method, evidently, may be employed 
 in dealing with any warped surface having a cone directer, in re- 
 gard to which the preceding problem may be proposed. In rela- 
 tion to the hyperboloid of revolution and the elHptical hyperboloid, 
 it is to be noted that since they are doubly ruled, the two elements 
 parallel to those cut from the cone directer at once determine the 
 tangent plane, and their intersection determines the point of con- 
 tact. If the auxiliary plane be tangent to the cone directer, 
 the two elements of the surface parallel to the line of contact, 
 will in the transverse section be tangent to the gorge curve 
 on opposite sides, and the point of contact will be infinitely 
 remote. 
 
 306. Problem 4. To draw a plane tangent to a liyperholiG 
 jparaholoid^ and perpendicular to a given rigid line. 
 
 Argument. This may be accomplished by the following series 
 of operations, viz. : 
 
 1. Draw the plane directors and find their line of intersection. 
 
 2. Draw a plane perpendicular to the gi^en line. 
 
 3. Find the intersection of this plane with each plane directer. 
 
 4. Find an element of the surface parallel to each of these 
 lines of intersection (226). 
 
 These elements will be of diiferent generations, w^ll determine 
 a plane perpendicular to the given line, and will intersect in the 
 point of contact. 
 
 307. Problem 5. To draw a plane tangent to any icarped 
 surface^ and perpendicular to a given rigid line. 
 
 Preliminary, In Fig. 243, let ZZ be the given line, 6'aS^' the 
 plane perpendicular to it. Through any point P in this plane, 
 draw JO^ parallel to the horizontal trace, and 7** 6^ parallel to the 
 
DESCRIPTIVE GEOMETRY. 215 
 
 vertical trace. The following is then applicable to any warped 
 surface whatever. 
 
 Argument. 1. Draw a series of sections of the given surface by 
 horizontal planes; draw a tangent to each section, parallel to MN^ 
 and find the point of contact. The line joining these points is the 
 curve of contact between tlie warped surface and a cylinder whose 
 elements are parallel to MN. 
 
 2. Draw a series of sections of the given surface by planes 
 parallel to V; draw a tangent to each section parallel to liO, find 
 the point of contact, and draw a curve through the points thus 
 found ; this is the line of contact with a second cylinder whose ele- 
 ments are parallel to BO. 
 
 3. The two curves thus determined wdll intersect in the re- 
 quired point of contact between the given surface and a plane per- 
 pendicular to the given line : which plane will of course contain 
 the rectilinear elements of the surface which pass through the 
 point. 
 
 Note, The lines MW^ RO^ have in the above argument been 
 mrxle parallel to H and V respectively, for convenience only ; it is 
 clear that any other lines in sSs' might have been used as well, the 
 elements of the two tangent cylinders being drawn parallel to them : 
 but in general the execution would be more laborious. 
 
 308. Problem 6. To draw a plane tangent to any vmrped 
 surface.^ through a given right line. 
 
 Argument. Let the given line be produced until it pierces the 
 surface ; then the rectilinear element through the point of penetra- 
 tion, and the given line itself, determine a plane which in general 
 wdll be tangent to the surface at some point of the element. 
 
 If two elements pass through the point, each will determine a 
 tangent plane; so again, if the given line pierce the surface in 
 more than one point, there will be more than one tangent plane. 
 If the given line be parallel to a rectilinear element, the point of 
 penetration will be infinitely remote, and the tangent plane is de- 
 termined by that element and the given line itself. The point of 
 tangency is, in all cases, the intersection of the rectilinear element 
 with the curve, if there be one, cut from the warped surface by 
 the tangent plane. 
 
216 DESCllIPTIVE GEOMETRY. 
 
 There are cases in wliicli there is no such curve — for instance, 
 the given line may He in a plane tangent to a conoid or cylindroid 
 along an element; — or, the plane determined by the given line and 
 an element of a hyperbolic paraboloid, may be parallel to a plane 
 directer. 
 
 309. The problems of passing a j^lane tangent to a warped sur- 
 face, and either parallel to a given right line, or through a given 
 point without the surface, are indeterminate. 
 
 In the first case, by making a sei-ies of sections of tlie surface 
 by planes parallel to the given line, and drawing a tangent to each, 
 also parallel to it, a cylinder may be constructed, tangent to the 
 surface. Any plane tangent to this cylinder satisfies the conditions. 
 
 In the second case, a series of sections of the warped surface 
 being made by planes containing the given point, let a tangent be 
 drawn to each through that point. These tangents are elements of 
 a cone tangent to the given surface, whose vertex is the given 
 point : and any plane tangent to this cone satisfies the conditions. 
 
 310. Use of Auxiliary Surfaces. The operation of drawing a 
 plane tangent to a w^arped surface may sometimes be facilitated by 
 the use of another warped surface. It w^ill presently appear that, 
 as stated in (144), one such surface may be tangent to another one 
 all along an element. In that case, any plane tangent to either is 
 tangent to both, if the jjoint of contact lies on the common ele- 
 ment ; and as has already appeared, it may be much easier to draw 
 a plane tangent to the one than to the other. 
 
 TANGENCY OF WARPED SURFACES. 
 
 311. Two warped surfaces are tangent to each other, like any 
 others, wdien they have at any point a common tangent plane. In 
 order that they may be tangent all along a common element, they 
 nuist have a common tangent plane at every point thereof. And 
 this will be the case, if that condition be satisfied for any three 
 points of the given element. 
 
 For, in Fig. 244, let LL be an element common to two given 
 w^arped surfaces, which have a common tangent plane at each of 
 the three points A^ B^ C. Any intersecting planes passed through 
 theso points will cut from one surface the three curves i>, E^ F^ 
 
DESCRIPTIVE GEOMETRY. 217 
 
 from tlie other tlie curves G^ 11^ Z, and from the tangent planes 
 tlie riglit hnes R^ S, T. 
 
 The curves D^ G, being tangent to each otlier, have two con- 
 secutive pomts in common ; and the same is tnie of the other 
 pau-s, E^ H, and F^ I. Consequently, if LL be moved eitlier upon 
 D^ E^ F^ or upon G^ H^ /, as directrices, into its consecutive posi- 
 tion, it will lie in both surfaces* which, therefore, have two con- 
 secutive rectilinear elements in common. Any plane cutting 
 tliese, evidently, will cut from the surfaces two lines which have 
 two consecutive points in common, or in other words are tangent 
 to each other : tlie two surfaces, then, are tangent all along LL. 
 
