^^V.' ''^ P Copyright, 1909 BY VICTOR T. WILSON COMPOSITION BY PRINTED BY RIPLEY & GRAY, PRINTERS THE SCIENTIFIC PRESS LANSING, MICH. NEW YORK PREFACE. Descriptive geometry is essentially a mathematical subject. The application of its principles to the making of working drawings, however, and the modifications which are made to suit the contingencies of practice, have had a tendency to obscure this fact, and like other theo- retical subjects it has suffered mutilation in the interest of short cuts to immediate practical uses. But does not technical education, after all, consist chiefly in an equip- ment of sound theory? It has been the author's purpose to refrain from any attempt to hold the student's interest by clothing a few principles with some immediate pratical application, but instead, to present a sound theoretical treatment. How well he has succeeded he leaves others to judge. The principles are herein formulated under theorems, as in plane and solid geometry; illustrative problems are solved in accordance with these theorems and special constructions discussed. The plan of,' at least, one well know text is followed of dividing all problems into two parts ; the first of which is a statement of the geometrical principles and the theoretical solution called an analysis ; the second is a description of the graphic solution, accom- panied by a drawing. An important feature is added, however, of giving the statement of the geometrical con- ditions and the solution in the analysis in a general form, instead of being made to refer to a certain kind of problem exclusively. iv PREFACE As an illustration of the generalized treatment throughout, attention may be called to the discussion of the cone. A common conception of a cone is that of a right circular cone, or cone of revolution. A generalized definition is given, however, to include all the surfaces which may be generated by a right line moving so as to pass through a fixed point and touch a curve. Further, it is stated that every cone is generically a right cone, and may be specifically named from the shape of a cross section so taken that a perpendicular, let fall from the fixed point or apex to the plane of the section, will pierce the latter in the center, focus, or other characteristic point, as a cusp, point of inflection, etc., etc. While the third angle is undoubtedly to be preferred for working drawings, it is not thought that descriptive geometry, as mathematics, has any concern with a partic- ular angle. The illustrative problems used deal indiffer- ently with all so that the emphasis can be laid upon sound theory. Exercises for students are grouped in the back of the book and suitably designated as belonging to a certain part of the text. An appendix deals with the subject of approximate methods, which has no proper place in the body of the book. The author wishes to acknowledge his indebtedness to the well known texts, notably those of Church and of McCord, and also in particular to Arthur G. Hall, pro- fessor of mathematics at the University of Michigan, for valuable service in examining the mathematical treatment of the text. TABLE OF CONTENTS. CHAPTEK I. THE POINT. Sections l to 12. PAGE Definition.— The planes of projection.— Lines and planes, not limited.— Parallel projecting lines.— Perspective and descriptive geometry.— Unfolding of the coordinate planes.— THEOREM I. The two project- ions of a point fully determine the position of the point in space.— Axes of reference for locating problems.— Exercises. .. 1 CHAPTEK II. THE LINE AND PLANE. Sections 13 to 88. The projections of a plane defined.— The alphabet of the plane.— Notation used.— Conventions recommended.— THEOREM IL— If through anylUne in space, a plane is passed perpendicular to a coordinate plane, it will Intersect that coordinate plane in a line which is the projection of the given line, upon the coordinate plane.— Corollaries.— To locate a point on a line.— THEOREM HI.- The projections of the point of intersection of two lines are the points of intersection, respectively, of the project- ions of the two lines.— Change of coordinate planes for a point.— The trace of a line.— The trace of a line on the coordinate planes, including the end plane.— To draw a line in any dihedral angle.— The alphabet of the line. — Revolution of a point about a line. — Change of one coordinate plane for a line.— To find the true length of a line _ _ _..ll CHAPTER III. PROBLEMS IN POINT, LINE AND PLANE. Sections 89 to 83. THEOREM IV. If a line lies In a plane, the traces of the line lie in the cor- responding traces of the plane.— THEOREM V. A given line is parallel to a plane when it is parallel to a line in the plane, or when it lies in a parallel plane— A plane through three given points.— The traces of a plane which shall contain a given line.— To assume a point in a plane. —THEOREM VI. The traces of parallel planes with a third plane are parallel. — Change of both coordinate planes with respect to a line.— vi CONTENTS PAGE Change of coordinate planes with respect to a plane,— A line parallel to a given plane through a given point. — A plane through one line and parallel to another line.— Within any given plane, perpendicular to a coordinate plane, to draw a line making a given angle with that co- ordinate plane.— To pass a plane through a given point parallel to two given lines.— Line of intersection of two planes.- Trace of a line with any plane.— THEOREM VII. If a line is perpendicular to a plane, the projections of the line are perpendicular, respectively, to the traces of the plane. — Through a given point, to pass a plane perpendicular to a given line.— Distance of a point from a plane.— Definitions.— THEOREM VIII, If two lines are perpendicular to each other in space, and one of them is parallel to a coordinate plane of projection, their projections on that plane are perpendicular to each other.— The angle between two lines which intersect.— Projections of any desired division of an angle between two lines.— Through a point in a plane, to draw two lines mak- ing a given angle with each other.— To project a line on any plane.— The angle a line makes with a given plane.— The angle between two planes. — Given a line in a plane, to draw another plane intersecting the first, In the given line, and making a given angle with the given plane.— The common perpendicular of two non-intersecting lines. — Given one trace of a plane, and the angle the plane makes with that coordinate plane, to find the other trace.— Distance from a point to a line.— Given one trace of a plane, and the angle it makes with the corresponding plane of projection, to find the corresponding trace. — The traces of a plane making given angles with both coordinate planes, — Given the angle a line makes with both coordinate planes, to draw its project- Ions. —To draw a regular pyramid, with base in any given oblique plane.— To draw a circle through any three given points.— Through a point to draw a line making a given angle with a given plane.— Through a given line to pass a plane perpendicular to a given plane.— Loci.— Review questions 84 CHAPTER IV. GENERATION AND CLASSIFICATION OF LINES AND SURFACES. Sections 84 to 98. A line is the path of a moving point.— Classification of lines.— Definitions. —Surfaces as generated by moving lines.— Classification of surface*.— Projections of curves.— THEOREM IX. Two projections of a curve being given, the curve will, in general, be completely determined, THEOREM X. If two lines are tangent in space, their projections on the same plane are tangent to each other.— A normal to a curve.— To draw a tangent to an irregular curve.- To find the point of tangency of an irregular curve and its tangent. — Tangent plane to a surface 83 CONTENTS vii PAGE CHAPTEB V. SINGLE CURVED SURFACES. Sections 99 to 185. THEOREM XI. A plane, which contains two consecutive rectilinear elements of a single curved surface, will be tangent to the surface throughout these elements, and the converse. — Cylinder defined.— To assume a point on any single curved surface.— A tangent plane to a cylinder through a point on the surface, a point outside and parallel to a line— A plane normal to a cylinder at a point on the surface, through a point outside, parallel to a given line, at a given point on the cylinder and parallel to a line.— Elements of contour,— The curves of intersection of planes and surfaces.— The development of single curved surfaces— The intersection of a cylinder and plane, and to draw a tangent to the curve.— To develop a right circular cylinder and curve of intersection of a plane— To develop an oblique cylinder. —The cone.— To assume a point on its surface, and to draw a tangent plane at the point, through a point outside, and parallel to a line outside.— To draw a normal plane, at a point on the surface, through a point outside and parallel to a line outside, intersection of a cone by any plane.— To develop a right cone, a cone In general— The con- volutes.— The helix and the helical convolute.— To assume a point on the surface and to draw a tangent plane at the point.— Intersection of the surface with any plane.— Tangent plane through a point outside, parallel to a line outside.- To develop the surface _ 98 CHAPTEE YI. WARPED SURFACES. Sections 136 to 156. Warped surfaces.— Table of classification.— The hyperbolic parabolold- THEOREM XII. The projections of all the elements of one system of generation of a hyperbolic paraboloid upon the plane directer, inter- sect each other in a point, known as a 'point of concourse', THEOREM XIII. The section of a hyperbolic paraboloid, by a plane parallel to the two rectilinear directrices, is a straight line.— Corollaries.- To assume a point on the surface, to draw a tangent plane, at a point on the surface.— To pass a plane through a line and tangent to it.— Its vertex, axis.— The conoid.— To assume a point on the surface, to pass a plane tangent at the point.— Right and oblique conoids.— The right and oblique helicoid, to assume a point on the surface, to draw a tangent plane at the point.— Its intersection by a plane.— THEOREM XIV. The trace of an oblique helicoid, with any plane parallel to the plane directer, is anarchimedian spiral _ _ _ 139 viii CONTENTS PAGK CHAPTEE VII. DOUBLE CURVED SURFACES AND SURFACES OF REVOLUTION. Sections 157 to 168. Double curved surfaces.— Tangent planes.— Laws of generation.— Surfaces of revolution.— To assume a point on the surface and draw a tangent plane.— THEOREM XV. A plane which is tangent to a surface of re- volution at a given point, is perpendicular to the meridian plane through the point.— Plane tangent to a sphere.— THEOREM XVI. If two surfaces of revolution, having a common axis, are tangent to each other or intersect. It will be in the circumference of a circle whose plane is perpendicular to the axis and center in the axis.— Intersection of surface of revolution with a plane. The hyperboloid of revolution. —THEOREM XVII. The hyperboloid of revolution, of one nappe, has two systems of generation and every element of the one system inter- sects all those of the other system.— THEOREM XVIII, The meridian curve of a hyperboloid of revolution, of one nappe, is a hyperbola.— To assume a point on the surface, and to draw a tangent plane at the point, its intersection with any plane, tangent plane through any line outside - - 165 CHAPTEK VIII. INTERSECTIONS OF SURFACES. Sections 169 to 182 Intersections of surfaces.— Intersections of bodies bounded by plane faces. —Complete or partial penetration. — Intersection of two pyramids, two cylinders, two cones, cone and cylinder, cylinder and convolute, cone and convolute.— Conditions of intersection of surfaces of revolution.— Intersection of those having a common axis, axes in the same plane, axes not in same plane.— Intersection of a single curved surface and surface of revolution.— Intersection of cone or cylinder with a warped surface — __ _ 188 APPENDIX. Discussing practical projection.— Third angle.— Approximate processes 197 EXEECISES. Series of graded theoretical problems, and also practical problems for students to solve. Total number. _ 199 DESCRIPTIVE GEOMETRY CHAPTER I. THE POINT. 1. Descriptive geometry is the science of representing forms, plane and solid, by projecting them upon two or more planes at right angles to each other with the aid of projecting lines perpendicular respectively to these planes, and it also consists of the solution of problems relating to the properties and magnitudes of the forms. 2. The planes upon which objects are projected are known as the coordinate planes of projection. They con- sist, in general, of vertical plane's, and a horizontal plane. The vertical planes may be at right angles to each other and are then distinguished as the fi'ont vertical and the end vertical planes respectively. Working drawings for guidance in construction are made according to the general principles of descriptive geometry. The projection upon the horizontal plane cor- responds to the plan view or simply plan; the projection upon the front vertical plane corresponds to the front view, front elevation, or simply elevation; the projection upon the end vertical plane corresponds to the end view or end elevation, as these terms are variously used. 2 DESCRIPTIVE GEOMETRY 3. Lines and planes in descriptive geometry, are not assumed to be limited. A line, which may be designated by letters at two of its points, is nevertheless possessed of the property of direction which does not stop short of infinity. A plane, likewise, extends to infinity. Hence parallel lines are said to have two points in common at infinity, and parallel planes to meet each other in a common line at infinity. 4. The parallel projecting lines spoken of in section 1 have a common point at infinity, hence it is said that the ''center of projection'' is at infinity. This point is also called the ''point of sight ^* because, were it possible for the eye to be in its position, the forms projected from it as a center would have their projections identical with them- selves, point for point, line for line. 5. Perspective is a branch of descriptive geometry in which the 'center of projection' is at a finite distance from the plane of projection. It gives the kind of pictures, for example, that would be seen if what was beyond a window were traced upon it by a person standing on the opposite side. The eye is the 'center of projection,' the window the 'plane of projection.' 6. Descriptive geometry and perspective together are parts of that broader subject projective geometry, in which the properties of figures are studied by projecting them upon any plane or planes from any center of projec- tion and where there is found to be a correlation of the *The 'center of projection' and 'point of sight' are both equally good designing terms used by different authorities. THE POINT FIGUBB 1 properties of the different projections of a figure upon the different planes and from the different centers of projec- tion. 7. The two fundamental planes of projection together with two end planes are shown in pictorial form or perspec- tive in Figure 1. The front vertical and the horizontal planes [V and H planes] form together four dihedral angles. That which lies in front of V and above H is the 1st; that which lies back of V and above H is the 2nd, and so on, continuing in the same direction. The line of intersection of these two planes is known generally as the ground line or abbreviated the G.L. Objects are projected upon the front vertical plane by means of projecting lines such as shown at v which means that the center of projection for the object is at infinity in a direction in front of the vertical plane, i. e., opposite to 4 DESCRIPTIVE GEOMETRY that of the pointed end of the arrow. This center may be thought of as either above or below the H plane as the object is above or below it. It is in fact, at infinity in the horizontal plane, and in a direction perpendicular to the G. L. Objects are projected upon the horizontal plane by means of vertical projecting lines such as at Ji, whence the center of projection is considered as infinitely distant above H in the V plane. Objects are projected upon the end planes by means of horizontal projecting lines in the directions of the double headed arrow e, for the right hand plane [with forms in the first angle] , from a center of projection which is at the left and for the left hand plane from a center of projection which is at the right, infinitely distant in the H plane. 8. The V, H and end planes arc unfolded for the graph- ical representation of forms until they form one and the same plane, see Figure 2. Either V is revolved* about the G.L. as an axis until it coincides with H, or H is revolved about the G.L. until it coincides with V; the result in either case is the same. The curved arrows in Figure 1, connected with the H plane, show the direction of the revolution. The end plane, which is at the right of any object, t is revolved about the ground line with the vertical plane or *An object Is said to 'rotate' upon Its own axis and to 'revolve' about any other axis not its own. t In the theoretical treatment of the subject. THE POINT / E plane above H and in front of V V above H H bacl£ of V E plane above E plane above H and back H and back of V of V "^ { E plane above \ H and in I front of V L. r \ E plane below ) H and in 1 &X)ntofV E plane below E plane below H and bock H and back of Y H In Front of V of V V below H Le E plane below / H and in ( front of V \ FlQUBB 2. GbLb until it lies in V, that part which is in front of V going toward the right, that part which is back of V going toward the left. Similarly, the end plane, which is at the left of any object, is revolved into coincidence with V, the portion which is in front of V going towards the left. 9. Let A be any point in space in front of V and above H, i.e., in the first dihedral angle. Conceive a plane to be passed through it perpendicular to both H and V. It would correspond to an end vertical plane. This plane would cut V in a vertical line and H in a horizontal line, both perpendicular to the G.L. at the same point. When V and H are unfolded, these lines would become one and the same straight line perpendicular to the G.L. The point A is projected on V by a projecting line perpendicular to V and lying in this end plane. It is a line parallel to H. A is projected on H by a projecting line perpendicular to H, also lying in this end plane. It is a line parallel to V. The piercing points of these projecting 6 DESCRIPTIVE GEOMETRY lines with the coordinate planes constitute the respective projections of A on those planes. From whence: The two projections of a point are on a common perpendicular to the G,L. When the Y and H planes are unfolded, see Figure 3, a is the projection on H of the point A and oa is the projection on H of the projecting line of A on V. oa is equal to the distance of the point A from V. Also a' is the projection on V of the point A and oa' is the projection on V of the projecting line of A on H, and oa' is the distance of the point A from H. This is true for a point in either dihedral angle or when lying in either or both coordinate planes. 10, Descriptive geometry does not deal with forms themselves but with the projections of the forms.* Fig. 4 illustrates the projections of a point when variously placed with respect to the coordinate planes. A is in the 1st angle; B is in the 2nd; C is in the 3rd; D is in the 4th; E is in V above H; F is in V below H; Gr is in H in front of V; I is in H back of V; J is in the G. L. whence both projections coincide with the point itself. When the pro- FI QUBE 8 ♦Hence the words 'the projection of are unnecessary and when hereafter a form is spoken of, it means the projection of the form. THE POINT «> I I I 0' 33 L. dl s% c'i di FlGTJBB 4. jections and the forms happen to coincide, it avoids confusion to consider the projections only. The positions shown in Figure 4 are known as the alphabet of the point, being the total of the various positions a point can have relative to the coordinate planes. 11 THEOREM I. The two projections of a point fully determine the posi- tion of the point in space. Proof; — If at the V projection of the point, a perpen- dicular is erected to the V plane, by hypothesis it will pass through the point in space. If at the H projection of the point, a perpendicular is erected to the H plane, it will also pass through the point in space. Hence the point itself must lie at the intersection of these two perpendiculars. But the perpendiculars are the two projecting lines of the point on the coordinate planes. Hence the point is fully determined. 8 DESCRIPTIVE GEOMETRY If the point is in the H plane, the length of the per- pendicular to the H plane is zero, and the point is its own projection. Likewise, if the point is in the V plane, it is its own V projection. If the point is in the G.L., both perpendiculars are zero and it is its own.V and H pro- jection. 12. Axes of reference may, for convenience in solving problems, be taken to coincide with the lines oa and oa' of Figure 3, the line oa', as a portion of a i/ axis and oa as a portion of an x axis. Hence, the position of the point A can be specified numerically by coordinates. For example, referring to Figure 3, A = 1, |, meaning that it is 1 inch above H and f inch in front of V, or its V projec- tion is 1 inch above the G.L. and its H projection is f of an inch in front of the G.L. The G.L. may also be taken as a ^ axis, being perpen- dicular to the plane of the x and y axes at their intersec- tion or origin, so that the exact location of points on a drawing can be specified by three coordinates, z, y and x^ given in the order named ; the z can be a distance from any convenient reference point, preferably one to the left as the border line of a sheet of drawings, the 2/ as + or — according as the point is above or below H and x as -\- or — according as the point is in front of or behind V. For example, in Figure 4, C can be specified as C = 1, — 1t6, — 1, where 1 means the z measured from the dotted line connecting a' and a; A can be specified as A = 0, i, f. The other points specified similarly, are: B = i, 1, - h D = If, - i, 1; E == II, i 0; F = 2, - i, 0; G = 2i, 0, f ; I = 2f , 0, - f ; J = 3i, 0, 0. THE POINT 9 When the z coordinate for any point is given as zero, let it be understood that the xy plane for the point may be taken anywhere, and the other points following are then measured from this plane as origin. If either the y or the x coordinate of any point is miss- ing and in its place is a question mark or a dash, it indicates that the coordinate omitted is to be found through other data in the problem. Exercises. 1 Locate the following points, designating the pro- jections by small letters, as a' for the V projection and a for the H projection, and state how each is related to the coordinate plane of projection: A = 0, 1, —\\ B = J, — i, -1; C = 1, l,i; D - U, -1, 1; E = 2, 1, 0; F = 2i, 0, -1; = 3,0,1; 1 = 31,0,0. 2 Locate the following points: A = 0, f, ^; B = i, - i - i; C = 4, f, - i; D = i, - i, f ; F = f, 0, f; G = f, I, 0; I = li, 0, 0. Project the same upon an end ver- tical plane at the right whose GbLe cuts the Gr.L. at the point = 2i, 0, 0. Construction. Figure 5 shows the result. Since the end plane is perpendicular to V and H, the point A is projected upon it by means of a projecting line parallel to V and H. This projecting line will pierce the end plane in a point which is as far from V and H as the point A is distant from those planes, i. e., as far from the ground line in each case between the Y, the H, and the end planes. When the end plane is folded into coincidence with the V plane, the projection of A upon it is a point at a distance oa from the 10 DESCEIPTIVE GEOMETRY Figure 5. G-K Le and a distance oa' above the G.L. The G.L. is coincident with the revolved position of the ground line between the end plane and the H plane. That which is at the left of Ge Le is that portion of the end plane which lies beyond the V plane. So when considering the pro- jections upon the end plane alone, Ge. Le is the V plane in end projection and the G.L. is the H plane seen in end projection so formed by looking at the end plane toward the right in the direction of the G.L. of the two coordinate planes. Likewise the end projection of the point B is found at a distance oh' below the G.L. and a distance oh to the left of the Ge Lb , being in the 3rd, dihedral angle, and so on for the other points. 3. Draw the projections of the following points: A, 1' ' below Hand I' ' in front of V; B, in the 3rd dihedral angle W ' from both V and H; C, U" behind V and f ^ below H; D, in H, If ' behind V; E, in V, If ' below H. Project these also upon an end plane whose Ge. Lb. may be chosen at pleasure. CHAPTER II THE LINE AND PLANE. 13. A plane is projected upon the coordinate planes only by the projection of points or lines lying in it or by draw- ing its lines of intersection with the coordinate planes, designated generally as traces, and specifically the V and H traces. Since a plane can intersect a line in only one point, the H and V traces of a plane will intersect upon the G.L. If a plane is oblique to H and V and also to the G.L., its H and V traces are inclined to the G.L., meeting it in a finite point. If a plane is parallel to the Gr.L. but oblique to H and Y, its traces are parallel to the G.L. and intersect the latter at infinity. If a plane is parallel to V its H trace is parallel to the G.L. and its V trace is at infinity. Similarly, if a plane is parallel to H, its Y trace is parallel to the G.L. and its H trace is at infinity. If a plane passes though the G.L. then both of its traces coincide with the G.L., and, referred only to its traces, the plane is indeterminate. Figure 6 shows the several positions. 1 and la are 12 DESCRIPTIVE GEOMETRY . / / ^/ \o,- RV SV 4 <"/ ^^9^ 7 2 UV 5 UH L \ Figure 6. the traces of a plane oblique to H and V and to the G.L. 2 is a plane parallel to the G.L. 3 is a plane parallel to V. 4 is a plane parallel to H. 5 is a plane passing through the G.L. 14. The alphabet of the plane is an expression used to designate the possible positions of a plane with respect to the coordinate planes. There are thirteen such positions. (1). A plane oblique to the G.L. as in 1 or la of Figure 6. (2) . A plane parallel to the G.L. and cutting through the 1st, 2nd and 3rd dihedral angles. (3) . A plane parallel to the G.L. and cutting through the 1st, 4th and 3rd dihedral angles. (4) . A plane parallel to the G.L. and cutting through the 2nd, 1st and 4th dihedral angles. (5) . A plane parallel to the G.L. and cutting through the 2nd, 3rd and 4th dihedral angles. (6) . A plane perpendicular to H and oblique to V. (7) . A plane perpendicular to V and oblique to H. THE LINE AND PLANE 13 (8) . A plane perpendicular to both H and V. (9). A plane passing through the G.L. (10) . A plane parallel to V and passing through the 1st and 4th dihedral angles. (11). A plane parallel to V and passing through the 2nd and 3rd dihedralangles. (12). A plane parallel to H and passing through the 1st and 2nd dihedral angles. (13). A plane parallel to H and passing through the 3rd and 4th dihedral angles. 15. The following notation will be followed throughout the text and in the exercises at the end of the book. A point in space is designated by a capital letter, as A, its projections by a small letter; if the V projection a' and the H projection simply a; if the end projection ae. A line is specified in general by two points on the line. A plane is designated by lettering its traces with a capital letter followed by Y, H or E according as it is the vertical trace, horizontal trace, or a trace with an end plane; the last letters of the alphabet will, in general, be used for this purpose. The letter does not specify any point on the trace. The line of intersection of the V and H planes is called the O.L. The line of intersection of V or H and an end plane is called the Ge.Le. 16. The following line conventions are recommended: The two projections of a point are connected by a dotted line, the dots to be about iV inch long, with A inch spaces between the dots. The G.L. and given lines, light, solid. 14 DESCRIPTIVE GEOMETRY The Required lines, heavy, solid. The projecting lines, construction and hidden edges of solids, dotted lines. Auxiliary lines, dash with dashes about i inch long and iV inch spaces between dashes. Traces of given planes, dash and dot; \ inch dashes with T6 inch dots and sV inch spaces. Traces of required planes, the same, heavy. Axes of revolution and axes of solids, dash and two dots,- It will be found helpful in the earlier part of the work to show all parts of forms in the 1st angle as solid lines and those in the other angles dotted. 17. THEOREM II. If through any line in space a plane is passed perpendic- ular to a coordinate plane, it will intersect that coordinate plane in a line which is the projection of the given line upon the coordinate plane. Proof: — For this projecting plane, by definition of pro- jection, contains the projecting perpendiculars of all points of the line upon the coordinate plane, the locus of their intersections with the coordinate plane constitutes the trace of the plane upon that coordinate plane. 18. Corollary 1. A line is, in general, fully determined when its projections are given, because if, through one of the projections of the line, a plane is passed perpendicular to that coordinate plane, it will contain the line, and if through the corresponding* projection a plane is passed ♦Note the use of the word 'corresponding' occurring throughout l"he book; Jt facilitates the generalized treatment, makes processes applicable to either coordinate plane. THE LINE AND PLANE 15 perpendicular to the corresponding coordinate plane, it will also contain the line. Hence the line itself must be identical with that line which is common to both planes, namely their intersection. 19. Corollary 2. If the projections of two points upon one of the coordinate planes are connected by a line, it is that projection of the line joining the two points in space. Similarly, if the corresponding projections of the points are connected by a line, it is the corresponding projection of the line connecting the two points in space. A line connecting the projections of two points on a coordinate plane is indeterminate if it lies in an end plane for the two projecting planes of the line coincide. 20. To designate a point on any line, assume either pro- jection of the point upon the same projection of the line, the other or corresponding projection is found upon the corresponding projection of the line, at the intersection with it, of a perpendicular to the G.L. through the assumed projection of the point. 21. The projecting planes of a line are useful in the solution of problems. Since the coordinate planes are perpendicular to each other and the projecting plane of a line is perpendicular to one of them, it follows that its corresponding trace is a line perpendicular to the G.L. Figure 7 shows the projections of a line AB. SH and SVare the traces of its H projecting plane; TH and TV are the traces of its Y projecting plane. That part of SV which is above the G.L. is the intersection of S with Y 16 DESCRIPTIVE GEOMETRY Figure 7. above H and that part which is below the G.L. is the intersection of S with V below H. Similarly, that part of the H trace which is in front of the G.L. is the intersection of S with H in front of V* and that part of the H trace which is back of the G.L., is the intersection of S with H back of y. The traces of a plane are not limited by the G.L, 22. THEOREM III. The projections of the point of intersection of two lines are the points of intersection respectively of the projections of the two lines. Proof: — The point of intersection of two lines, being common to both, can have but one projection on V and one projection on H. And since the point lies on both lines it must lie on the projection of both lines; the only point which satisfies these conditions is the point of intersection of the projection of the lines. *It Is well to speak of the H projection of a point as in front of the G.L., not 'below,' also back of and not ' above.' THE LINE AND PLANE 17 23. The form of proof to theorems, in descriptive geometry, differs in some respects from that nsed in math- ematical subjects heretofore studied by the student, and a failure to readily grasp the significance or meaning of the various steps is a common source of difficulty. As in other mathematics, each step in descriptive geometry is capable of demonstration and proof, and should be so treated in all study. The proofs of steps should be followed mentally until the processes involved are performed uncon- sciously. A faithful adherance to this direction will re- jnove much of the difficulties in the study of the subject. \ i' ? 1 >r \\ 1 1/ Y G. '\ ^j^^ ! Figure 8. 24. The change of a coordinate plane is many times useful. Consider a change of the V coordinate plane. Let A, B and C be three points shown in projection in Fig. 8. Project the same upon a new V plane or Vi, cut- ting H in the line Gr i .L ^ . The Vi plane is revolved about the Gti .Li . into coincidence with H, that which is above H going downward toward the left. The two projections of a 18 DESCRIPTIVE GEOMETRY point are on a common perpendicular to the G.L., hence A is projected on a perpendicular to the Gi .Li . through a. The distance of A above H remains the same, hence a/ is a distance from Gi.Li. that a' is above the Gr.L. The point B is in the 2nd angle, but Gi .Li . cut H back of h, therefore, when 6/ is located on a perpendicular to the Gi .Lj . at a distance above it that h' is above the G.L., it brings the point B in the 1st dihedral angle with respect to V 1 . C lies in V and its H projection is at the intersection of the Gi .L 1 . with the G.L., hence, it also lies in the new V, plane at c' a distance from c that c' if from c. The new H projections do not change; the distances of points from H are constant, hence the distances of their new projections in front of or back of the new Gi. Li. are constant. 25. A new coordinate plane, or Hi plane, may be used with respect to either V or Y^. This is not a horizontal plane but is called an Hi plane for convenience and to associate it with the V or Vi planes. Let the ground line of the Hi and Y or Vi planes be called G2:. Lg . The Hj projection of a point is found upon a perpendicular to the G2. L2. through the Y projection and at a distance from G2. L2. that the H projection is from the G. L. Or the Hi projection is found upon a perpendicular to the G2. L2. through the Yi projection and at a distance from G2. L2. that the H projection of the point is from the Gi. Li. In this change the Y or the Yi projection does not change, nor the distance of the point from the Y or Yi plane. The distance of a point from the Hi plane may be different from it distance from H. THE LINE AND PLANE 19 FIGUBB 9. Let A and B be two points in projection in Figure 9. Project them also upon a Vi plane whose Gi.Li. cuts the G.L. as shown. A new Hi plane also cuts Vi in a line G2. Lg. at right angles to the Gi. Li. a^' is found at a distance above the Gi.Li that a' is above the G. L. and «! is found on a perpendicular to the G2.L2. through ^i' and at a distance from it that a is from the Gi.Li. Simi- larly &^ is a distance in front of the Gi L^ , i. e., to the left, that B is from the H plane shown Sith\ and 6. is a distance in front of G2 L2 that h is from the Gi Li . 26. If a line is parallel to either coordinate plane, the trace of the projecting plane of the line on the other or corresponding plane of projection is a line parallel to the G.L. If a line is parallel to both coordinate planes of projec- tion then the traces of both projecting planes of the line are parallel to the G.L. Hence, when a line is parallel to 20 DESCKIPTIVE GEOMETRY y its H projection is parallel to the G.L. and when it is parallel to H its V projection is parallel to the G.L. If it is parallel to both H and V it is parallel to the G.L. and both projections are parallel to G.L. If a line is parallel to a plane all points of it are equally distant from that plane, hence the projection of a line on a coordinate plane to which it is parallel will show the true length of the line, and, by its angle with the G.L., the angle the line makes with the corresponding plane of projection. If a line is perpendicular to a coordinate plane its projection on that plane is a point and its projection on the corresponding coordinate plane is a line perpendicular to the G.L. and equal in length to the line itself. If a line is parallel to an end plane, it is necessary to project the line, by two of its points upon the end plane, to determine its length, direction, and its angles with either coordinate plane. 