 312. If the two surfaces have a common plane directer, and 
 
 common tangent planes at two points upon a common element, 
 they will be tangent all along that element. For in this case the 
 motion of L^L in Fig. 244 would be completely determined by two 
 of the pairs of curves there shown ; and the argument is otherwise 
 the same as above. 
 
 313. If in the same figure the generatrix LL be moved upon 
 the tangent lines B^ /S, T, as directrices, instead of upon either set 
 of curves, it will generate a third warped surface, tangent^ all, along 
 the element, to both the others: this surface having three recti- 
 linear directrices, must be eitlier a hyperbolic paraboloid, or a 
 warped hyperboloid. If the three intersecting planes passed 
 through J., B^ and (7 are parallel, then R^ S, and 7^ are all parallel 
 to one plane, and the surface is a hyperbolic paraboloid ; if they 
 are not, the surface will be either a circular or an elliptical hyper- 
 boloid. The relations and directions of the intersecting planes 
 
218 DESCRIPTIVE GEOMETRY, 
 
 being entirely arbitrary, any number of sets of planes, parallel or 
 not, may be drawn tlirougli those points ; consequently, any num- 
 ber of hyperbolic paraboloids, and any immber of warped hyperbo- 
 loids, may be constructed, all of which shall be tangent along the 
 element LL to both the given surfaces. 
 
 314. Since the directions of the tangents 7?, 8^ T, are deter- 
 mined by either of the two given surfaces independently of the 
 other, it might seem that the above would be true of any element 
 of any warped surface. But there are exceptions. It has been 
 seen that a plane may be tangent all along certain elements of a 
 warped surface, as in the cases of the cylindroid and some forms of 
 the conoid. To such a surface, evidently, no other warped surface 
 can be tangent along those elements, except one which possesses 
 the same peculiarity, and this is conspicuously not the case with the 
 hyperboloids : single-curved surfaces, however, may be so, of any 
 kind and of any number. 
 
 The normals at all points of such an element, being perpendicu- 
 lar to one plane, are parallel to each other, and thus determine a 
 plane normal to the surface all along the element. 
 
 315. Tlie normals to a warped surface at various points of a 
 given element are not in general tlius perpendicular to any one 
 plane. But if they are not, tlien, whatever tlie nature of the given 
 surface, these normals are elements of a rectangular hyperbolic 
 paraboloid. 
 
 This may be shown as follows : In Fig. 245, let Z>, E^ F^ be the 
 curves cut from a warped surface by planes perpendicular to the 
 element LL at the points A^ B^ C\ let JT, Y^ Z be the normals at 
 those points, which will be perpendicular to the tangents /^, S^ T 
 lying in the same intersecting planes. If LL be moved upon the 
 tangents to any new position MN^ it will, as has already been seen, 
 generate a hyperbolic paraboloid ; of which one plane director will 
 be any plane F perpendicular to LL^ and the other will be any 
 plane H parallel to both MN and LL^ and consequently perpeii- 
 dicular to Y. This tangent paraboloid, then, is rectangular ; and 
 if it be revolved about LL through an angle of 90°, the elements 
 ^, S^ T will coincide with the normals X, Y^ Z, the line MI^ 
 taking the position TJW. The paral)oloid now has for one set of 
 
DESCRIPTIVE GEOMETRY. 
 
 219 
 
 elements the series of normals, and for one plane director the plane 
 V to which they are all parallel ; the other plane directer will be 
 any plane P parallel to both UW and ZZ, and therefore perpen- 
 dicular to V: and since the revolution was through an angle of 
 90°, this plane P will also be perpendicular to ZT. 
 
 Note. In this illustration, the conditions have for the sake of 
 clearness been so chosen that the plane directors F, Zf, P, are 
 respectively parallel to the vertical, horizontal, and profile planes; 
 and for further elucidation, the positions of the tangents i?, S, T, 
 
 
 N 
 
 M 
 
 c 
 
 B 
 
 T—sN 
 
 
 S \ 
 
 \ 
 
 A 
 
 L 
 
 
 \ 
 
 R 
 
 
 Tig. 246 
 
 Fig. 245 
 
 and the line MN^ before and after revolution, are represented by 
 tlieir projections, in Fig. 246, of which no explanation is needed. 
 
 316. Applications of the Preceding. The hyperbohc paraboloid 
 is readily drawn, and a plane tangent to it easily determined ; it is 
 therefore natural that this surface should be the one usually em- 
 ployed as an auxiliary, in constructing tangent plaues to other and 
 more intractable warped surfaces, as suggested in (310): the two 
 following examples illustrate its use for this purpose. 
 
 317. Problem 1. To draw a plane tangent to the Cow^s ZZorn 
 at a given point of the surface. 
 
230 
 
 DESCRIPTIVE GEOMETRY. 
 
 Construction. In Fig. 247, X, , 1\ , parallel to V, are the cir- 
 cular directrices, whose centres JT and Y, as well as tlie rectilinear 
 directrix ^iT, lie in the plane JJ parallel to H ; it is required to 
 
 Fia. 24:7 
 
 draw a plane tangent to the surface at the point P, upon the given 
 element FD, 
 
 This element and the rectilinear directrix, being two lines of 
 the surface, determine a plane gQm\ tangent thereto at their inter- 
 section K Draw in this plane a line ^^ parallel to V, and at D 
 and F draw tangents to the circulai- directrices ; these three lines 
 
DESCRIPTIVE GEOMETRY. 221 
 
 are elements of one generation, and ED is an element of the other 
 generation, of a hyperbolic paraboloid liaving V for one plane 
 director. 
 
 Through any point B of the tangent at i>, pass a plane con- 
 taining the tangent at F\ this plane cuts EK\vl (7, and CTJR is an 
 element of the paraboloid. 
 
 Through P pass the plane // parallel to T, cutting Cli in O^ 
 then PO and EB determine a plane tangent at P to the auxiliary 
 paraboloid and therefore to the given surface. ED pierces H in 
 G^ and PO pierces it in S\ also ED j)ierces V in J[f : consequently 
 tT^ the horizontal trace of the required plane, transverses g and «, 
 and Tt\ the vertical trace, passes through m' — being, moreover, 
 parallel io jp'o' the vertical projection oi PO. 
 