27. The trace of a line is the point of intersection of the line with any plane. Commonly, in descriptive geom- etry, it means the point of intersection of a line' with one of the coordinate planes of projection. Eef erring to Fig. 10, every point in the vertical pro- jection of A B, being the vertical projection of some point in the line, shows that the point D in which this projection cuts th^ G. L. is the vertical projection of the point of the line which is in H*, that is, it is the vertical projection of * The G. L. contains the vertical projections of all points lying in the H plane, in fact it is the vertical projection of the H plane. It also contains the H projections of all points in the V plane, and it is likewise the H projection of the V plane. THE LINE AND PLANE 21 Figure the H trace of the line AB. The H projection of the H trace is on the H projection of the line. Similarly the point of intersection of the H projection of AB with the G. L., e, is the H projection of the V trace; the V project- tion of the V trace is on the V projection of the line. Hence may be derived the following: 28. To find the trace of a line with either coordinate plane, prolong the corresponding projection of the line until it cuts the G. L. It is the corresponding projection of the trace desired. The other projection of the trace will be found upon the other projection of the line. The preceding is a general statement. It may be di- vided more specifically into a rule for finding the H trace of a line and one for finding the V trace of a line. The first of these can be stated as follows : Prolong the Y projection of the line until it intersects the G. L., which point is the V projection of the H trace; the H projection of the H trace is on the H projection of the line at the intersection of a perpendicular to the G. L. through the V projection. Observe that neither of the two coordinate points alone is the trace sought, but the two together constitute the projections of the trace. 22 DESCRIPTIVE GEOMETRY Figure 11. 29. Let AB and CD, in Fig. 1 1, be two intersecting lines the first of which is parallel to H and the second parallel to V. The point E is the V trace of the line AB. The H trace is at infinity. The V projection of the H trace will be on the G. L. at infinity and the H projection of the H trace will be on the H projection of the line at infinity. Similarly S is the H trace of the line CD. The H projec- tion of the y trace is on the G.L. at infinity, the V projection of the V trace is on. the V projection of the line at infinity. Hence to make a general statement : When a line is par- allel to either coordinate plane the projection of its trace on that plane is on that projection of the line at infinity; the corresponding projection of the trace is on the G. L. at infinity. 30. If a line is parallel to both coordinate planes its H and V traces are identical both upon the G. L. and the line at infinity. If a line goes through the G. L. at a finite point, that point constitutes both projections of both traces. THE LINE AND PLANE 23 FlQUBB 12. If a line lies in an end plane its traces are in the ground line or Ge . Lb . of the end plane with the Y or H plane. The rule is modified to suit the condition. For ex- ample see Fig. 12. AB is a line in an end plane. Eevolve this plane about its Ge .Le . into the V plane and get the end projection of AB at ae he . To obtain the H trace of the line prolong the end' projection of the line until it inter- sects the G. L. which is the revolved line of intersection of the end plane with the H plane. Let Se be the end projec- tion of the H trace. The H projection of the H trace will be on the H projection of the line at a distance above the G. L. that Se is to the left or back of the Ge . Lb . The V projection of S will be on the G. L. at its intersection with the Ge . Lb . Similarly to obtain the V trace prolong the end projection of the line until it intersects the Ge Lb which is in y. The H projection of the V trace is in the G. L. at the intersection with Ge . Le . 24 DESCRIPTIVE GEOMETBY 31. To find the traces of a line by means of an end plane, project the line upon the end vertical plane, then prolong this projection, if necessary, until it intersects the V plane in the Ge.Lk., and the H plane in the G.L. The former is the V projection of the V trace, the latter is the revolved H projection of the H trace. The H projection of the V trace is in the G. L. and the H projection of the H trace is on the H projection of the line, a distance in front of or back of the G. L. that the point is in front of or back ofV. 32. The proper study of descriptive geometry requires that every problem be divided into two parts, first analysis, second, solution or graphic demonstration. The first is the consideration of the geometrical principles involved, together with a description of the method of solution if it is a problem or of laying out the data if it is simply the graphic interpretation of the geometric conditions; its treatment should be general. The second concerns the graphical part and consists in solving the problem graph- ically according to the principles decided upon in the ana- lysis or graphically interpreting the data if it be only the interpretation of descriptive geometry conditions. The student is cautioned never to hasten to the second without doing full justice to the first. The analysis may involve the statement of descriptive geometry conditions, the geometric principles involved and a division of the main problem into parts each of which has its separate solution. However simple or complex the problem is, it should have its analysis preceding the graphical interpre- tation. Verify geometrically all steps or processes by going THE LINE AND PLANE 25 over the most fundamental and elementary principles in- volved in each problem until the interpretation of the fundamentals no longer gives any trouble, but is dealt with as unconsciously as letters are formed to spell words in writing. The entire subject depends upon a few compara- tively elementary concepts. These should be fully grasped before their varied application can be intelligently handled. 33. To draw a line containing a point in each of two given dihedral angles:— Locate the points in the specified angles and draw the projections of the line to contain the respective projections of the points. A line can also be made to pass through any one of the dihedral angles severally, by a consideration of its traces and their proper location. For example, if a line goes from left to right, and from the 1st angle through the 2nd into the 3rd, its V trace must be above the G.L., and its H projection must touch the G.L. to the left of the V projection; its H trace must be back of the G.L., i.e., the perpendicular from the Y projection of the H trace must meet the H projection of the line back of the G.L. If a line having the same direction goes from the 1st angle through the 4th into the 3rd, then the H trace must be in front of the G.L., i.e., the V projection must intersect the G.L. to the left of the H projection, and the perpendicular from the H projection of the V trace must intersect the V projection of the line below the G.L. A line lying in four dihedral angles is a line lying in either coordinate plane and crossing the G.L. in a finite point. 26 DESCRIPTIVE GEOMETRY 34. The alphabet of the line is an expression used to designate the possible positions of a line with respect to the coordinate planes. The directions of a line are seven in number. 1. Inclined to H and V. 2. Parallel to H and inclined to V. 3. Parallel to V and inclined to H. 4. Parallel to H and V or parallel to the G.L. 5. Perpendicular or normal to Y. 6. Perpendicular to H. 7. Inclined to H and V and perpendicular to the G.L. With direction (1) a line may go (a) through the 1st, 4th and 3rd angles, (b) through the 1st, 2nd and 3rd angles, (c) through the 4th, 1st and 2nd angles, (d) through the 4th, 3rd and 2nd angles, (e) through the G.L. and from the 1st to the 3rd angles, or (f) through the G.L. from the 2nd to the 4th angles. With direction (2) a line may go (a) through the 1st and 2nd angles, (b) through the 4th and 3rd angles, or (c) it may lie in H and hence cut the G.L. With direction (3) a line may go (a) through the 1st and 4th angles, (b) through the 2nd and 3rd angles, or (c) it may lie in Y and hence cut the G.L. With direction (4) a line may lie (a) in the first angle, (b) in the 2nd angle, (c) in the 3rd angle, (d) in the 4th angle. With direction (5) a line may go (a) through the 1st and 2nd angles, (b) through the 4th and 3rd angles, or (c) it may lie in H and go through the G.L. With direction (6), a line may go (a) through the 1st and 4th angles, (b) through the 2nd and 3rd angles, or (c) it may lie in Y and go through the G.L. THE LINE AND PLANE 27 . . S' i ! ii ! ^ i 1 1 j a j / V' \ o^ L 1 L T Figure 18. With direction 7, a line may go (a) through the 1st, *4th and 3rd angles, (b) through the 1st, 2nd and 3rd angles, (c) through the 4th, 1st and 2nd angles, (d) through the 4th, 3rd and 2nd angles, (e) through the G.L., the 1st and 3rd angles, or (f) through the G.L., the 2nd to the 4th angles. The student is advised to draw out these positions for himself as a drill in determining traces and as a reference chart for the earlier work in the subject 35. A point is said to revolve about a line as an axis when the path of the point is a circle whose plane is per- pendicular to the axis and whose center is in the axis. A simple case is that of the revolution of a point about a line which is perpendicular to either coordinate plane. If the line or axis of revolution is perpendicular to H, the conditions are as shown in Fig. 13. A point A revolves about the line O P. The path of the point being a plane perpendicular to OP, which in turn is perpendicular to H, must be parallel to H, hence has its center at and is a true circle in H projection. Its vertical projection is a 28 DESCBIPTIVE GEOMETRY o' G. Q' a's o' ai P' L Figure 14. limited straight line perpendicular to OP or parallel to the G. L. with center at o' . If the point A moves through any given angle, the angle in its true value is laid off with as a center and the vertical projection of the point is found upon the vertical projection of the path. Let the point move toward V from the position A until it lies in the V plane, ^i is its H projection, where the H projection of the path of the revolution crosses the G.L. and ^i 'is the vertical projection upon the vertical projection of the path. Another illustration is, when the point revolves about any line lying in either coordinate plane. Let A in Fig. 14 be such a point and QP the given line. The path of the point, being perpendicular to the axis, is projected on H in this illustration as a limited line perpendicular to o p for OP lies in H and a plane perpendicular to H will be projected on H in a straight line. The center O of the path is horizontally projected where the H projection of the path cuts the H projection of the axis. The radius of re- THE LINE AND PLANE 29 volution is the perpendicular distance of the point from the axis. This perpendicular is oblique, in this case, to both H and V. To state it in general terms it is the hypotenuse of a right angled triangle of which one side is the perpen- dicular distance of the projection of the point on the plane of the axis from the axis and whose other side is the distance of the point from the plane of the axis. In the example under consideration the first side mentioned is the distance oa, and the second side is the distance of a' from the G. L. This radius may be derived by auxiliary construction but it is better to get it in the following way: Conceive the plane of the path of the point to be revolved about its H projection into H that is about the line oa, A falls at a^ ; a ai is the second side of the triangle just mentioned and the revolved position of the path is a circle with oa^ as a radius and diameter ^2 ^3 • If A moves from the given position, in the direction of the arrow in revolved position, it falls in the H plane at 0^2 » a distance from equal to ai . If it continue motion it will go through the 4th angle again falling into H at ^3 . Every point in the path of A shown in the figure as re- volved into H is horizontally projected in the diameter ^2 ag and vertically projected on perpendiculars through these points as shown in the figure. The lengths of the ordinates to the diameter ^2 ^3 are the distances respect- ively that the points lie above or below H according to the direction of revolution of the path into coincidence with H. In this case all the points in the circle to the right of the diameter ^2 ^3 stand for positions above H and those to the left for positions below H. If it is desired to revolve the point A through a given 30 DESCRIPTIVE GEOMETRY angle about the line QP the angle can be laid off with o as a center and subtended by an arc of the circle ^3 a^ a^ . The new position of the point would be horizontally pro- jected by an ordinate to the axis and vertically by laying off the length of this ordinate above or below the Gr. L. as the case may be on a perpendicular to the G. L. through the H projection. Let A, Fig. 15, be a point in space to move about a line PQ in V, in the direction of the arrow shown in revolved position. The radius of revolution is the hypotenuse ai' and the revolved position of the path is shown at a\ h' I d\ , A is in the 1st angle; if it moves to the position hi' shown as revolved into V the V projection of the point is h' and the H projection at b is a distance below h' equal to the length of the ordinate hi' h' , The point is in H in front of V. If the point moves from the position B to the position C it is in the 4th and 3rd angles. C is a point in V below H and hence c, in the G. L., is its H pro- jection. If the point moves from the position C to the posi- tion D, it lies in the 3rd and 2nd angles, di' \s> the re- THE LINE AND PLANE 31 volved position of the point lying in H back oiY, d' is its V projection and d is a distance from the G. L. equal to the length of the ordinate (^'c^i'. From D to E the point is in the 2nd angle. At E the point has come into V, hence its H projection is in the G. L. The student is advised to copy the figure upon a some- what larger scale and locate the projections of a few intermediate points in the revolution to familiarize himself with the principles involved. 36. To revolve a point about a line which is oblique to both coordinate planes is a much more difficult problem than the foregoing.. If it should occur in any practical problem, a good solution is to change the coordinate planes; project the axis and point upon a Vi plane parallel to the axis, or to project them upon a plane through the axis, and then proceed in the manner outlined. 37. A change of one coordinate plane for a line may be made as follows: A line may be projected upon any new Yi plane (see figure 16) by finding the new Vi projections of points of the line. If the G i .L ^ . is taken perpendicular to the H projection of the line obviously the Vi projections of all points of the line will be in a common perpendicular to the Gi.Li., and we have indeterminate projections of the line as we have when the line lies in an end plane. Project the line upon a new vertical plane or Vg (see Figure 16) to distinguish it from the one just mentioned, whose G2 . L 2 . is parallel to the H projection of AB. Obvi- ously V2 is parallel to the line since G2 . Lg. is parallel to ah which is the H trace of the H projecting plane of AB. Finding the projection of AB on Va in al h\, we have a 32 DESCRIPTIVE GEOMETRY Figure 16. projection which is parallel to the line and of the same length as the line. This shows one of the uses of a change of coordinate plane. To project a line upon a new Yi plane whose Gi . L^ . is parallel to the H projection of the line results projec- tively in the same way as considering the H projecting plane of the line. And since the Vi projection is revolved about the G-i. Lj. into coincidence with H, and further since a revolution about the Y projection into Y is sim- ilar to a revolution about the H projection into H, we may state as the analysis for finding the true length of a line by projection: Analysis:— Bevolve the line about either projection into the coordinate plane of that projection. The revolved position is the true length required. 38. To find the true length of a line by revolution. Ansiiysis:— Revolve the line about the projecting line on V or H of any one of its points until the line is parallel to H or V, Its projection on the plane to which it is parallel, will give the true length required. THE LINE AND PLANE 33 FlQUBK 17 AND 18. Construction: — A convenient point is one extremity of the limited line. See Figure 17. Revolve the line AB about the H projecting line a' o' of the point A on H* until a& is parallel to the G.L. The point h travels in the arc of a circle with a as a center. This circle being parallel to H, is projected on V as a straight line parallel to the G.L., whence })\ the inter- section of such a parallel with a perpendicular to the G.L. through 6i is the position of the point in vertical projection when the line is parallel to V. a'6'i is then the true length of the line. Similarly, the line might be revolved about the V pro- jecting line of either A or B until the line AB is parallel to H, whence, in its revolved position, the H projection would show the true length of the line. (See Figure 18.) *a'o Is the vertical projection of the H projecting line. CHAPTER III. PROBLEMS IN POINT, LINE AND PLANE. 39. THEOREM IV. If a line lies in a plane the traces of the line lie in the corresponding traces of the plane. Proof:— By geometry, a line lies in a plane when every point of the line lies in the plane. The trace of a plane with a coordinate plane is a line lying wholly within the plane. It must intersect every other line of the plane, each in some point. Such a point being on another line of the plane, must be the trace of that line with the coordinate plane. The converse of this is not always true, for example, if a line goes through the G.L. at the point of intersection with the G.L. of any plane, the traces of the line will coincide with each other and lie upon the traces of the plane but the line will not necessarily lie in the plane unless some other point also does. 40. THEOREM V. A given line is parallel to a plane when it is parallel to a line in the plane or when it lies in a parallel plane. PROBLEMS IN POINT, LINE AND PLANE 35 Proof:— Since from geometry parallels between par- allels are equal, and conversely, if from any two points in the line parallels can be drawn which also terminate in the given plane and are of equal length, they will pierce the given plane in points which lie on a line parallel to the given line. Hence the given line is par- allel to the plane. And since what is true of one line will also be true of either line, a line is parallel to a plane when it lies in a parallel plane. 41. Corollary. If a line is parallel to either coordinate plane, then any plane passed through the line will have its trace on that coordinate plane parallel to the projection of the line on that plane. 42. Every point in the V trace of a plane is a point in V as well as a point in the plane, hence the corresponding projection is in the G.L. Similarly every point in the H trace of a plane is a point in H as well as a point in the plane, hence the corresponding or Y projection of this point is also in the G.L. To locate any other points lying in the plane conceive of a line or lines lying in it, passing through the point, and limited by the traces of the plane. 43. To pass a plane through three given points. Analysis:— Since the points lie in a plane they will lie upon lines of the plane passing through them. Hence if lines are drawn connecting the points the traces of two such lines are sufficient to locate the traces of the plane of the lines, that is the plane of the points. 36 DESCRIPTIVE GEOMETRY Q. ^ / I I, r/ \u' 1 IS t^^i:^ \Q -^ / / yaf^ / -^-' \ I / / / x/ Figure 19. Construction:— (See Fig. 19). Let A, B and C be three points in projection. Connect A and B by a line, also A and C. Q is found to be the V trace of AB (by Sec. 28) and E the V trace of AC. The V trace of the plane of the three points passes through the points r' and q' , Similarly U is the H trace of the line AB and S is the H trace of the line AC, and the H trace of the plane of the three points goes through the points u s. If the work is done accurately the H and V traces will meet on the G.L. Hence, if one trace is obtained by the method described only one point is required in the other trace. If one of the lines drawn connecting two of the three points is parallel to the G,L, then any plane passed PROBLEMS IN POINT, LINE AND PLANE 37 through this line is parallel to the G.L. by Theorem V, Sec. 41, and the traces of the plane are parallel to the G.L. Hence, it is only necessary to locate one point in each trace. If the traces of lines connecting three points in space do not come within available limits on the drawing, then any auxiliary lines may be taken to intersect those connecting the three points. These can be so chosen that their traces can be readily obtained. If the lines connecting the projections of the points are perpendicular to the G.L., it is necessary to project the points upon an end plane; if one line connecting two points is perpendicular to the G.L., then an auxiliary line may be used. The problem to pass a plane through a line and a point does not differ from the preceding. Draw an auxiliary line through the point to intersect the given line and proceed as before. A convenient auxiliary line to use is one which is parallel to the given line. 44. To draw the traces of a plane which shall contain a given line. Analysis: — Since an infinite number of planes can be passed through a line, draw, through the V and H traces of the line, the V and H traces of any plane which will intersect each other in the G,L, 45. The horizontal and the vertical of a plane are respective lines in the plane, the first of which is parallel to H and the second is parallel to Y. 38 DESCRIPTIVE GEOMETRY Figure 20. For example, see Figure 20, the line AB is a horizontal of the plane T for its H projection is parallel to the H trace of the plane T, and its V projection is parallel to the G.L. And similarly, CD is a vertical of the plane T for its V projection is parallel to the Y trace of the plane and its H projection is parallel to the G.L. These lines are useful in the solution of problems. It is useful, sometimes, to consider a line as the intersection of two planes, for example, the line AB of Figure 20 is the line of intersection of the plane T with a Vi plane which is parallel to the H trace of T and whose Gi. Li. coincides with ah. Its vertical trace will be the line W . In like manner the line CD is the line of inter- section of the plane T with a new Hi plane which is parallel to the V trace of T and whose G-g. L2. coincides with the line c' d' , Its H trace will be the line d' d. 46. To assume a point in a plane. Analysis: — Since lines lying in the plane and passing PROBLEMS IN POINT, LINE AND PLANE 39 through the point have their traces in the respective traces of the plane, one projection of such a line may he assumed at will to contain one assumed projection of the point, and hy means of the traces of the line, the corresponding projection may he found, which in turn contains the corres- ponding projection of the point. The horizontal or vertical of the plane are convenient lines to use for this purpose. E is a point in the plane T, Figure 20, found by the method described. 47. THEOREM VI. The traces of parallel planes with a third plane are parallel. Proof: By hypothesis the traces each lie in the re- spective planes and also in a third plane together. By Theorm V, if through any two points of one of the lines or traces parallels can be drawn, lying in the same plane, and of the same length the line connecting their extremities will be parallel to the line or trace. But the two such lines which may be drawn to connect the points on the second trace are themselves in a third plane together with the first line, and the line connecting their extremities lies in the third plane and is parallel to the first line or trace. 48. Both coordinate planes may be changed with respect to a line. Let AB, Fig. 21, be the line to project upon any new Vi plane and Hi plane. Find the Vi projection by Sec. 24. Let the new Hi plane be chosen by its G2 .Lg , it is perpendicular to the V, plane. The points A and B are 40 DESCRIPTIVE GEOMETRY Figure 21. projected upon it by drawing perpendiculars from their Vi projections to the G2-L2- The distances of the points from Vi are fixed and their projections upon any Hi plane are as far from the ground line of that plane with Y^ as the H projections are from the ground line with Vi hence ^2 and h^are on perpendiculars to G-g .L2 . through a\ and h\ and distant respectively from G2.L2. the amounts that a and h are from the G 1 .L i . 49. Change of one coordinate plane with respect to any oblique plane: Let T be any plane, see Fig. 22. Choose / / F IGUBE 22. any new Vi plane with Gi .Li . cutting TH in the point A. The plane Vi cuts a line from V which is perpendicular to PROBLEMS IN POINT, LINE AND PLANE 41 the G. L. because Vi is perpendicular to H, this line is hb' . h' is also a point in T because it is in the V trace of T. The Vi projection of the point B is on a perpendicular to the Gi.Li. through &, namly &'i . The Vi projection of A is a because the point lies at the intersection of TH and Gi .Li . It must moreover be a point in the new Vi trace of the plane T. Hence ab\ is the Vj trace of the plane T. If Gi.Li, had been taken perpendicular to TH, by in- spection, we can see that the plane T would have been per- pendicular to Vi . / / Figure 23. 50. Change of both coordinate planes with respect to a plane: Let T be any plane, Fig. 23. Find its V, trace as in the preceding paragraph. Now take any new Hi plane cutting V, in the line G^.L^. Where TV^ cuts G2.L0. is one point in the H^ trace of the plane, i. e., C. The plane Hx cuts a line from H which is perpendicular to V^^ , because both planes are perpendicular to V; this line is DE. E is also a point in T because it is in the H trace of T. The H projection of E is on a perpendicular to Gi . Li . 42 DESCRIPTIVE GEOMETRY through e'. The Hi projection of E is on a perpendicular to G2 .Ls ., a distance from it equal to d e. It is a point in the new Hi trace of the plane T. C is another point in the H trace. Hence, the line THi connecting C2 and e^ will be the Hi trace of the required plane. \ ^^ Figure 24. Consider another case of change of coordinate planes, with respect to any oblique plane. Let T, Fig. 24, be the given plane and let it be required to take a new Vj plane perpendicular to it and a new Hi plane parallel to it. The Gi Li of Vi is perpendicular to TH, for if a plane is per- pendicular to y its H trace will be perpendicular to the G. L. By preceding paragraph we find TVi . The new Hi plane, to be parallel to T must have its G^.La. parallel to the Vi trace of the plane or TV i . The plane Hi cuts a line from H which is perpendicular to Vi but this perpendicular by construction will be parallel to TH, meeting it at in- finity. Hence since TVi is parallel to G2 L2 , THi will also be parallel to and at an infinite distance from G2 .L2 . PROBLEMS IN POINT, LINE AND PLANE 43 51. To draw a line parallel to a given plane through a given point Analysis:— From Theorem F, a line is parallel to a plane when it is parallel to a line lying in the plane. There can be an infinite number of lines through a point parallel to a plane, each will be parallel to a line in the plane, hence draw any line lying in the given plane and through the point draw the projections of a line parallel to it. Figure 25. Construction:— Let T, Figure 25, be an oblique plane and C the given point. Draw any line AB in T (whose traces will lie in the traces of the plane). Through c draw a line parallel to ab and through c ' a line parallel \jQ> a'b' , The line through C is parallel to the plane T. Since an infinite number of lines can be drawn through C and parallel to T, they will constitute a plane, so if two lines are drawn through C parallel respectively, to two lines lying in T, they will lie in a plane parallel to T through the given point. But such a plane, by theorem VI, would also have its traces parallel to those of T, hence it is not necessary to draw but one line throuiorh C to locate it. 44 DESCRIPTIVE GEOMETRY 52. To pass a plane through one line and parallel to another line. Analysis: — By Theorem F, a line is parallel to a plane when it is parallel to a line lying in the plane. Hence draw through a point in the first line, a line parallel to the second. The plane of the two intersecting lines will he parallel to the second line. Figure 2( Construction:— In Figure 26, let AB be a line through, which to pass a plane parallel to the line CD. Through any point of AB as O draw an auxiliary line EF parallel to CD. The plane of AB and EF is the plane required. If the two given lines are parallel to each other, i.e., their projections respectively parallel then any plane through the one will be parallel to the other. If either one of the two given lines is parallel to the G, L., the required plane will be parallel to the G.L. and its traces will be parallel to the G.L. If either one or both of the lines have their projections perpendicular to the G,L,, then the traces are obtained by use of an end plane. / PROBLEMS IN POINT, LINE AND PLANE 45 There can be but one plane containing a given line and parallel to another line unless the two lines are parallel to each other. 53, Within any given plane perpendicular to a coordi- nate plane, to draw a line making a given angle with that coordinate plane. \ / Analysis:— j5^ TJieorem II, one projection of the line will coincide with the trace of the plane, that projection on the coordinate plane to which the given plane is perpen- dicular; hence we can assume that projection at will, Bevolve the assumed projection together with the trace of the given plane into the corresponding coordinate plane about the corresponding trace of the plane as an axis. The angle can here be constructed its true value. Bevolve the line back again to its original position, and both projections of the line and the angle will be shown. Construction: — Let Fig. 27, show a plane perpen- dicular to V. Required a line AB lying in it and making an angle of 60' with V. Assume a'b' with A a point in V. 46 DESCRIPTIVE GEOMETRY Revolve AB about TH as an axis until A fall into H. The angle at a^ with the the G.L. as one side will be the true angle and h^ can be obtained in the manner shown. Eevolve the line back again into its original position, a^ moves along the G.L. as in the figure to a at the inter- section with a perpendicular from a' to the G.L. and h^ moves parallel to the G.L. to the position & on a perpen- dicular to the G.L. through h' , whence ah is the H projection of the line. If the plane T has hut one trace, which would of course be parallel to the G.L. then one projection of the required angle would coincide with the given trace of the plane and the other projection of the angle would at once show its true value. If the given plane is perpendicular to hoth coordinate planes the angle is obtained by revolving the given plane into Y about its V trace. The projection of the angle on the CT'd plane would show its true value. 54. Through a given point to pass a plane parallel to two giv^ auxiliary plane that is perpendicular to the line of intersection of the two given planes and cuts from each plane a line, cuts also a line from the projecting plane of the line of intersection which in turn pierces the coordinate plane in the trace of the auxiliary plane with the latter. Hence a trace of the auxiliary plane may he assumed and the piercing point tvith it found of the line of intersection of the two given planes. The 64 DESCRIPTIVE GEOMETRY remainder of the solution is the same as for Method No. 1. Construction:— Let the conditions be given again as they were in Figure 38, see Figure 39. Assume the H trace of an auxiliary plane P perpendicular to the line of inter- section of the planes.* The point q in which this trace cuts the H projection of the line of intersection is the H projection of the H trace of the line OQ cut from the pro- jecting plane of the line of intersection by the auxiliary plane. Eevolve the line of intersection CD about its H projection into H and at q draw a perpendicular to it qo^. O is the piercing point of the line CD with the auxiliary plane P and o is its H projection. Whence aob is the pro- jection of the angle. By revolving the vertex into H at O2 » it is found that ao 2 & is the true value of the angle. If both planes are perpendicular to a coordinate plane, the angle between their traces on that plane is the angle between the planes. If one of the planes has its traces in the G,L., or if *Its corresponding trace Is not needed In the construction. PROBLEMS IN POINT, LINE AND PLANE 65 both planes have their traces in or parallel to the G.L., then it is necessary to find their intersection with an end plane. In the first case the intersection of the traces of both planes on the end plane is a point in the line of inter- section of the two planes. In the latter two cases the angle between the traces on the end plane is the true angle between the planes. If the traces of both planes intersect each other on the G.L., it is necessary to either use auxiliary planes perpen- dicular to a coordinate plane or to pass an auxiliary plane parallel to one of the given planes in order to get the pro- jections of the line of intersection of the planes. In the latter case the projections first obtained are those of a line parallel to the line of intersection. If the angle between any oblique plane and a coordinate plane is desired, the auxiliary plane will be perpendic- ular to the coordinate plane and hence cut from the oblique plane a line of greatest declivity with respect to the coordinate plane and from the latter a line of greatest declivity with respect to the oblique plane, the angle between which is^the desired angle. 69. Given a line in a plane, to draw another plane inter- secting the first in the given line and making a given angle with the given plane. Analysis:— T^e given line will be the line of intersection of the two planes, hence to establish the plane angle which is given, through any point of the given line pass a plane perpendicular to it and note its intersection with the given plane. Revolve the line of intersection about a trace of the auxiliary perpendicular plane into a coordinate plane and construct the angle required. Revolve it bach again and 66 DESCRIPTIVE GEOMETRY the second side of the angle as derived and the given line determine the vlane desired. 70. To find the angle between two planes by the method of change of coordinate planes. Analysis:— 7/ one coordinate plane is taken parallel to the line of intersection of the given planes and the corres- ponding one perpendicular to it, this line will he projected upon the latter coordinate plane as a point and the angle between the planes will he projected upon the same planCy its true value, heing the angle between the traces. Figure 40, Construction: — Let the planes T and S be given as in Figure 40. Take a Gi .L i . parallel to the H projection of the line CD, which is the line of intersection of the planes, giving c' id' 1 as the V projection of the line on the Vi plane parallel to it; on this plane also find the V traces of T and S by using C as a point in each plane and drawing PROBLEMS IN POINT LINE AND PLANE 67 horizontals of the planes respectively. TiV and SiV are the traces so found. Next take a G2.L2. perpendicular to the Yi projection of CD getting the new Hi projection c^d 2 as the projection of the line on this plane. This is also a point in the new H traces of the planes T and S. Hence the angle between TgH and S2H is the true angle between the planes. Figure 41 71. To find the common perpendicular of two non-in- tersecting lines. Analysis: — If an auxiliary plane is passed through one of the lines and parallel to the other the common perpendic- ular will he projected on this plane as a point , i.e., the point of intersection of the one line and the projection of the 68 DESCRIPTIVE GEOMETRY other. Hence at this point erect a perpendicular to the auxiliary plane to meet the other line. It will he the line desired. Construction: Let the lines AB and CD be given as shown in Fig. 41. Pass a plane through the line CD and parallel to AB, (by Sec. 52) using an auxiliary line EF parallel to AB and intersecting CD. TH and TV are its traces. Project AB on the planq. T by using perpendicu- lars from A and B to the plane T (by Sec. QQ) their piercing points are Ai and Bi respectively. Aj Bi is the projection of AB on the plane T. At O, the point of intersection of the projections, draw the projections of a perpendicular to the plane T. It intersects the line AB in the point P. OP is the required perpendicular o'px\ is true length of this perpendicular. Method No. 2, Analysis:— ^^ change of coordinate planes^ project the lines upon a new V^ plane tvhich is parallel to one of theyn and upon a new H^ plane perpendicular to the same line. The required distance is the perpendicular distance between the two lines as projected on H^, one of them being projected as a point. 72. Given one trace of a plane and the angle the plane makes with that coordinate plane, to find the other trace. Analysis: — If the trace of an auxiliary plane -is draivn which cuts out the lines of greatest declivity of the coordi- nate plane and the given plane tvith respect to each other, its trace will he perpendicular to the given trace and the projections of the lines cut out of each plane will coincide ivith it. Hence assume a line of greatest declivity of the given plane to he revolved ahout its projection on the coordi- PROBLEMS IN POINT, LINE AND PLANE 69 nate plane, until it lies in the latter. It will then make the angle with its projection that the required plane is to make with the coordinate plane. Construct this angle and find the revolved position of the piercing point with the other coordinate plane, of the line of greatest declivity. The true projected position of this piercing point is a point in the desired trace. \ \ X \ TV !P \o' v/L. \ ^- ./ "\ y^ X^ /' y y F IGUKE 42. Construction:— Let the trace TH of a plane be given as in Fig. 42 and let its required angle with H be 60** , SH is the trace of any auxiliary plane perpendicular to both T and H. The portion of it, op, coincides with the H projec- tion of the line cut from the plane T and also that of the line cut from H. Assuming the former to be revolved into H, it falls at Oi9 1 , making an angle of 60** with op. The point p 1 , being on a perpendicular from op to the line op i , is the revolved position of the piercing point of the line of 70 DESCRIPTIVE GEOMETRY greatest declivity of the plane T with the V plane, hence the projection of the piercing point is a distance above the G.L. on a perpendicular at p that the point p, is from p. TV is the required trace of the plane. 