 318. Problem 2. To draw, a plane tangent to the cylindroid 
 at a given i)oint of the surface. 
 
 Construction. In Fig. 248, let 1", Z, be the circular directrices, 
 V the plane directer, and P, on the element CD^ the given point. 
 Draw, at C and Z>, tangents to the circles ; these tangents are the 
 directrices, and CD is the generatrix, of the auxiliary hyperbolic 
 paraboloid. The tangents pierce y m E and E respectively, and 
 EE is another element of the auxiliary surface. Pass through P 
 a plane parallel to Z^jB'and CE\ its vertical trace is r's'^ which cuts 
 ef in o\ horizontally projected in o on AB. Then PO and CD 
 determine the required tangent plane; whose vertical trace is 
 l^arallel to c'd' the vertical projection of CD. 
 
 319. The normals to any warped surface at points of a given 
 element thereof, determine, in general, a hyperbolic paraboloid 
 (315), and belong to the same generation. If any element of the 
 other generation of this paraboloid be taken as an axis, the given 
 element by revolving around it will generate an hyperboloid of 
 revolution, which will be tangent to the given surface all along the 
 element : of which the following exhibits a useful application «in 
 mechanism. 
 
 320. Problems. To construct two hyperholoids of revolution^ 
 tangent to each other along an element. 
 
 Construction. In Fig. 249, let the axis of one hyperboloid be 
 vertical, c being its horizontal and o'c' its vertical projection ; let 
 
222 
 
 DESCRIPTIVE GEOMETRY. 
 
 CO be tlie radius of its gorge circle, and OP^ parallel to T, its gen- 
 eratrix : the projections of this surface are then drawn as in Fig. 
 209. The gorge radius of which o' is the vertical and co is the 
 horizontal projection, is evidently normal to the surface ; the nor- 
 mal at P must lie in a plane perpendicular to OP^ therefore c'p' 
 perpendicular to o';p' is its vertical and cj) is its horizontal projec- 
 
 tidh : these normals are elements of one generation, and the verti- 
 cal axis and the generatrix OP are elements of the other generation 
 of the normal hyperbolic paraboloid, — of which one plane director 
 is V, and the other is perpendicular to OP. Any plane as JJ 
 parallel to Y is seen in the horizontal projection to cut co produced, 
 in 6, and cp produced, in ^; e is vertically projected in o' ^ and d 
 
DESCRIPTIVE GEOMETRY. 
 
 223 
 
 in d\ on the prolongation of c'])\ therefore ed is tlie horizontal, and 
 o'd' the vertical, projection of another element of the normal parab- 
 oloid : which may be taken as the axis of the second hyperboloid. 
 
 321. Make a supplementary projection on a plane perpendicular 
 to this axis, looking in the direction of the arrow w. In this view^ 
 
 eo is seen in its true length as ^^<9, , and the common element OP 
 as (9i^j tangent at o, to the circle of the gorge, which in the verti- 
 cal elevation appears 2,^ fk! perpendicular to o'd' . The radius of 
 the upper base, passing " through P and also perpendicular to the 
 
224 DESCRIPTIYE GEOMETRY. 
 
 inclined axis, is e^p^ , to which accordingly g'V in the vertical pro- 
 jection is made eqnal. Any point TJ on OP^ is projected to %i^ on 
 o^])^ , and s'li't' perpendicular to o'd! is eqnal to e(ii^ ; and in like 
 manner any desired number of points on the required hyperbola 
 may be found. 
 
 In the foreshortened horizontal projection of the inclined hyper- 
 boloid, both the gorge circle and the upper base appear as ellipses ; 
 as will also any intermediate transverse sections^. A portion of one 
 such section through B^ is projected at y ; the visible contour zx is 
 the enyelope of all these ellipses, and not, as sometimes supposed, a 
 curve through the extremities of their major axes : in fact it passes 
 through the extremity of only one of them, viz. , that of the gorge 
 circle at 2, at which point the contour line has a tangent perpen- 
 dicular to ez the gorge radius. 
 
 INTERSECTIONS OF WARPED SURFACES. 
 
 322. The intersection of any warped surface with a plane may 
 
 be determined by finding the points in which its elements pierce 
 the plane. Many such intersections have already been illustrated ; 
 and in any case the problem is simple in principle, though the 
 necessary repetition of the same process may render it tedious in 
 execution. 
 
 The intersection with any other surface may be determined by 
 the general method, of passing a series of auxiliary planes cutting 
 both surfaces, and joining the points in which the lines cut from 
 each surface intersect each other. Just what system of auxiliary 
 planes will be most convenient, must in the nature of things dej^end 
 largely upon the peculiarities of any given case, and be decided by 
 the judgment of the operator. Attention to the following points 
 may, however, sometimes be of service ; — 
 
 1 . If one of the given surfaces he a cylinder^ Planes may be 
 passed through the elements of the warped surface, parallel to 
 those of the cylinder. 
 
 2. If one of the s 117 faces he a cone; Planes may be passed 
 through the elements of the warped surface and the vertex of the 
 cone. 
 
 3. If hoth surfaces are war][)ed^ hut have a common plane di- 
 
DESCRIPTIVE GEOMETRY. 
 
 325 
 
 recterj A system of planes parallel to this plane directer may be 
 used. 
 
 In either of these .cases, the auxiliary planes will cut right lines 
 from both surfaces ; but it does not necessarily follow that these 
 wdll give the most satisfactory determinations, since they may in- 
 tersect each other very acutely. 
 
 323. The intersection of a helicoid with a surface of revolution 
 having the same axis, is of special interest as being frequently met 
 with in the construction of screw-propellers ; a few illustrations of 
 it are therefore appended. That particular form of the helicoid 
 only is here considered in which the generatrix cuts the axis ; bo- 
 
 rn' 
 
 \d" y" h" 
 
 cause in practice it is used, if not exclusively, at least more exten* 
 sively than any other. 
 