73. To find the distance from a point to a line. Method No. 1, Analysis:— 7/ a plane is passed tJirougJi the point perpendicular to the line, the distance from the given point to the piercing point of the line, with this plane, will he the required distance. Method No. 2, Analysis:— Pass a plane through the point and the line, revolve both about either trace of their plane until they lie in the coordinate plane; the perpendicular distance between point and line will be projected its true length, 74. Given one trace of a plane and the angle it makes with the corresponding plane of projection, to find the corresponding trace. Analysis:— The plane will be tangent to a right circular cone whose apex is on the given trace with its base in the corresponding coordinate plane and axis perpendicular to the latter; the elements of this cone will make the same angle with the corresponding coordinate plane that the required plane makes. Hence assume any point upon the given trace and draw the projection of the cone upon that coordinate plane; it will be an isosceles triangle with base in the G.L., and base angles equal to the given angle. Draw the corresponding projection of the cone which will be a circle. The trace of the required plane upon this cor- responding plane will be tangent to this circle. PROBLEMS IN POINT, LINE AND PLANE 71 Construction:— Let TH be given as in Figure 43 and let the required angle with V be 60°. Assume any point on it as O. Draw OP and OQ the contour elements of the pro- jection of the cone on H whose base angles are 60°. qp is the H projection of the base in V. Draw a circle with o' as a center and o'p' as a radius. Tangent to this circle draw TV, the required trace. Note:— The problem could be solved by considering only the triangle formed by a plane passed perpendicular to V and the plane T. o'o\ is the revolved position of the line cut from H and could be drawn from o ' in any direc- tion, o'p/ is also the revolved position of the line cut from V and will be perpendicular to o' o\. Lastly, the line o' ip' I is the revolved position of the line cut from the plane T and can be drawn to make the required angle with o'p' I, The required V trace, then, is tangent to a circle of radius o'p/ The true revolved position of the triangle is when op' 1 is perpendicular to TV. TO DESCRIPTIVE GEOMETRY 75. Given the angles a plane makes with both coordinate planes to locate its traces. Analysis:— T/^e required plane can he considered as tangent to a sphere of any radius whose center is in the G.L, If planes are assumed which are perpendicular to the given plane and to the coordinate planes respectively, thus cutting out lines from each which will he measures of the angles the required plane makes with the respective coordi- nate planes, and if further these planes are passed through the center of the sphere they will cut lines respectively from the required plane which are tangent to the sphere. There- fore the triangles formed hy the assumed planes cutting the required plane and hoth coordinate planes in each case can he assumed separately — dealt with as the one triangle was dealt with in Section 72, Ohserving the fact that the revolved position of the lines cut from the required plane in each case will he tangent to a circle ivhich is the projection of the sphere and have each one vertex at the center of that sphere. Construction.— Let the required angles of a plane with V and H be a and P respectively. See Fig. 44. Draw the projections of a sphere with any radius and center at any point O. Next draw an assumed revolved position of the lines which would be. cut from the V plane, the required plane and the H plane, by a plane assumed to be perpen- dicular to the first two. o ' a ' &i is the triangle made by these lines, oa' is the line cut from Y, a'h^thQ line cut from the required plane and o'h^ the line cut from the H plane. h^a' o' is made equal to a and a'h^i's, drawn tangent to the sphere. The V trace of the plane is tangent to a circle with radius o' a' and the H trace goes through a point h on PROBLEMS IN POINT, LINE AND PLANE 73 Y "k /' FiaUBB 44. a perpendicular to the G.L. at O, and distant from equal to o&, . Likewise, and constructed similarly, ocd\ is the triangle formed by lines cut from the H plane, the required plane and the V plane by a plane perpendicular to the first two, with the angle 5 as shown. Ocis the line cut from H, cd\ , the line cut from the required plane and 06?' i, the line cut from V. The H trace of the required plane is tan- gent to a circle of radius oc and the. V trace goes through the point 6?', a distance od\ above H. 76. Given the angle a line makes with both coordinate planes to draw its projections. Note:— In this problem the analysis for the general so- lution is difficult to follow hence a special solution is added after the general form. General Analysis: — Consider the line as limited by its 74 DESCEIPTIVE GEOMETKY traces. Assume either projecting plane of the line to he revolved into either coordinate plane about its intersection with that coordinate plane, Bevolve the other projecting plane of the line bodily with it into the same coordinate plane and about the line of intersection of the two project- ing planes, which is the line itself. The revolved position of the line is the hypotenuse of two right angled triangles, an angle of each of which being oppositely directed, is the respective angle the line makes with the coordinate planes. Specific Analysis:— Assume a line limited by its V and H traces. Bevolve the H projecting plane, bounded by the line, its H projection and the V trace of the projecting plane, into V; next revolve the V projecting plane of the line bounded by the line, its V projection and the H trace of the projecting plane, into V about the revolved position of the line when it lies in V. In the first triangle, the angle between the G.L. and the revolved position of the line is the angle with H. Opposite to that, with the revolved position of the line as a side, is the angle the line makes with V, from which, by completing the triangle, the distance of its H trace from the G.L. can be found. Next revolve back into proper projective position the first triangle, to do which the H trace must be located at the distance from the G.L. which it is found to be from the revolved position of the second triangle. Construction: Assume oa'b^ , Fig. 45, to De the re- volved position of the H projecting plane of a line of which a '&i is the line itself. On the latter as an axis, assume the V projecting plane to be revolved into Y. The perpendicular distance of c' i from a'biis the revolved pos- ition of the distance of the H trace from the V plane, i. e. PROBLEMS IN POINT LINE AND PLANE 75 from the G-. L. Next, lay off the distance c'l &i on H from the G. L. and draw a horizontal line. Where an arc of a circle, with radius ahi and center at a cuts this horizontal in h, is the H projection of the H trace of the line and ah is the H projection of the required line. The V projection is h'a\ 77. To draw a regular pyramid of given base and altitude with its base in a given oblique plane. Analysis:— r^.y' Assume the base as revolved about either trace of the given plane until it lies in the coordinate plane of that trace, where it can he drawn its true shape. Hevolve it hack again into the given plane hy any convenient method. Next, to find the projections of the apex of the pyramid, draw a perpendicular to the plane at the center of the base, and hy any convenient method, get the projected distance laid off on this perpendicular, i.e., the projection of the altitude of the pyramid. The apex connected with the corners of the base completes the pyramid. Anaiy sis:— fJ2. J Assume the pyramid to he projected 76 DESCRIPTIVE GEOMETRY upon a plane perpendicular to that of the base, i,e., parallel to the axis. Assume this projection as revolved about the trace of its plane with the coordinate plane until it lies in the latter. Here the axis will appear its true length, since it lies in the coordinate plane the base will be projected as a straight line. By revolving, next, the base about its inter- section with the coordinate plane into the latter, its true shape will appear and the relative position of its vertices be ascertained. By counter revolution the projection of the pyramid may be constructed. Construction:— According to Analysis 2. Let a hexa- gonal pyramid with diameter of base of 2 inches and altitude 2| inches, have its base in any oblique plane T, see Figure 46, and an edge of the base lying in H. Assume the pyramid to be projected on any plane S per- pendicular to H and to T. The line ab^, is the revolved projection of the base on S, being also the revolved posi- tion of the line cut from T by S and p^ is the apex also revolved into H. Next assume the base of the pjn^amid to be revolved about a& i , as a diameter, giving 1 1 , 2 1 , 3 1 , etc., as the vertices of the base In, 2n, 3„, etc., are the true projected positions of the vertices on the plane S when the latter is revolved into H. The horizontal distances of the vertices from the diameter are constant, hence the edge of base in H can be assumed anywhere and by counter revolution the vertices 1 ,2 ,3, etc., found. The vertical projection of the pyramid is obtained by noting that the distances of points above the G.L. are respectively equal to the distances of the vertices 1 1 , 2i , 3 1 , etc., above SH. It is to be noted that the method just described is PEOBLEMS IN POINT, LINE AND PLANE 77 FiGUBE 46. really that of a change of coordinate planes where SH is the G.L. of a new Vi plane parallel to the axis of the pyramid. 78, To draw a circle through any three given points. Analysis:— T/^e three points are the extremities of chords of the circle passing through them. Hence find the 2)lane of 78 DESCRIPTIVE GEOMETRY tJie three points; revolve them ahout either trace of their plane into the coordinate plane and draw the circle passing through them. Bevolve the circle into its true projective position hy any convenient method. Note:— A convenient method is to use parallel chords perpendicular to the trace of the plane about which the three points are revolved. Figure 47. PROBLEMS IN POINT LINE AND PLANE 79 Construction:— Let the three points A, B and C be given as in Figure 47. The traces of their plane are TH and TV respectively. Revolve the points about the H trace of their plane into H to fall at a i , &i and Ci , respect- ively, whereupon the circle passing through them may be constructed. To revolve the circle back into its true projective position, the following method may be employed. Pass a series of auxiliary planes Q, R, S, etc., perpendicular to the H plane and the plane T to cut the circle lying in the latter plane, each in two points. The lines of intersection of these several planes with T, will pass through these points, respectively ; further, the points revolved into H about TH, will lie on a perpendicular to TH as shown. By counter revolution these may be located on the respective lines of intersection of the auxiliary planes with T. 79. Through a given point, to draw a line making a given angle with a given oblique plane. Analysis: — The locus of the lines which can he drawn from the given point to the given plane j making a certain angle with it, form the elements of a right circular cone whose hase is in the plane and whose apex is the point. Therefore draw the cone with elements at the required angle with the base hy a similar method to that for the hexagonal pyramid in Section 77, Choosing any one element, revolve it hack into true projective position. The construction is left to the student. 80. Through a given line, to pass a plane perpendicular to a given plane. Analysis:— The required plane must contain a perpcn 80 DESCRIPTIVE GEOMETRY dicular to the given plane, hence, through any point of the given line, draw a perpendicular to the given plane. The plane of the two lines is the plane required. The construc- tion is left to the student. 81. LociL When a point moves so as to satisfy a given condition or conditions, the path it traces is called its locus under these conditions. As an example, the parabola is the locus of a point moving in a plane according to the conditions of the equa- tion y''=4px. This may be defined, in graphical terms, as the locus of a point which moves in a plane so that its distance from a fixed line is equal to its distance from a fixed point. The line is known as the directrix, and the point as the focus of the parabola. The ratio of the two distances is, in mathematics, called the eccentricity (e) ; for the parabola, it is equal to unity. It has different values for the different conies. The ellipse is the locus of a point, which moves in a plane, so that the ratio of its distance from a fixed line to its distance from a fixed point is greater than unity, in X^ 11^ other words satisfies the equation -^ + 7^ = 1. a The hyperbola is the locus of a point which moves in a plane, so that the ratio of its distance from a fixed line to its distance from a fixed point is less than unity, in other x^ v^ words satisfies the equation -^ — -?2 = 1. a The circle, defined in the same terms, is the locus of a point, which moves in a plane, so that the ratio of its distance from a fixed line to its distance from a fixed PROBLEMS IN POINT, LINE AND PLANE 81 point is equal to infinity. In more generally understood language, it is the locus of a point, which moves in a plane, so as to be equally distant from a fixed point. Graphical locii are a legitimate part of descriptive geometry. 82. To draw the locus of points equally distant from two given planes. Analysis:— T/^e locus of points equally distant from two straight lines in a plane is the bisector of the angle between the two lines, hence the locus of points equally distant from two planes, each containing one of the lines, is the plane bisector of the dihedral angle between the two planes. Therefore find the angle between the two planes, bisect it and pass a plane through the bisector and the line of inter- section of the planes. This plane ivill be the required locus. 83. To draw the locus of points in space equi-distant from three given points. Analysis: - The locus of points in space, equally distant from two points, is a plane perpendicular to and bisecting the straight line connecting the tivo points. Hence the locus of points, equally distant from three points, will be the common line of intersection of three such planes. Therefore find the plane of the three points and by revolving them about either trace of their plane find the perpendicular bisectors of the lines connecting the points; these will meet in a point which is the center of the circle passing through the three points. Find the true projective position of this point in the plane. Through it draw a line perpendicular to the plane of the points which will be the locus required. 82 DESCRIPTIVE GEOMETRY REVIEW QUESTIONS If two lines are parallel in space, are their projections parallel ? If two lines are perpendicular to each other in space, are their projections perpendicular? If two planes are parallel to each other in space, are their respective traces parallel ? If two planes are perpendicular in space, are their respective traces perpendicular to each other ? If a line is parallel to a plane, are its projections parallel respectively to the traces of the plane ? If a line is perpendicular to a plane, are its projections, respectively perpendicular to the traces of the plane ? What is the locus of points equi-distant from a given straight line 1 What is the locus of lines which make a constant angle with a straight line at a common point in it ? What is the locus of points in space equi-distant from a fixed point ! What is the locus of points in space equi-distant from two intersecting lines I CHAPTER IV. GENERATION AND CLASSIFICATION OF LINES AND SURFACES. 84. A line is mathematically the path of a moving point under fixed conditions, i.e., the locus of the point under those conditions ; it may also be an element of a surface which latter also has a mathematical interpretation as the path traced by a moving line under certain conditions. A line may also he defined, according to various properties of the line, one of which, for example, is the intersection of two surfaces or solids. A straight line, or right line, is the path of a point which moves in a fixed direction, or again, the intersection of two planes. Any two consecutive positions of a moving point, being infinitely close, may be regarded as the extremities of an infinitesimal element of the path traced, or an elementary line of it, and in going from the one position to the next, no matter what the ultimate direction of the path, the elementary path traced is straight. All lines may be grouped into three classes according to the law governing the motion of the generating point. (1.) Eight lines. (2.) Single curved lines. (3.) Double curved lines. 84 DESCBIPTIVE GEOMETRY If a point moves in a fixed direction, the path generated is a straight line. If a point moves in a constantly changing direction the path is a curve; whether it be a curve of single curva- ture or a curve of double curvature depends upon the following conditions : If any four consecutive points, that is, points in the curve infinitely near to each other, or any three con- secutive elements, that is, right lines connecting the consecutive points as defined, lie in the same plane, then all the points of the path lie in the same plane and the curve' is a curve of single curvature, i.e., a plane curve. If no four consecutive points nor any three consecutive elements, passing through the four points, lie in the same plane then the curve is a curve of double curvature, i.e., it is not a plane curve. A curve of any kind may be further considered as gen- erated by the motion of a line intersecting itself in succes- sive positions in which case the curve is known as the en- velope of the successive positions of the line. For example, if two intersecting lines as in Figure 48, are divided into a number of equal parts and the division on one farthest from the vertex is joined to the first division from the vertex on the other and so on until all the divisions are connected by lines, it will be seen that these lines are tangent to a curve. This curve is the parabola and is the envelope of its tangents. A curve of double curvature may be considered as generated by the motion of a plane whose successive positions intersect each other in lines which constitute suc- cessive elements of the curve. For example, just previously LINES AND SURFACES CLASSIFIED 85 Figure 48 . it was stated that if no four consecutive points, or three con- secutive elements connecting these points, lie in the same plane the curve is a curve of double curvature. Conceive any four such points connected by elements, one through points 1 and 2, another through 2 and 3 and a third through 3 and 4. A plane can be passed through the first and second elements, and another through the second and third, but the two planes will intersect each other in the second element, i.e., 2 and 3 and this is an element of the curve also ; in like manner the successive planes will inter- sect each other two and two in successive elements of the curve. To sum up the classification of lines and give illustra- tive examples of them, however by no means complete, the following table is appended: — 85. Table of classification of lines. 86 DESCRIPTIVE GEOMETRY ^ eg II i 0) jg ?H •3.& <» • rH 1 O ^ ^ ^ 2 ® S-2 C3 O i ^ o cS o a; J3 o I— 1 o C3 S "2 1.1 S ^ PM m ^ ^ m GQ H OQ Q EH . " .2 (—1 .B .a" 1;^ § o § CO 13 a o o CO 1 o ^1 •a • 1— 1 .1 eg ^3 § ^3 rt o ft p-^ ?-i o ?:; as Q ^ H H p^ EH 02 1 i-:i 'S ' 'S s en- dicular to the axis of the helical directrix is an archimedian spiral Figure 76. Proof. — Referring to Figure 76, suppose an element a'h' c' to move half way around the cylinder. It will occupy a position e' c' f\ parallel toh' d\ since the pitch is constant. Since h' c' = 2 o'h\ we will have d' f = 2 o' d' = a' d' * And, since the revolutions of points of the element are uniform, and the axial advance is also uniform, the trace with a plane perpendicular to e'o', is an equable spiral, of which d' f is the radial expan- sion, in half a revolution; and for any other position of the element, it is directly proportional to the axial advance, and, since these properties are identical with the archimedian spiral, the curve traced is identical. 164 DESCBIPTIVE GEOMETRY 155. To ascertain the point of tangency, of any given plane, with a oblique helicoid, along a given element Analysis — Since the tangent plane contains hut one element of the surface, it will pierce the other elements in voints, which lie upon a curve, cutting the given element in the point of tangency. 156. The hyperboloid of one nappe is a unique warped surface in that it is also a surface of revolution, and the only one of the latter class which is, at the same time, a warped surface. Since surfaces of revolution have special characteristics, controlling their treatment in projection, and as the hyperboloid has no particular char- acteristics, as a warped surface, beyond those which have already been discussed, it will be treated fully latter in connection with surfaces of revolution. CHAPTER VII DOUBLE CUEVED SUEFACES AND SITRFACES OF REVOLUTION. 157. Double curved surfaces. Double curved surfaces are those which can only be generated by the motion of curves ; they have no straight line elements, for, if they did, they would belong either to the single curved surfaces or warped surfaces. But the converse is not true, for, every surface generated by the motion of a curve is not a double curved surface. A double curved surface may be fully defined as one gener- ated by the motion of any curve, such that no two consecutive points of the curve travel in straight line paths. There is an infinite variety of double curved surfaces, since there is both an infinite variety of curves, plane and double curved, and an infinite variety of ways in which the curves may move to generate surfaces. But the most familiar double curved surfaces, and those of interest to the student, are also surfaces of revolution. No particular interest attaches to, or value to be gained by, the discus- sion of double curved surfaces, in general, except to note the following points : Planes can he tangent to such surfaces only in a point, or isolated points. A plane may he tangent to a double curved surface. 366 DESCRIPTIVE GEOMETRY and also intersect the surface in a curve, which may or may not contain the point, or points, of tangency. A tangent plane, if it also cuts the surface, or a secant plane will intersect it in a curve, or curves, never in a straight line. The law of generation of a douUe curved surface is not limited to the motion of any particular curve in any par- ticular surface, for the curve formed by any secant plane, can be conceived of as one which could generate the surface, by moving and changing its shape, according to a law. The best definition would be that involving, at once, the simplest curve of section, and the simplest law of motion, to be decided by an inspection of the particular surface in question. 158. A surface of revolution is one generated by the revolution of a line about a right line as an axis. The revolving line may be a right line or it may be a curve, it may cut the axis or it may not.* If the revolving line is a right line, it generates either the cone, when it intersects the axis or the only other surface of its class, known as a hyperboloid of revolution. If the revolving line is a curve, it generates a double curved surface, and has no straight line elements. There may be an infinite variety of surfaces of revolution depending upon the form of the revolving curve, the simplest being the following: The sphere, formed by the revolution of a circle about one of its diameters; a cone, formed by the revolution of a right line about another right line which it intersects; a cylinder, formed by the *A surface of revolution Is not strictly limited to one formed by the revolu- tion of a plane curve, although In general such are the ones considered. SURFACES OF REVOLUTION 167 revolution of a right line about another right line to which it is parallel; an ellipsoid, formed by the revolution of an ellipse about one of its axes;* a paraboloid, formed by the revolution of a parabola about its axis ; a Jiyperboloid of revolution of two nappes, formed by the revolution of a' hyperbola about its transverse axis, etc., the surface, in general, deriving its name from that of a curve which revolves. It is obvious that any plane, perpendicular to the axis of a surface of revolution, will cut circles from the surface whose centers are in the axis, therefore, it is possible to conceive of such a surface being generated by the motion of a circle, such that the center moves along the axis of the plane of the circle while the latter changes its shape according to a law. If planes are passed through the axis, they will cut equal curves from the surface, known as meridian curves. That meridian curve which is parallel to a coordinate plane, the axis being also parallel to the coordinate plane, is known as the principal meridian curve A In any surface of revolution, which is re-entrant, the path traced by the point of the revolving curve, nearest the axis, is known as the circle of the gorge, and the plane of this circle, as the gorge plane. 159. To assume a point upon a surface of revolution and to draw a tangent plane to the surface at the point * This Is sometimes called a spheroid; a prolate spheroid, If the revolution Is about the major axis, an oblate spheroid if the revolution is about the minor axis. tin discussing surfaces of revolution, let it be understood that the axis of the surface is taken parallel to one of the coordinate planes and perpendicular to the other, which is the most convenient method of representation for the purpose of analysis. 168 DESCRIPTIVE GEOMETRY Analysis:— T/^e point will lie on a meridian curve of the surface, which can he drawn, and a tangent plane to the surface, at the point, will contain a tangent to the meridian curve, through the point. Therefore, draw the meridian plane through the point, and if it is not the principal meridian plane, revolve it, and the point about the axis, until it becomes the principal meridian plane. Its projec- tion, upon one coordinate plane, will lie upon the contour of the surface. Revolve the point bach again into its original position; the projection of the path, on one coordinate plane, will be a circle, that on the other will be a parallel to the G,L, The tangent plane contains, in addition to the tangent to the meridian curve, through the point, the tangent to the circular path of revolution of the point of tangency, which, in turn is parallel to one of the coordinate planes; and therefore, one trace of the tangent plane will be par- allel to one projection of this line. Construction:— Let the surface be given as an oblate spheriod, shown as in Fig. 77 with axis parallel to Y and perpendicular to H. Either projection of the point can be assumed as a'. Its path of revoluton, into the principal meridian plane, is a horizontal line in V projection, and a' falls at a^' , and its H projection at a^ , Revolving the point back again to its originally assumed position, a i moves in the arc of a circle, with o as a center, to the posi- tion a, on a perpendicular through a' , a', as assumed, could stand for two positions of the point on the surface, the other being 0^2 . The tangent to the surface at A can be drawn, at the revolved position of A, the revolved V projection of its H trace being bx' . The H trace, B, re- SURFACES OF REVOLUTION 169 FlQUBB 77. volves back to the original position, and falls at &' in V projection; and & in H projection, and the H trace of the required plane, T, is tangent to the circle of the path of B, at 6. The V trace is obtained in the usual manner, or it goes through the V trace of the tangent to the surface at A. The tangent to the meridian curve, in revolving from the one position to the other, generates a portion of the surface of a right circular cone, whose apex is in the axis and its line of tangency with the surface may be con- sidered its base. 170 DESCRIPTIVE GEOMETRY 160 THEOREM XV. A plane which is tangent to a surface of revolution at a given point is perpendicular to the meridian plane through the point. Proof;— The tangent plane contains the tangent to the circle of revolution of the point. The tangent, being perpendicular to the radius of this circle, is also perpen- dicular to any line connecting the point of tangency with the axis of the surface, since the radius is perpendicular to this axis. But the radius and the axis lie in a plane, hence the tangent is perpendicular to this plane which is the meridian plane. 161. To pass a plane tangent to a sphere, and through a given line outside. Analysis — TJie given line will he the intersection of the two possible planes tangent to the sphere, through the line. Each of these planes will he perpendicular to a radius of the surface through the point of tangency. The plane of these normals will he perpendicular to the required planes. Therefore, pass a plane through the center of the sphere, and perpendicular to the given line. From the point in which it is pierced hy the line, draw a tangent to the great circle cut from the sphere hy the plane. This line, and the given line, constitute one of the possible tangent planes. Construction:— Let the sphere be given, as in Fig. 78, and the line as AB. Pass a plane through the center of the sphere, and perpendicular to the line AB by using a horizontal of the plane through O. The line AB pierces this auxiliary plane in the point P. Revolving the center of the sphere O and the point P into H, P falls at p^ . One SURFACES OF REVOLUTION 171 Figure 78. tangent to the great circle cut from the sphere, by the aux- iliary plane, is Pi r, the other is Pi q. Hence, the plane of AB and PR, in the one case, and AB and PQ in the other, determine the required tangent planes, E and T, respect- ively. 162. THEOREM XVI. If two surfaces of revolution, having a common axis, are 172 DESCRIPTIVE GEOMETRY tangenMo each other, or intersect, it will be in the circum- ference of a circle, whose plane is perpendicular to the axis, and center in the axis. Proof:— If a plane is passed through any point of the curve of intersection, and the common axis of the sur- faces, it will cut a meridian curve from each surface, intersecting each other in a point. If each meridian curve is revolved about the axis, it will generate the surface to which it belongs, while the point, common to both the meridian curves, will describe the circum- ference of a circle, common to both surfaces, with center in the axis, and plane perpendicular to the axis. 163. To find the curve of intersection of any surface of revolution by an oblique plane. Analysis: — Planes , passed through the axis of the surface, will cut meridian curves from it, and right lines from the secant plane, the points of intersection of which lie on the curve of intersection of the surface and the plane. Or again, planes perpendicular to the axis, will cut circles from the surface and lines from the secant-plane. In construction, the latter are to be preferred because the circles are easily drawn, if the axis of the surface is perpendicular to a coordinate plane, and the lines cut from the secant plane will be horizontals or verticals of the plane. Construction:— Let the surface be a torus form, a name derived from its use in architecture, a surface formed by the revolution of a circle about a line lying in its plane, but not passing through the center. It is shown in Fig. 79, and the secant plane as T. SURFACES OF REVOLUTION 173 FlQUBK 79. Pass a series of secant planes perpendicular to V, and to the axis of the surface, as shown at 1, 2, 3, etc. Their V traces will be parallel to the G. L., and the horizontals cut, thereby, from the plane T will intersect the respective circles, cut from the surface, in points of the curve of inter- section. The V traces of these lines are identical with the traces of the secant planes respectively. 164. The hyperboloid of one nappe, or hyperboloid of rev- 174 DESCRIPTIVE GEOMETRY olution, as it is called, is a surface generated by a straight line, which moves so as to touch three windschief lines, which are equi-distant from a fourth, and make equal angles with a plane, perpendicular to the fourth. Or it may be considered as having two rectilinear directrices, and a right circular cone directer. This surface is distinguished from the h3rperbolic para- boloid by the conditions that the three windschief direct- rices shall be equally distant from a fourth, and shall make equal angles with a plane perpendicular to the fourth. Its construction, however, is best understood when made according to another statement of the law of genera- tion, namely, a surface generated by a line revolving about another line, which it does not intersect.* Let Fig. 80 represent a hyperboloid of revolution, con- structed as follows : — Let the axis of revolution, OP, be perpendicular to H, and AB be a limited portion of the re- volving line, shown here as parallel to V, and moving around OP, in the direction of the arrow. Its perpendic- ular distance from the axis is ep. Every point of the line AB describes a circle, whose center is in the axis, and whose plane is perpendicular to the axis; hence, when any point, as Gr, has revolved to a position G i , in a plane with the axis OP, which is parallel to V, it will be a point in the contour of the surface ; for the contour is the line of tan- gency of a cylinder of projection, and since the surface is one of revolution the contour is in a plane parallel to Y. All points of the line AB will take, successively, positions in the principal meridian plane, in the revolution about the axis, and other points in the contour may thus be plotted. *A law shortly to be proved. SURFACES OF REVOLUTION 175 F IGUBB 80. This surface is symmetrical on its axis, so that for every point, as G, there will be one symmetrical with it upon the opposite side of the axis, and the Y projection, in this case, will be symmetrical with respect to the line o'p' , For convenience, in the illustration chosen, AB is bisected at E, which describes the smallest circle of revolu- tion of any point of the line; it is the circle of the gorge y and its plane is the gorge plane. 176 DESCRIPTIVE GEOMETRY 165. THEOREM XVII. The hyperboloid of revolution of one nappe has two systems of generation, and every element of the one system intersects all those of the other system. / A \ a'j \ \ '"' \ / F I QUBE 81. Proofi — If any point of an element AB as E in Fig. 83., revolve through an angle of 180°, it will occupy the SURFACES OF REVOLUTION 177 position F, and the element, that of the position of CD, making the same angle, with the plane perpendicular to the axis, as AB does. If, now, through the point F, another line KL is drawn making the same angle with this plane, and lying in a plane parallel to V, it will, if revolved about the axis, generate the same surface, for, any plane passed perpendicular to the axis, will cut both these elements in points, respectively, G and M which are equally distant from the axis, and these each, in revolution, will describe the same circle perpendicular to the axis, and so for any other piercing points of the elements with any plane perpendicular to the axis; hence, the surfaces generated, by the elements, will be identical. And, since this is true, through any point of the surface, two right lines can be drawn, which are elements of the two systems of generation, respectively. Since the point G, of the element AB, and the point M, of the element KL, generate the same circumference, it follows that, at some position of the element AB, the point G coincides with the point M. This position is shown by the dotted line AiBj, and so for any other point of AB. Hence, if either generatrix remain fixed, the other one, in revolution, would intersect it in all points successively.* And, if any three generatrices of one system, are taken as directrices, and a generatrix of the other system moves, so as to touch the three, it will generate the surface, hence the definition. *Thl8 property Is one characteristic of warped surfaces In general bavins double sj'stems of generation. 