 324. Example 1. Intersection of a helicoid with a right cir- 
 cular cone having the same axis. 
 
 Construction. In Fig. 250 are given, on the left an end view, 
 on the right a side view^, of a portion of a right helicoid ; ad^ ad\ 
 being the helical directrix. In the end view, ca^ cb^ cd^ etc., equi- 
 distant radii, represent elements; if these be revolved into the ver- 
 tical plane ca^ they will in the side view appear as the equidistant 
 lines G'a\ r'h'\ w'd'\ etc., perpendicular to the axis; these pierce 
 the cone m'o'n' at the points h\ e'\ d'\ etc. Set off on the radii 
 
226 
 
 DESCRIPTIVE GEOMETRY. 
 
 in tlie end view the true distances of these points from the axis, as 
 
 cJi =: c'h\ ce = r'e'\ ct = s't" ^ etc. Project the points thus 
 located, back to the elements in the side view, as d Xo d\ t to t\ 
 e to e\ etc. ; the curves deli^ d'e'K ^ thus determined, are the re- 
 quired projections of the intersection. 
 
 Tig. 252 
 
 In the case of the oblique helicoid, Fig. 251, the elements when 
 revolved into the vertical plane ca^ appear in tlie side view as equi- 
 distant parallels inclined to the axis. 
 
 Tlie construction is the same as before, with the exception that 
 the points d\ t\ e' ^ etc., are located, not upon tlie elements, but 
 upon perpendiculars to the axis from the points d'\ t'\ e'\ etc. ; 
 because each point must revolve in a plane pb/pendicular to the 
 axis. 
 
DESCRIPTIVE GEOMETRY. 
 
 227 
 
 Fig. 252 represents tlie intersection of the cone with a hehcoid 
 of varying pitcli. In this case the revolved elements appear in the 
 side view as lines of different inclinations, dividing into the same 
 number of equal parts the distances a'd'\ c'w'\ which distances 
 are the same fractions of the ]^itches at tlie outer circnrnforence and 
 the axis respectively, that tlie arc ad is of the whole circumfer- 
 ence : otherwise the construction is the same as in Fig. 251. 
 
 325. Note. The arcs hg^ ef^ etc., in the end views, represent 
 concentric cylinders, which cut all these helicoids in true helices — 
 two of them shown in the side views as k'g\ e'f. The outlines 
 
 ym. 253 
 
 y'g' ^ e"f'^ of these cylinders must pass through the points Tc"^ e'\ 
 i" ^ etc., in all three side views; in the first two these points are 
 equidistant, therefore in the end views the points A, g^ y, are also 
 equidistant, and the curve deli in Figs. 250 and 251 is an Archi- 
 medean spiral : but in Fig. 252 this is not so, for the distances hg^ 
 gf^ etc., are not equal, but continually increase as the points A, ^,/*, 
 recede from the centre. 
 
 This problem is encountered in determining the form of the 
 trailing edge of a propeller-blade having what is technically called 
 an ' ' overhang ' ' ; when it will often be found more satisfactory to 
 
228 DESCRIPTIVE GEOMETRY. 
 
 ascertain the distances ck^ ce^ etc. , by calculations based upon tlie 
 law of tlie spiral, since accuracy in tlie end view is most essential. 
 
 326. Example 2. Intersection of a helicoid with an annular 
 torus having the same axis. 
 
 Explanatory. A propeller-blade fashioned as above would re- 
 volve within a surface of revolution whose meridian section is 
 o'd"a' in either of the three preceding figures: it would also have 
 an objectionable sharp corner at d^ d' , The imaginary "box" 
 within which the blade revolves is therefore sometinies ' rounded 
 off at the angle,' as by the arc x'\(!'n" in Fig. 253, This arc is a 
 part of the circumference of a circle whose centre is z ; and the 
 complete circumference is the meridian section of an annular torus 
 having. the same axis as the helicoid. 
 
 Construction. The w^arped surface in this figure is a right lieli- 
 coid, and the mode of operation is substantially the same as in Fig. 
 250. Thus, dividing the arc ad and the distance a'd" in the same 
 proportion, the elements are represented by radii in the end view 
 and ])y perpendiculars to the axis in tlie side view, drawn through 
 the points of division; and the curves deh., d'eh\ are found as in 
 (324). The section of the torus is tangent at ti" to the element 
 s'y" ; set off on cy, the distance cu = s'u'\ and project u back to 
 s'y" at It' ; this determines a limiting point of the curve in each 
 view. The circular section is tangent to o'7ri at v'\ and an ele- 
 ment through this point cuts the circumference also at t" : the 
 distances of these points from the axis being set ofl: on the corre- 
 sponding radius, the two points t and v are determined — of which 
 the latter is the point of tangency betw^een the spiral aeh and the 
 new intersection vtx. Ry repeating this process as many points as 
 are deemed requisite may be found, and the entire curve of pene- 
 tration constructed : only that portion is here shown wdiich would 
 form part of the contour of the blade of a propeller. 
 
 327. Example 3. Intersection of a helicoid v&ilh a cylinder 
 whose elements are parallel to the axis. 
 
 Explanatory. The form of a propeller blade is sometimes fixed 
 by the condition that it shall present a given outline in the end 
 view : the drawing of the other views then involves the problem 
 above mentioned. 
 
DESCRIPTIVE GEOMETRY. 
 
 229 
 
 Construction. In Fig. 254, let the warped surface again be a 
 right heHcoid, and let mon be the base of the cylindrical surface. 
 
 The operation is simply the converse of the preceding ; the 
 points in which the elements of the helicoid pierce the cylinder are 
 seen directly in the end view, and are projected to the correspond- 
 ing elements as seen in the side view, — as e to e' on w'd" ^ o to o' on 
 s'y" , etc. Then in order to find the outline of the surface within 
 
 FiGf. 254 
 
 which the blade revolves, set up w'e" = ce^ s'o" = co, and so or>, 
 thus determining the required curve e"o"x" . 
 
 328. These last problems have been illustrated only in connec- 
 tion with the riglit helicoid, merely for tlie sake of simplicity. 
 But by attention to the explanations given in (324), there will be 
 no difficulty in dealing in a similar manner with the others ; and 
 indeed substantially the same methods might be applied if the gen- 
 eratrices of the helicoids were curved, as they sometimes are : in 
 whicli case, however, the surfaces are no longer warped, but are of 
 double curvature. 
 
230 
 
 DESCmPTIVE GEOMETET. 
 
 CHAPTEK YII. 
 
 ISOMETRICAL DeAWING, CavALIER PROJECTION, AND PsEtJDO- 
 
 Perspective. 
 
 ISOMETRY. 
 
 329. In Fig. 255, (7 is a top view of a cube so placed that in 
 the front view A tlie diagonals cc/, ah, of its upper face are respec- 
 tively parallel and perpendicular to the paper. The cube is shown 
 
 as cut by a plane pp, perpendicular to the paper in view A ; the 
 gection thus made, as seen in the perspective sketch F, is bounded 
 "by the three face diagonals ah, ad, hd : it. is, then, an equilateral 
 triangle, to the plane of which the three equal edges ca, ch, cd are 
 
DESCRIPTIVE GEOMETRY. 231 
 
 e' iially inclined. And as seen in view ^, this plane is perpen- 
 dicular to the body diagonal ch of the cube, which pierces it at o. 
 
 In the view i>, which is an orthographic projection upon the 
 plane ^j?, the three face diagonals are seen in their true lengths, 
 forming the equilateral triangle a'Vd' . Since the three edges 
 which meet at g are equally inclined to the plane, they will be 
 equally foreshortened : therefore c' is the centre of the triangle, 
 a'G\ h'c'^ d'c' are equal to each other, and the three angles at c' are 
 each equal to 120°. 
 
 Every other edge of the cube being equal and parallel to one of 
 these three, each visible one will appear equal and parallel to one 
 of those already drawn ; thus the apparent contour of the entire 
 cube will be a regular hexagon, the representation of each face be- 
 ing a rhombus. 
 
 Because the edges of the cube are thus foreshortened in the 
 same proportion, so that they and all parallels to them may be 
 measured l)y the same scale, such a view as D is called an Isometric 
 Projection ; a'c' ^ h' g' ^ d'c\ are called the isometric axes ; the planes 
 which they determine, and all planes parallel to them, are called 
 isometric planes / and all lines parallel to the axes are called iso- 
 metric lines. 
 
 330. Drawings made in this manner possess the advantage of 
 conveying, in one view, ideas of the three dimensions, as do those 
 made in perspective ; and in many cases they exhibit the peculiarities 
 of structure more clearly than ordinary plans, sections, and eleva- 
 tions. They are readily understood by those who are not familiar 
 with common projections ; and in making sketches this system is 
 very useful. 
 
 Obviously, however, the advantages of isometry are more pro- 
 nounced Avhen the objects to be represented are bounded by right 
 lines, of which the principal ones are parallel and perpendicular to 
 each other. It is not well adapted for the general drawing of 
 machinery, since it involves an unpleasant distortion, and also be- 
 cause in most cases the circles are projected as ellipses. 
 
 331. Distinction hetween isonietrical projection and isometri- 
 col drawing. In Fig. 255 the actual length of the edge of the cube 
 is cd\ its apparent length in view D is c'd' ^ equal to od in view A, 
 
23^ DESCRIPTIVE GEOMETRY. 
 
 Suppose cd to be one unit in length — an incli for example : tlien 
 by taking od as a unit it is possible to construct an isometriG scale^ 
 by wliicli all tlie isometric lines in D might have been set off ; and 
 such a scale could be used in constructing any isometrical projection. 
 
 This is a matter of purely abstract, theoretical interest, and not 
 of any practical use whatever. Since tlie isometric lines are all 
 equally f-oreshortened, there is no reason why they should be repre- 
 sented ac foreshortened at all. Consequently an Isometric Drawing 
 of the given cube is made as shown at E^ each edge being drawn of 
 its true length. This is the method always adoj^ted in practice, 
 the scales in common use being alone employed. The man who 
 should construct a true projeetio7i, and send it to the workman to 
 be measured, by an isometric scale, would simply make a record of 
 his own stupidity ; he who sliould teach others to do so, would 
 commit a blunder of much more serious importance. For, to use 
 the words of another, '' the value of isometry as a practical art lies 
 in the applicability of common and known scales to the isometric 
 lines. "-^ 
 
 332. We proceed, then, just as in making ordinary working 
 drawings, setting off the dimensions on those lines either ' ' full 
 size," or with the 3-inch scale, the li-inch scale, etc., as the case 
 may require. Naturally, lines which are vertical are so repre- 
 sented ; the other isometric lines are then drawn with great facility 
 by the aid of the T-square and the triangle of 60° and 30°. 
 
 Figs. 256-261 are simple exercises, composed wholly of iso- 
 metric lines, the construction being so obvious that no detailed 
 explanation is required : the method of locating the foot of the 
 cross in Fig. 257, and the mortise and the tenon in Fig. 261, by 
 measui'ing along the lines ea, ch, or parallels to them, is sufficiently 
 shown by the dotted lines. 
 
 It is to be distinctly understood that these figures are illustra- 
 tions merely : the student is not to coj>y them, but to construct 
 them or others of similar character, with such variations of dimen- 
 sions, arrangement, or design as may be suggested by his ingenuity, 
 which should be given full play. 
 
 * W. E. Wortben. 
 
DESCRIPTIVE GEOMETRY. 
 
 !33 
 
 333. Shadow Lines. In making mechanical drawings, on any 
 system of projection, those lines which, being the intersections of 
 
 Fig. 257 
 
 snrfaces which are illuminated w^ith others which are not, intercept 
 the light and thus cast shadows, are usually emphasized by making 
 
 Fm. 258 
 
 them heavier than the other outlines. By this means an effect of 
 relief is produced ; the drawing is more easily read, and its ajDpear- 
 
J34 
 
 DESCRIPTIVE GEOMETRY. 
 
 ance greatly improved : and tlie lines tliiis emphasized are called 
 shadow lines. 
 
 In orthographic drawings, the direction of the light is by com- 
 mon consent assumed as follows : Suppose the observer to be 
 standing in a cubical room, facing one of the walls as the vertical 
 plane ; then the light comes from behind, the rays going downward 
 to the right, in the direction of the body diagonal of the cube, each 
 projection making an angle of 45° with the ground line. 
 
 And in isometric drawings reference is also made to the cube 
 as a standard. Thus in E^ Fig. 255, the light is supposed to have 
 the direction of the body diagonal af^ so that the faces eo^ cg^ are 
 
 Fig. 261 
 
 illuminated, and shadows are cast by the edges ed^ dc^ cb^ and Ig, 
 In drawing, the first of these lines should be made the heaviest, 
 the last one the lightest, and the other two of equal and medium 
 thickness. 
 