178 DESCEIPTIVE GEOMETRY 166. THEOREM XVIII. The meridian curve of a hyperboloid of revolution of one nappe is a hyperbola. Figure 88 . Proofi— Let Fig. 83 be a hyperboloid with G and N, any two points upon it. The horizontal distance of n^' from^' is less than that of g^' from g\ Still further removed from G, the corresponding projection of the SURFACES OF REVOLUTION 179 distance of any point, on the meridian curve, from the projection of the generatrix, when parallel to one coordi- nate plane, is less. This distance, in any case, as g/g', is the difference between the radial distance of the point from the axis, sls g o, and its distance in H pro- jection from the point of tangency of the element AB, to the circle of the gorge. The first distance is the hypotenuse of a right angled triangle of which the second distance is the base, and the distance of the point of tangency, in H projection, from the axis is the al- titude. The hypotenuse, and the base vary, for every point on the element, while the altitude remains con- stant. Let the hypotenuse be represented by h, and the base by h, and the altitude by a. Then the conditions may be expressed algebraically, as a' = }i- — y 0Ta'= (h-{-h) ih-b) whence The left hand member diminishes in value as the denominator of the right hand member increases, and becomes equal to zero when the denominator becomes infinite. Therefore, the element AB is an asymptote to the curve. Again, when h == zero, h^ = a% and the radius of the circle of the gorge is equal to a, the nearest point of the moving element to the axis. Referring also to Fig. 80, and remembering that the line through a b may stand for the H projection of an element of each system, assume any other position of AB as AiBi. It will cut one element of the other system in a point M., and a second element in a point N, 180 DESCRIPTIVE GEOMETRY on the opposite side of the circle of the gorge. Now, since this element has but one point in common with the meridian curve, the vertical projection of AiBi will be tangent to this curve at K. But n k = k m, then m' ¥ = ¥ n' , that is, the tangent to the meridian curve is bisected at the point of contact, a well known property of- the hyperbola. Hence, the meridian curve is a hyperbola; in the figure, it has a horizontal trans- verse axis and a vertical conjugate axis. The surface could be generated, therefore, by revolu- tion of a hyperbola about its conjugate axis. If the circle of the gorge diminishes in diameter to zero, the generatrices will go through its center and the surface becomes a cone. If the angle of the generatrices with the plane directer diminishes to zero the surface become a plane. 167. To assume a point on the surface of a hyper- boloid of revolution of one nappe, and to draw a plane tan- gent to the surface at the point Analysis:— T/^e elements of both systems of generation, which pass through the assumed point, will each he tangent to the projection of the circle of the gorge, on a plane per- pendicular to the axis of the surface, and intersect any assumed bases of the surface in points, which can he found. And the tangent plane will contain the elements of the surface — one of each system— passing through the point of contact, will he tangent to the surface at a point, and cut the surface in two straight lines. SURFACES OF REVOLUTION 181 Figure 83, Construction:— Let Fig. 83 be a hyperboloid of revolu- tion. First, assume P, in H projection, in which case either or both elements can be drawn through it, tangent to the circle of the gorge, and cutting the base in the circle of contact with H, from which their V projections may be ob- tained. Or the V projection of P may be assumed. In this case, revolve the point about the axis until it falls into the principal meridian plane, and after its H projection has been obtained in this position, then revolve it to its original position and obtain the true H projection. The tangent plane contains the elements, one of each system, passing through the point of contact, audits traces 182 DESCRIPTIVE GEOMETRY pass through the traces of these elements. TV and TH are the traces as shown. 168. To find the curve of intersection of the hyper- boloid of revolution with any plane. Analysis: — The most convenient method of finding the curve of intersection^ is to pass a series of auxiliary secant planes perpendicular to the axis of the surface, cutting circles from the hyperholoid, and right lines from the giveri plane, the respective points of intersection, of which, give points upon the curve of intersection of the two. CHAPTER VIII INTERSECTIONS OF SURFACES. 169. Two surfaces will intersect each other in curves or straight lines, according to the character of the surfaces. And the method of ascertaining the form of the curve, will depend also upon the properties of the surfaces con- sidered; no general procedure can be laid down, except the following: Pass a series of secant planes to cut from- both surfaces, the simplest lines that can he drawn, the intersections of the curves cut from both surfaces by the secant planes, respectively, will give points in the curve of intersection of the surfaces. The subject of intersections is best investigated by- working out a series of problems involving different kinds of surfaces. The intersection of the simpler single curved surfaces, and surfaces of revolution are of particular interest to the student, and furthermore, intersections, together with the development of the surfaces intersected, form a class of problems of most frequent occurrence in practical descriptive geometry, and deserve to be dwelt upon. As the student has by this time become accustomed to the fundamental processes, it is proposed to curtail the analyses, somewhat, and to cover only the special features involved in each new problem. 384 DESCRIPTIVE GEOMETRY 170. To find the intersection of two bodies bounded each by plane faces. General Analysis: — The curve of intersection will connect the points of intersection of the various edges of the one form with the surfaces of the other form. Hence, (i), find the various lines of intersection, of the planes of the one form, with those of the other, respectively, or, (2), find the traces of the edges of the one form with the faces of the other, the latter extended if necessary to contain the traces. Figure 84 . Construction: — Let the two forms be two prisms, square and triangular, as in Figure 84. Starting with any INTERSECTIONS OF SURFACES 185 edge of one of the prisms, as 1, of the square prism, see if it intersects the surfaces of the triangular prism, by using the H projecting plane of the edge, and finding its line of intersection with the faces of the triangular prism, and so on, systematically, each edge in turn, and then find, similarly, the piercing points of the edges of the triangular with the square prism, wherever the intersection has not been determined. 171. One form will completely penetrate the other when two parallel, tangent and diametrically opposite, planes to the one form cut lines out of the second form. This construction is very useful sometimes, in determining the sphere within which to work with least expenditure of effort. Two sets of tangent planes, one set to each form, will determine at once the limits of the curve, or curves, of intersection. To make the statement for complete or partial pene- tration general, for all surfaces, it may be said that one completely penetrates the other when two tangent, and opposite, planes to the one form, and constructed in accordance with the limitations imposed hy the character of the surface, cut the second form. If one completely penetrates the other, there will be two distinct curves of intersection, if not, the curve of intersection will be single and continuous. 172. To find the lines of intersection of two pyramids. Analysis:— jPmc? the piercing points, of the edges of one, with the surfaces of the other, hy means of auxiliary planes, passed through the edges of each, and, when possible, pass them through the apexes of loth pyramids; whence, the 186 DESCKIPTIVE GEOMETRY common line of intersection, of the several secant planes, is the line connecting the apexes of the two pyramids, and the lines cut from both pyramids are elements of them respec- tively, 173. To find the curve of intersection of two cylinders^ Analysis:— 7/ the planes of the bases of the two cylinders are parallel, choose the secant planes parallel to the bases. Similar curves to the bases will be cut from each, and their respective intersections give points on the curve desired. If the bases are not in parallel planes, pass secant planes through the elements of one, which are parallel to those of the other, hence, cutting elements, if at all, out of both. Construction:— Let two cylinders be given, as in Fig. 85, having axes AB and CD respectively. Since the second cylinder has its axis parallel to the G. L., any auxiliary planes, passed parallel to its elements, will be parallel to the G.L. Hence, to find the intersections of such a series of auxiliary planes with both cylinders, project the cylind- ers upon and end plane as shown. The respective inter- sections of the contour of both cylinders, with any one aux- iliary plane, are the piercing points of elements of the cylinders, with their respective bases, and these can be transferred to the coordinate planes. One such auxiliary plane is shown, cutting the elements EF and IJ, respec- ively, out of both cylinders. Note. In the practical application of descriptive ge- ometry, in such problems as these, it is found convenient to choose one coordinate plane parallel to the axes of both solids, so that the elements cut out will be projected their true length on that coordinate plane, since the main object INTERSECTIONS OF SURFACES 187 Figure 85. of ascertaining the intersection is that the forms may be developed, such for example as two intersecting pipes. 174. To find the curve of intersection of two cones. Analysis:— Unless the bases are circular and in parallel planes, pass a series of secant planes to cut elements from both cones. These planes will intersect each ether in a line through the apexes of both cones, and their traces, moreover, will go through the traces of this line. The respective intersections of the elements, cut from the cones, will be points in the curve of intersection sought. 188 DESCRIPTIVE GEOMETRY Figure 86. Construction:— Let two cones be given, as in Figure 86. The circular base of one is in V, with apex at O, and the circular base of the other is in H, with apex at C. The line OC, connecting the apexes, has its V and H traces in A and B, respectively, and the secant planes used will con- tain these traces. Planes T and R, drawn tangent to the first mentioned cone, show that T cuts elements out of the INTERSECTIONS OF SURFACES 189 second, but R does not, likewise planes S and U, drawn tangent to the second show that S cuts elements out of the first, but U does not ; hence, neither cone completely pen- etrates the other. The construction of points by use of another auxiliary plane is shown. If both bases are in the same coordinate plane, only one trace of a secant plane is necessary to determine whether that plane cuts both cones. Its corresponding trace may be found with a horizontal, or vertical, of the plane, or it may not be needed in construction. If both cones have the same apex, either trace of any secant plane can be assumed at will, the corresponding trace being determined by the fact that the apex is a point in the plane. If the line connecting the apexes of the cones, is parallel to their bases, the traces of the auxiliary planes, with those bases, will be parallel lines. 175. To find the curve of intersection of a cone and a cylinder. Analysis:— T/^e secant planes, to cut elements out of each, must pass through the apex of the cone, and be parallel to the elements of the cylinder; hence, draw a parallel line through the apex of the cone, as the common line of inter- section of the secant planes. Construction:— Let the cone and cylinder be given as in Figure 87. Draw a line through the apex of the cone, as the line of intersection of the secant planes. In this case, it is perpendicular to H. The secant planes T, E, S, etc., are, therefore, perpendicular to H, and cut both surfaces in straight lines. The H traces can be assumed at will, to pass through the H trace of the auxiliary line, 190 DESCRIPTIVE GEOMETRY to' FIGTJBE 87. through the apex of the cone. The points of intersection, D and E, of the elements cut out of each, by one auxiliary- plane R, are shown. Where the bases of cone and cylinder are in the same plane, the traces of the secant planes, with that plane, can be assumed at will to go through the trace of an auxiliary line, through the apex of the cone, and parallel to the elements of the cylinder. If this auxiliary line is parallel INTERSECTIONS OF SURFACES 191 to the plane of the bases, the traces of the secant planes, on that plane, are parallel. If the bases of cone and cylinder are in or parallel to different coordinate planes, then either of the traces of the auxiliary planes may be assumed at will, to go through the similar trace of an auxiliary line, through the apex of the cone, and parallel to the elements of the cylinder. The proper traces to assume, of the auxiliary planes, are those with the coordinate plane in which the bases re- spectively lie, for the traces show, at once, the elements cut from that solid whose base lies in that plane. 176. To find the curve of intersection of a cylinder and a convolute. General Analysis:— 7/" the base of the cylinder lies in or parallel to the plane of section of the convolute, used to represent part of the contour of the latter, then secant planes, parallel to the base of the cylinder, will probably give the simplest construction, although cutting curves out of both. If the base of the cylinder is not so related to the convolute, then, through any point of an element of the latter, draw an auxiliary line parallel to the elements of the cylinder; the vlane of these two lines determines the direction of one auxiliary plane, 177. To find the curve of intersection of a cone and a convolute. General Analysis: — If the base of the cone lies in, or parallel to the plane of section of the convolute, used to represent part of the contour of the latter, then secant planes, parallel to the base of the cone, will give the simplest construction, although cutting curves out cf both. 192 DESCRIPTIVE GEOMETRY If the base of the cone is not so related to the convolute, the general solution involves the cutting of a series of differing curves from the convolute, while cutting curves or elements from the cone. 178. Two intersecting surfaces of revolution may come under one of three groups. 1. When they have a common axis. 2. When the axes are in the same plane. 3. When the axes do not intersect. If two intersecting, or tangent, surfaces of revolution have a common axis, they will touch each other on a circle or circles; for, having a common axis, any point, common to the two, must lie on a meridian curve, and, in each case, if revolved, will generate ihe surface, every point travel- ing in the path of a circle, whoes center is in a plane per- pendicular to the axis. 179. To find the intersection of two surfaces of revolu- tion having a common axis. Analysis: — If the axis is taken parallel to one coordinate plane, and perpendicular to the other, then secant planes, parallel to the latter, will cut both surfaces in circles, which intersect in points desired. If the axis is not so placed, a change of coordinate plane or planes would be most desirable, 180. To find the curve of intersection of two surfaces of revolution whose axes are in the same plane. Analysis:—// the axes are parallel, pass a series of auxiliary planes perpendicular to the axes, the curves cut from each surface will be circles. If the axes are not parallel, the convenient method is INTERSECTIONS OF SURFACES 193 to intersect both surfaces with a series of spheres, of dif- ferent radii, and with centers at the point of intersection of the axes. These spheres will cut circles, respectively, from both surfaces. If the plane of the axes is taken parallel to a coordinate plane, the projection of these circles on this coordinate plane will be limited and inter- secting straight lines, the points of intersection of which, respectively, are the common chords of the circles cut from each surface, one point each, as a common chord to the two circles cut from the surfaces by any one sphere. FIGUBE 88 Construction: — Let the two surfa'ces be an ellipsoid of revolution, or prolate spheroid, and a right elliptical cone, whose apex C, is on the surface of the spheroid. See Figure 88. The center for a series of auxiliary circles is at 194 DESCBIPTIVE GEOMETRY O, the intersection of the axes of the surfaces. A sphere, such as 1, and another such as 3, since they go through the points of intersection of the contours of the solids, and since the plane of the axes is parallel to V, will cut out chords of zero length, i.e., be the piercing points of those elements of the cone which are parallel to V. Sphere 2 will cut out a circle from each intersecting in a common chord CD, the circle cut from the spheroid is vertically projected in the line i'g' and the circle cut from the cone, is vertically projected in the line e'f. In like manner other points are ascertained. 181. To find the curve of intersection of two surfaces of revolution who axis are not in the same plane. Analysis:— Pa55 secant planes to cut the simplest series of lines out of each surface, circles out of one, for example, hy planes perpendicular to the axis, and curves out of the other. No general analysis leading to a shnple solution under these conditions is possible. 