 334. Isometrical Drawing of the Circle — In the ellipse repre- 
 senting the circle inscribed in the face of the cube. Fig. 262, the 
 axes coincide with the diagonals, and are at once determined by 
 representing the parallels through Z, m, n^ 6>, in the elevation shown 
 at the left. Describe a semicircle upon cd as a diameter, divide it 
 into four equal parts by the points 1, 2, 3, draw 3 3', 1 1' perjDen- 
 dicular to cd^ and through 3' and V draw parallels to hc] tliese 
 
DESCRIPTIVE GEOMETRY. 
 
 235 
 
 will cut the diagonals at m, <?, and n^ I, thus limiting the major and 
 minor axes. As a check, note that hn and on should be parallel 
 to cd. 
 
 The sides of the rhombus being equal, tins construction may be 
 made upon either one at pleasure. And, since all the faces of the 
 cube are exactly alike, it follows that all circles lying in isometric 
 planes are represented by similar ellipses. 
 
 By drawing tangents at the points ?, in^ n, o in the elevation the 
 circle is circumscribed by a regular octagon, the isometric represen- 
 tation of which is therefore made by drawing at the corresponding 
 
 Tig. 262 
 
 points Z', m', n\ o\ in the upper face of the cube, perpendiculars to 
 the diagonals, terminating in the sides of the rhombus. 
 
 335. Graduation of the Isometric Circle. First Method. At 
 the middle point d of gh^ in Fig. 263, erect a perpendicular de 
 equal to dh^ and draw eh., eg ; about ^ as a centre describe with any 
 radius the quadrant r^, divide it as desired by the points 1, 2, 3, 
 etc., through which draw radii and produce them to cut gh. 
 From these intersections with gh draw lines to ^, the centre of the 
 ellipse : these will cut its circumference in the required points of 
 division, 1,, 2, , etc. 
 
 Second Method. Describe a semicircle upon the major axis ttio 
 as a diameter, divide it in the desired manner by the points 1', 2', 
 
236 
 
 DESCRIPTIVE fiEOMETRY. 
 
 etc. , tlirongli wliicli draw perpendiculars to mo^ cutting tlie cir- 
 cumference of the ellipse in 1', 2', etc. : these will be the points 
 required. 
 
 Fig. 263 
 
 An application of the above is shown in the drawing of the 
 bolt, nut, and washer, Fig. 264. About ^, the centre of the outer 
 ellipse, describe an arc with radius j96> = semi-major axis, set ofi the^ 
 
DESCRIPTIVE GEOMETRY. 
 
 237 
 
 arc oa = 60°, erect tlie vertical ab^ and draw h]) cutting the inner 
 ellipse (circnmscri})ing the base of the nut) in c. 
 
 336. To draw Angles to the Sides of the Isometrical Cube (Fig. 
 265). Draw a square cg^ whose side is equal to the edge of the 
 cube ; about one of its angles, say <?, as a centre, describe the quad- 
 rant db^ graduate it, and produce the radii through the points of 
 division to cut the sides of the square. The scale of tangents thus 
 formed may, bj cutting out the square, be applied to any side of 
 the isometrical cube, thus determining the direction of a line in its 
 face which shall represent a line making any required angle with 
 its edge. For example, make a'e'=ae^ and h'f'= hf\ then a'o'e'^ 
 
 Pig. 265 
 
 y 
 
 Vcf are the isometrical representations of the angles ace^ hcf. 
 The same angles are represented on the left-hand vertical face of 
 the cube by making d'e"= ae^ G^f"= ^? and drawing a^e^'^ (^'f* 
 
 An application of this is found in making the isometrical draw- 
 ing of the piece shown in plan and elevation at 6^ and A^ Fig. 266, 
 in which the angles a?, y are assigned : also in C the distance ad 
 is given, dke is perpendicular to ad^ and ef parallel to ok : the 
 thickness is uniform, and equal to ah in view A. The isometric 
 drawing is lettered to correspond, and should require no further 
 explanation. 
 
 337. Another method of dealing with lines which, though 
 lying in isometri 3 planes, are not parallel to either of the isometric 
 axed, is by means of ^' offsets." Thus in Fig. 267 the slope of the 
 diagonal brace is determined by measuring the distances cd, d^, 
 
238 
 
 DESCRIPTIVE GEOMETRY. 
 
 along tlie isometric line ca^ and setting np the vertical ef^ of the 
 values ascertained from tlie elevation shown on a reduced scale. 
 This really amounts to the same thing, the angle being constructed 
 bj laying off the base and altitude of a triangle of which the re- 
 quired line is tlie hypothenuse, — which is in perhaps the majority 
 of cases the most convenient means. Another illustration is given 
 in the drawing of the box, Fig. 268 ; the outline of the end of the 
 partially opened lid being set out by means of the vertical meas- 
 urements <?«•, co^ cb, ch^ and the offsets ae^ ol^ hcl^ hg^ taken directly 
 
 Fig. 268 
 
 from the transverse section shown at the left. 
 
 338. This principle may be extended, and is applied to the 
 determination of lines which do not lie in isometric planes ; as illus- 
 trated in Fig. 269, representing the roof of a cottage, cf the fonn 
 and proportions shown in plan and elevations on a smaller scale at 
 the right. The sloping lines of the roof at the nearer end are 
 found by setting' up the heights ah^ ad on the vertical through a,, 
 and drawing the isometric lines JA, df: then the points ?', Jc are the 
 intersections of hf^fh with the isometrical line through c. A simi- 
 Jar construction may be made at the farther end, thus fixing the 
 line of the ridge ^', on which the point g is located by setting olf 
 
DESCRIPTIVE GEOMETRY. 
 
 239 
 
 fg^ its distance from the plane de : we can then draw gi^ gh^ which 
 do not lie in any isometric plane. 
 
 The same process is applied in drawing the wing roof, the 
 heights n^ v, r^ being set up on the vertical through the nearer 
 corner ??^, and the distance oq measured from the plane mo. The 
 ridge line w^ill pierce the main roof at a point t^, which may be 
 thus located : Set up at = mr, draw the isometric line ts cutting 
 ^"in 6', and through s draw a parallel toff: this will cut the ridga 
 line of the wing in the required point. 
 
 It will be observed that the lines fh^ gl^^ and ff\ differ very 
 
 Pia. 269 
 
 JP ^ -. 
 
 m 
 
 little in direction, and qz., ou^ differ still less. I'his simply shows 
 that the farther side of the main roof is very nearly, and that of 
 the w^ing roof almost exactly, perpendicular to the plane upon 
 which the isometric drawing is made ; and it will be perceived that 
 in such cases this not a peculiarly eligible mode of representation, 
 — as indeed it is not for architectural subjects of any description. 
 
 339. Thus far one of the isometric axes has been made vertical. 
 But inasmuch as it is the relative direction of the lines among 
 themselves which determines whether a drawing is an isometric one 
 or not, there is no necessity that any of them should be vertical. 
 In Fig. 270, for example, the principal lines are horizontal ; but 
 the drawings of the die and its matrix, and of the timber with its- 
 
340 DESCRIPTIVE GEOMETRY. 
 
 mortises and its tenon, are at once recognized as isometric, and 
 are just as easily understood as if tliej stood upiight. 
 
 For convenience in constructing the drawings by means of the 
 T-square and triangles, it is preferable in most cases, of course, that 
 one of the isometric axes should be either vertical or horizontal, but 
 should there be any reason for selecting other positions, there is 
 nothing in the principle of isometry to prevent their adoption. 
 
 340. It will be noted that the correspondence of the die to the 
 matrix in Fig. 270 is made much more obvious than it otherwise 
 would be, by exhibiting the opposite ends of the tw^o pieces. By 
 
 V 
 ^^i VA\ 'CIA 
 
 Fig. 270 
 
 merely turning the page around, it will be apparent that this could 
 have been done equally w^ell if the two pieces had been drawn in a 
 vertical position. 
 
 For this reason isometry affords a means of illustrating in a very 
 clear and striking manner many subjects in which views of the 
 lower surfaces are desirable : a good example is shown in the draw- 
 ing of the small shelf with its supporting bracket. Fig. 271. 
 
 In making such a drawing, as will readily be seen, the process 
 is equivalent to constructing the projection of the cube, Fig. 255. 
 upon the plane ^j!?, as seen from the lower left-hand side, and look 
 ing in the direction opposite to that indicated by the arrow. 
 
 341. In Fig. 272 is shown an isometric. drawing of the ratchet- 
 wheel represented in the full-size views at the right. The backs of 
 
DESCRIPTIVE GEOMETRY. 
 
 241 
 
 the teetli not only terminate in, but are tangent to, tlie interior 
 circle ; and a test of the accuracy of the isometric drawing is found 
 in the tangency of the edges to the ellipse representing that circle. 
 And this embodies a principle capable of many other applications 
 — as, for instance, in laying out a wheel with radial tapering arms : 
 
 Fig. 271 
 
 the side outlines of each arm are tangent to a circle, which being 
 drawn in the isometric construction, it is seen that the breadths of 
 the a'rms at the outer ends only need be set out, thus fixing points 
 through which tangents are to be drawn to the ellipse. 
 
 342. It seems needless to multiply examples, as it is believed 
 that by the aid of the preceding any isometrical drawing likely to 
 be required in practice may be constructed. 
 
 In drawings of machinery, the circles of wheels, bearings, ends 
 of shafts, and the like, will usually lie in isometric planes. Should 
 occasion arise to represent one which does not, circumscribe it by 
 a square : the projection of this will be a parallelogram, within 
 -)Yhich the ellipse may be drawn by any convenient method. So, 
 
242 
 
 DESCRIPTIVE GEOMETRY. 
 
 too^ if it should be necessary to represent tlie section of a cylinder 
 or a cone by an oblique plane, the solid may be conceived as sur- 
 rounded by a square pyramid or prism, whose section by the given 
 plane, as well as tlie isometric drawing of it, will be a parallelo- 
 gram circumscribing the required ellipse. 
 
 343. In conclusion, it may be pointed out that many lines not 
 usually classed as isometric are strictly so in fact. This distinctive 
 term is technically restricted to lines parallel to the isometric axes, 
 which again are so called because they are equally foreshortened, 
 and this is the result of their equal inclination to the plane upon 
 
DESCRIPTIVE GEOMETRY. ?43 
 
 wliicli tliey are projected. Now, in Fig. 273 the cube is cut by 
 the plane pp as in Fig. 255, and in the projection D we at once 
 recognize' c a\ e'h\ c'd' as the isometric axes. If cd revolve around 
 €0 as an axis it will generate a cone dch^ all of whose elements make 
 the same angle with the plane j:^^; so that any one of them, as cr 
 (seen in D as c'r')^ would be foreshortened in the same proportion 
 as any otlier one. But in the isometric projection this fact w^ould 
 not be indicated by merely drawing c'r' : it is necessary to locate 
 the point r' by means of offsets — g'7i giving its distance from the 
 plane a'c'd' ^ c'l its distance from the plane a'o'h'^ and n^n its dis- 
 tance from the plane h' c'd' . This being done, s' is at once s.een to 
 be the foot of the perpendicular r's' from the point in question to 
 the plane last mentioned. 
 
 Again, the isometric projection of any frustum of a cone, xyuv, 
 whose bases are parallel to pp, would appear simply as two con- 
 centric circles, and without some auxiliary view that projection 
 would convey no definite information about the cone, which might 
 be of any altitude or have either base uppermost. 
 
 Since the whole value of isometry, in practice, lies in the power 
 of imparting in one view definite ideas of the three dimensions, the 
 above hints may serve a purpose as indicating possible relations of 
 parts for the representation of which this method of drawing is not 
 suitable. 
 
 CAVALIER PROJECTION. 
 
 344. In Fig. 274, let JlfiTbe a vertical glass plate represent- 
 ing the vertical plane ; let c be a point in this plane, and ca a line 
 perpendicular to it. Let ar be a visual ray, making an angle of 
 45° with the plane JO^, and piercing it at j9 : then cp is the repre- 
 sentation of ca upon the picture plane, and it is equal to ca^ be- 
 cause the angles cpa^ cap are each equal to 45°. 
 
 Suppose the eye to be at an infinite distance in the direction ar : 
 then all the visual rays will be j)arallel, and all lines perpendicular 
 to MW will be represented upon that plane by lines of their actual 
 length, and parallel to cp. 
 
 The fact tliat the projection is of the same length as the per- 
 pendicular line ca depends upon the condition that the picture 
 
244 
 
 DESCRIPTIVE GEOMETRY. 
 
 plane cuts the projecting lines at an angle of 45°. But the direc- 
 tion of tlie projection depends upon tliat of the visual ray. Thus 
 if the eye be still at an infinite distance, but in the direction at^ the 
 projection will have tJie direction co^ but its length will remain 
 equal to ca. 
 
 Thus the direction of the projecting lines may be parallel to any 
 element of the cone whose axis is ca^ the angle at the vertex a be- 
 ing 90°, since all these elements make angles of 45° witli the pic- 
 ture plane MN. 
 
 Fig. 275 
 
 345. ]^ow any line which lies in the picture plane is its own 
 projection. ^ In representing a cube, therefore, as in Fig. 275, we 
 may assume its nearer face to lie in that plane, and it will thus ap- 
 pear of its true form and size, that is, a square, as shown. From 
 the preceding it follows at once that the edges which are perpen- 
 dicular to MN may be represented by parallel lines of their true 
 
DESCRIPTIVE GEOMETRY. 
 
 245 
 
 lengtli, but having any direction at pleasure ; which enables us to 
 show, in addition to the front face, either the right face or the left, 
 the upper or the lower, as may best suit our purpose. And, as 
 the figure shows, either of these faces at will may be made more 
 conspicuous than the other by proper selection of the angles. 
 
 We have, then, a system of true ohlique projection : it is more 
 flexible than the isometric, always quite as easily executed and in 
 many cases more so, and like it exhibits the three dimensions in 
 one view. All lines lying in planes parallel to the paper are shown 
 in their true forms and relations ; and not only these but lines per- 
 
 FiG. 276 
 
 pendicular to the paper are shown of their actual dimensions, the 
 introduction of any such senseless appliance as the " isometric 
 scale " being prevented by the very nature of the process. 
 
 346. This system is well adapted for purposes similar to those 
 in which isometric drawing is employed — such as the representa- 
 tion of joiner- work, as exemplified in the case of the box. Fig. 
 276, and in that of the peculiarly notched and fitted pieces shown 
 in Fig. 277. In the illustration, and especially in the sketching of 
 small mechanical details, it possesses the decided advantage over 
 isometry that, as shown in Fig. 278, circles whose planes are 
 
246 
 
 DESCRIPTIVE GEOMETRY. 
 
 parallel to tlie paper are represented by circles, whicli greatly ex- 
 pedites the work of construction. Those lying in planes perpen- 
 dicular to the paper, however, must here too be represented by 
 ellipses : since each circumscribing square is projected as a rhombus, 
 the axes will coincide with the diagonals, and may be found as in 
 Fig. 262. 
 
 Fig. 277 
 
 The use of ordinates, or offsets, in determining lines whicli are 
 neither parallel nor perpendicular to tlie paper is substantially the 
 same as in isometric drawing. Thus in Fig. 279 the point /in the 
 plane eg is located by setting off gli^ its distance from the plane gn^ 
 and then hf^ its distance from the front plane gm ; the 2:)oint h is 
 
 ^^^zf 
 
 Fig. 278 
 
 determined by a5, its distance from the end plane gn.^ Ixl its height 
 above the plane cg^ and dh its distance in front of tlie rear plane, 
 which is invisible. The two points /"and h being tlius fixed, the 
 projection of the line y^ is determined; and the rest of the con- 
 struction can be readily traced without explanation. 
 
 The lines which cast shadows, and are therefore to be made 
 
DESCRIPTIVE GEOMETRY. 
 
 247 
 
 heavy, can usually be determined by inspection, — the hght, as in 
 the common orthographic projections, being supposed to come 
 from over the left slioulder, and to go downward to the right as it 
 recedes, as explained in (333). 
 
 347. There is, then, no need to pursue this subject farther: 
 the principles which have been thus briefly set forth are sufficient 
 for applying either of these modes of projection to any subjects 
 within the common range of practice ; and additional examples of 
 cavalier projection are found in the pictorial illustrations intro- 
 duced in the preceding chapters. Both are very useful, with 
 certain limitations which have been suggested, and the question 
 
 Fig. 279 
 
 whether either is suitable for any given case can be settled by ex- 
 perience alone. 
 
 But one thing has been decided by experience beyond all 
 question ; and that is, that the attempt to apply either isometric or 
 cavalier drawing in the construction of a general plan of any 
 complex machine is certain to result in a melancholy failure : the 
 distortion, less noticeable in the case of minor details and detached 
 pieces, becomes unendurable when the various parts are assembled. 
 
 PSEUDO-PERSPECTIVE. 
 
 348. For the purpose of producing a certain effect of relief, 
 and conveying at least some idea of the three dimensions, while at 
 the same time avoiding this distortion as well as the labor of con- 
 
248 
 
 DESCRIPTIVE GEOMETRY. 
 
 structing a true perspective drawing, a mode of re23resentation lias 
 been devised, to which the name of Pseudo-Perspective seems 
 appropriate, of which we give a single illustration in Fig. 280. 
 
 This is, in principle, a modification of cavalier projection ; in 
 that, as has been stated, the j^arallel projecting lines are inclined to 
 the picture plane at an angle of 45°. But, referring to Fig. 274, 
 it will be seen that if the cone of visual rays should have a less 
 angle at the vertex, the projection of ca would be shorter ; and by 
 properly choosing this angle, the projection may be made shorter 
 than the line in any desired ratio, while its direction is still entirely 
 arbitrary. 
 
 Fig. 280 
 
 The pseudo-perspective drawing, then, is made by representing 
 the lines which are parallel to the paper in their true size, while 
 those which are perpendicular to it are reduced to, say, one twelfth 
 of the actual length, but paratlel. This of course renders tlie result 
 valueless as a working drawing ; but it gives a sense of depth, and 
 such representations are in many cases well suited for illustrations 
 upon a small scale, such as cuts for encyclopaedias and the like. 
 
 The distortion in the true cavalier projection is due to the men- 
 tal recognition of the facts that the true representations of receding 
 lines ought to converge, and that equal distances upon them ought 
 to appear less as they recede. Both these errors are made less con- 
 spicuous by reducing the lengths of these representations, which is 
 accomplished by the method of drawing above explained. 
 
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