182. To find the curve of intersection of a single curved surface and a surface of revolution. Analysis: — Pass a series of secant planes normal to the axis of the surface of revolution, to cut circles from it and circles, ellipses, or other curves from the other surface. If the single curved surface is a cone, or cylinder, it is, in general, possible to get fairly simple curves of inter- section. If the surface is a sphere, also, the curves of intersection will be circles. If it is a convolute surface, the curves cut out will depend upon the position of the axis of the convolute relative to that of the surface of revolution. APPENDIX Practical projection is based upon descriptive geometry, but the short cuts, in common use, are, many of them, entirely at variance with it. They are correct in their way, but the practical and the theoretical should not be confused. For example, the real differ- ence between first and third angle projection has not always been followed by draftsmen, the two have been mixed. In the first angle, an object is projected upon the coordinate planes from a center of projection, which is on the same side of the plane as the object; in the third angle, the center of projection is on the opposite side of the plane from the object. This, in the case of the first angle, results in an end view of an object being placed upon the opposite side from the one the view is supposed to represent, while, in the third angle, the end view comes next to the side which it is supposed^to represent. Further than this, in the first angle, the revolution of the end view is away from the end it represents, and in the third angle it is toward it, theoretically, while in practical projection, for both angles, the revolution is away from the end it represents, thus, theoretically, dividing the end plane, on its intersection with V, and folding the front and rear parts on each other like the leaves of a book. If these distinctions are kept in mind, the student will not be likely to mix the first and third angles in his practical work. Since practical projection is but a language constructed to give expression to ideas, it should be allowed license in its mode of expression, to best suit its purposes, as long as there is consistency, and a common understanding and approval of what is meant. Practical projection ignores the ground lines between any coor- dinate planes of projection, and for the very good reason that they in no way affect the projections of forms or the relations of the projections to each other except as to distance apart; and, because ground lines are ignored, in various other ways, detail views are arbitrarily placed where they will be most explanatory. As a con- crete illustration of this, the section of a spoke of a wheel may be 198 APPENDIX revolved about the median line of the section, where taken, and shown as limited by the outlines of the spoke. Again, it may be said, that such arbitrary constructions are permissible if well understood and accepted by custom. A very large share of the practical problems involving descrip- tive geometry are those of ascertaining the direction and shape of sections of forms, and of the development of the forms. Not all peripheries of solids can be developed, in the true sense of the word, while, in reality, all forms are developed in practice. The only forms that can be developed, in conformity with the princi- ples of descriptive geometry, are» those having straight line elements and intersecting, at least consecutively, two and two. No double curved or warped surfaces are developable. In practical work these latter surfaces frequently must be shaped up, notably in the sheet metal worker's business, and interesting devices are resorted to. For double curved surfaces there are two general methods: 1. The zone. 2. The gusset. In the zone method, the solid is divided into zones, by planes per- pendicular to a principal axis, if there be any, and the zones again broken up into two or more parts, to get smaller pieces, more closely approximating the real form. The gusset method is best illustrated by reference to the periphery of an orange when divided into its natural lobes, or to a sphere, divided by meridians into the necessary number of slices, closely approximating the true form. In the sheet metal worker's business, the material is sometimes punched out or stretched to make it fill out the remain- ing difference between the true and the constructed forms. Other forms are developed by a system known as triangulation. The surface is divided up into triangles having their correspond- ing bases oppositely directed. In the development, each triangle is revolved about its edge of contact with an adjoining one, and the bases of the alternate triangles form a continuous develop- ment of the perimeter of a portion of the form; opposite to these, the bases of another set of alternate triangles go to make up another portion of the perimeter. The frustrum of an oblique cone is a surface which might be dealt with in this way, but the method can also be applied to other surfaces. EXERCISES Following Section 24, Page 17. 1. Draw the projections of the following points: A = 0, If, |; B = 7, If, — i; C = 1\, — 1, — i; D = 2^, — If, i; E = 3i, — If, 0. Project them upon a new Vi plane whose Gi-L^. cuts the G.L. at 3i, 0, 0, and makes an angle of 30° with it downward towards the right. 2. Draw the projections of the following lines: A B has A = 0, i^, — 1^; B = li, 1^, — li^. C D has C = 2, If, f ; the line pierces the G.L. and d' is at 3t^, — f , 0. Project both of these lines upon a new Vj plane whose Gi.Lj. is parallel to the H projection of C D and cuts the G.L. at If, 0, 0. Following Section 26, Page 19. 3. Given two points A = 0, li, f , and B = If , — 1\ — H; draw the projections of the line joining them. 4. Given two points, A = 0,-11, — 1, and B = 2, f , 1|; draw the projections of the line joining them. 5. A B, C D and E F are three lines. A = 0, 0, 0, and B = U, if, 1^; C = U, - f, - f, and D = 2t, - f, - 1; E = 3, - Ih - f, and F = 4, 0, — f . What are the relations of these lines to the coordinate planes? the angles CD and EF with them? 6. A B and C D are two lines. A = 0, 2^, li^, and B = U, 1, i^; C = f , 1, 1, and D = If, If, H. Draw their projections. 7. A B is a line with A = 0, f , U, and B = If, U, i. C and D are points on the line; C is in a vertical plane j inch to the right of A, and D lies in a horizontal plane f inch below B. Locate both projections of the points. 8. A B is a line, with A = 0,— U, — U, and B = H — i, — |. C is at If, — 1, — If. Through C, draw any line to intersect A B. 200 EXERCISES 9. A B is a line, with A = 0, 1|, — |, and B = U, If, — |. C is the middle point of A B. D is a point at i, i, f . Through D, draw a line to intersect the line A B, in the point C. 10. A B is a line, with A = 0, i, U, and B = If, U, i. C is at I4, i, If- From C, as one extremity, draw a line so that its middle point shall lie on the line A B. 11. Locate the projection, on a right hand end plane, of a line whose one end is } inch from V, and f inch from H, in the first angle, and whose other end is i inch from H, and f inch from V, in the third angle. The ends of the line are H inches apart hori- zontally. Draw the projections of the line in either of its two possible positions upon the H and V planes. 12. Draw the projections upon the H, V and Ve planes of the two lines joining the points AB and CD. A = 0, i, f, and B = 2, H, 1; C = i, i, — If, and D = If, — 1, f. Let Ge.Le. cut the G.L. at 3f , 0, 0. 13. Draw the projections, upon the H, V and Ve planes, of the two lines joining the points A B and CD. A = 0, H, — I, and B = 0, — t, U; C = i, — If, — 1, and D = 2, 0, 0; Ge.Le. cuts the G.L. at 3. 0, 0. 14. Given the line A B, with A = 0, li, |, and B = If, i. If; draw the projecting planes of the line. 15. A point A = 0, f , i and a point B = 0, U, 1; find the projec- jections of a line C D, 2 inches long and parallel to the line A B, and distance f inch from it, in a vertical plane through the line AB. 16. Draw the lines A B and C D which intersect each other in a point O, at 0, f , — 1 i. A B is parallel to V and oblique to H, and C D is parallel to H and oblique to V. Coordinate the extremities of the limited line used. 17. Given the line A B C, B being the middle point in it. A EXERCISES 201 = 0, U, I, and C = 0, i, 1\. Draw a line through B parallel to a line E F, in which E = f , i, i, and F = If, |, f . 18. Given a line A B; A = 0, f, |, and B = l-J, If, 1. Draw limited lines parallel to it lying in each of the four dihedral angles. 19. Prove that the lines of problem 12 do not intersect. To Follow Section 32, Page 24. 20. Find the traces of the lines A B and CD. A = 0, H, — |, and B = li, - f , li; C = If, If, i, and D = 3f , f , U. 21. Find the traces of the line A B. A = 0, — |, — If ; B = 2, 16» T6« 22. Find the traces of the line M N. M = 0, U, U; N =r 2f , 0, 0. Also find its traces with an end plane Ge.Le. cutting G.L. at If, 0, 0, showing same as revolved into V about Ge.Le . 23. A B is a line with A = 0, li, — U; B = 1|, i, — \. Locate its traces with V and H and also with an end plane whose Ge-Lb. cuts G.L. at 1, 0, 0, revolving the same into the V plane. 24. Find the traces of the broken line connecting successively the points. A = 0, - f , - i^; B = U, H, - i; C = 2f, f , If; D = To Follow Section 33, Page 25. 25. Draw a line A B, with its V trace at 0, f , 0, which passes through the 1st and 2nd angles only. 26. Draw a line A B with its V trace at 0, |, 0, which passes through the 2nd, 1st, and 4th angles. 27. Given a line A B with A = 0, i, — i; B =1, U, — 1\. What is the position of the line with respect to H and V; where are its traces? 28. Given a line A B, with A = 0. If, — i; B = U, i - If. Find its traces and also its traces with an end plane whose Ge.Le 202 EXERCISES passes through the point of intersection of the V and H projections of AB. 29. The V trace of a finite line A B is U inches below H and its H trace is H inches back of V. Draw the projections of such a line and find its traces with an end plane whose GeXe. cuts through the middle of the line. To Follow Section 38, Page 32. 30. A B is a line, with A = 0, i, i; B = 1|, U, H. Find the true length of the line, by revolving it about the horizontal projecting line of the point B until it is parallel to V. Show, by appropriate notation, the H and V projections of the arc of revolution of the point A. 31. A B is a line, with A = 0, |, U; B = U, If, |. Find the true length of the line by revolving it about the vertical projecting line of the point B until it is parallel to H. Show, by appropriate notation, the H and V projections of the arc of revolution of the point A. 32. A B is a line, with A = 0, H, 0; B = H, 0, f. Find the true length of the line, by revolving it about the H trace of its H pro- jecting plane into H. Also, find its length, by revolving it about the V trace of its V projecting plane into V. 33. AB is a line, with A = 0, If, f ; B = li, f,^. Find the true length of the line, by projecting it upon a supplementary Vi plane, taken in any convenient place, parallel to AB. 34. A line goes through the points A and B with A = 0, i, — i, B = 1, H, — H. Locate any two points, O and P, on it which are li inches apart. 35. Given the four points. A, B, C, and D. A = 0, i, |; B = If, U, H; C = i^.lf, If ; D = 2, 0, i. Find the distances between the extremities of the two lines AB and CD. Also, find the distances between P, on AB, | inch from A, and Q, on CD, i inch from C. 36. A, the V trace of a line AB, is at 0, — 2i, 0; B the H trace EXERCISES 203 of the line is at 2|, 0, — 2^; C, the H trace of another line inter- secting AB, is at 1^, 0, — If; D, the point of intersection has its z = li. Draw the lines AB and CD, and find the V trace of CD. Through D, draw a line whose V trace has z = f , and is i inch below H. 37. A = 0, f, i; B = I, If, U. Find the true length of the line by the following methods: (a) by revolving parallel to V; (b), by revolving parallel to H; (c), by revolving about its H projection into H; (d), by revolving about its V projection into V; (e), by projecting upon a new Vi plane parallel to the line, using a Gi. Li to cut the G.L. at |, 0, 0. 38. Find the true length of the line AB with A = 0, U, — U; B = li, 0, 0. To Follow Section 46, Page 38. Where TO is given it means the point in which the plane T cuts the G.L. Point 1 stands for a point of a plane in V, 2, for a point of the plane in H, i, e., points on the traces. 39. T is a plane with TO at 0, 0, 0. TH makes an angle of 30° with the G.L., and TV makes an angle at 60° with the G.L. both towards the right. A, B, C and D are four points in the plane T, and their vertical projections are as [follows: a' =l,f, 0; b' = — i, — I, 0; c' = i, — H, 0; d' =0, H, 0. Find the H projections of the points by using a horizontal of the plane, in each case. 40. T is a plane with TO at 0, 0, 0. TH makes an angle, with G.L. whose tangent is f ; TV makes an angle with G.L. whose tangent is |, both towards the right. A, B, C and D are four points in the plane and their horizontal projections are respec- tively as follows: a = 2, 0, |; 6 = 1, 0, — 1; c = — i, 0, — i, and d = H, 0, 0. Find the vertical projections of the points by using a vertical of the plane in each case. 41. Given the plane T, with TO = 0, 0, 0. TH has 2 at 2, 0, If; TV has 1 at 2, li, 0; apoint O, which is in the plane, has o = li, 0,i. Draw any two lines through the point O and lying in the plane T. 204 EXERCISES 42. Given the three points, A = 0, i, i^; B = 2i, 0, 2, and C = ¥, 1, 1^; to pass a plane through them. 43. Given two parallel lines AB and CD; A = 0, |, i; B = f, |, I; C = i, f , 1. Pass a plane through the two lines. Note. — Parallel lines have parallel projections. 44. The line AB has A = 0, i, |; B = |, ^, i. Pass a plane through the line AB and a point C = If, 1, — |. 45. The line AB has A = 0, U, i; B = If, — i, U. Pass a plane through the line AB and a point C = If, li, — i. 46. Given the vertices A, B and C of a triangle: A = 0, f , li; B = f , li, f ; C = If, i, 1. Find the lengths of the edges of the triangle by revolving it into H about the H trace of its plane. 47. Find the traces of the plane of the three following points: 48. The H projection of a line AB has a = 0, 0, — |; h = 2i, 0, U. It lies in a plane T, with TO = 2f , 0, 0. TV has 1 = 4f , 1|, 0. TH has 2 = 0, 0, |. Find the other projection of the line AB. 49. A = 0, — 1, — 1t¥. a plane T containing the point has TO = If, 0, 0. TH has 2 at 3f , 0, U. Find TV. Analysis:— Since the point lies in the plane, a line of the plane may be drawn to pass through it, and by means of its traces, the required trace of the plane can be found. Construction:— Draw any line through the point A, and lying in the plane T, preferably a vertical of the plane. Its H projection will be parallel to the G.L. Where it cuts TH prolonged is the H projection of its H trace; the V projection of its H trace is on the G.L. at the foot of a perpendicular through the H projection. The vertical projection, of the vertical of the plane, goes through the vertical projection of the H trace and the point a' . The V trace of T is parallel to this line and goes through TO. 50. AB and CD are two intersecting lines. A = 0, — |, | and B EXERCISES TV ?« 205 TH 'U- 51. 52. ia' r 53. 54. = 2, 2f , U; C = f , 2f , If, and D* = 3f , -, 0. A point O lying in the plane of AB and CD, has o' = If, |, 0. Find o without using the traces of the plane of the lines. To Follow Section 51, Page 43. 51. Draw a line through the point A and parallel to the plane T. 52. Draw a line through the point A and parallel to the plane T. 53. Draw a line through the point A and parallel to the plane T. 54. Draw a line through the point A and parallel to the plane T. a' c/ 55. 55. A is a point in the plane T. Draw a line through the point B and parallel to the plane T, which is not a horizontal line. To Follow Section 52, Page 44. 56. Pass a plane through the line AB and parallel to the line CD. 57. Pass a plane through the line AB and parallel to the line CD. * When a coordinate is left with a — , It means that It Is to be found. It Is impracticable to give by coordinates, four points accurately on two Inter- secting lines. 206 EXERCISES. oT f 69. 60. 58. Pass a plane through the line AB and parallel to the line CD. 59. Pass a plane through the line CD and parallel to the line AB. ' 60. Pass a plane through the line AB and parallel to the line CD. b ^pf a a' \ *' P d , \ V ^6 a df 61. 62. 63. 64. To Follow Section 54, Page 46. 61. Pass a plane through the point O and parallel to the lines AB and CD. 62. Pass a plane through the point P and parallel to the lines AB and CD. 63. Pass a plane through the point P and parallel to the lines AB and CD. 64. Pass a plane through the point P and parallel to the two lines AB and CD. To Follow Section 56, Page 48. 65. T and S are two planes with TV and TS at 45° to the G.L. EXERCISES 207 towards the right; TV intersects the G.L. at 0, 0, 0; SV intersects the G.L. at 1, 0, 0. The H traces of both intersect at 3i, 0, f . Find their line of intersection. 66. T and S are two planes. T is perpendicular to V; TV making 45° with the G.L. toward the left, cutting same at 3\, 0, 0. S cuts the G.L. at 2, 0, 0; SV makes an angle of 60° with the G.L. towards the left, and SH, 30° with the G.L. in front of, and to the right. Find the line of intersection of the two planes. 67. T and S are two planes. TV is perpendicular to the G.L., cutting it at 0, 0,0; TH makes an angle of 30° with the G.L. in front of and toward the right. SH is perpendicular to the G.L., cutting it at 3, 0, 0; SV makes an angle of 45° with the G.L., above and toward the left. Find the line of intersection of the two planes. 68. T and S are two planes. TH and SH are perpendicular to the G.L. and cut the latter, respectively, at f , 0, and 2\, 0, 0. Their V traces intersect each other at a point 0, 2i, 0. Find the line of intersection of the planes. To Follow Section 57, Page 51. 69. T is a plane intersecting the G.L. at 0, 0, 0. TV makes an angle at 30° with the G.L. and TH, 45° with the G.L., both toward the right. A line has both projections parallel to the G.L. and goes through the point A = 0, 1, li. Find where the line pierces the plane. 70. AB is a line, with A = 0, — li, - f , and B = li, U, |. Find where it pierces a plane T with TV perpendicular to the G.L. and cutting the latter at 2|, 0, 0, and TH making an angle of 30° with the G.L. toward the left. 71. A line AB has A = 0, If, i, and B = 0, |, 1. A plane T has TV and TH coinciding and at 30° to the G.L. above and toward the left. Find where the line AB pierces the plane T. 72. A plane T has both traces parallel to the G L. TV U inch 208 EXERCISES 6' b i 73. 74. 75. 76. above it, and TH, li inch in front of it. A line has A = 0, i, — i, and B = li, H, — 1^. Find where the line AB pierces the plane T. To Follow Section £9, Page 53. 73. Through the point O, pass a plane perpendicular to the line AB. 74. Through the point O, pass a plane perpendicular to the line AB. 75. Through the point O, pass a plane perpendicular to the lineAB. 76. Through the point O, pass a plane perpendicular to the line AB. To Follow Section 60, Page 55. 77. Plane T is perpendicular to H and V at 0, 0, 0. A point O is at f , f , \. Find the distance from the point O to the plane T. 78. Plane T is perpendicular to H, makes an angle of 20° with V, cuts G.L. at 0, 0, 0. A point O is at |, f , f . Find its distance from the plane T. 79. T has its V trace parallel to the G.L., | inch above it, its H trace is parallel to the G.L., \ inch in front of it. O is a point at 0, — I, — f . Find its distance from the plane T. 80. A plane T is perpendicular to V and makes 20° with H., cutting the G.L. at 0, 0, 0. A point O is at f, 0, f. Find its distance from the plane T. EXERCISES 209 81. A plane T has its V trace parallel to the G.L., f inch below H, its H trace parallel to the G.L., f inch back of V. O is any point on the G.L. Find its distance from the plane T. 82. A plane T has both traces parallel to the G.L. and coincid- ing; and I inch above it. A point O is at 0, — |, f . Find its distance from the plane T. To Follow Section 63, Page 58. 83. A = 0, 0, f ; B = U, U, U; C - 1, i^, |; D = A, I, -. Find the angle between the lines AB and CD. 84. A plane T has V trace at 45° with the G.L. above and towards the right, and H trace at 30° with the G.L. below and towards the left. Find the angle between the two traces. 85. A, B and C are the vertices of a triangle. A = 0, |, li; B = 4, H, f ; C = If, i, 1. Find its true shape and the angles between the sides. 86. A = 0, - t, - i; B = 1, - f , - i; C = i, - 1, - f. Find the angle between the lines AB and BC. 87. A =0,-1,-1; B = 0, - t, 0; C = i, - 1, — |. Find the angle between the lines AB and BC. 88. A = 0, If, — 1; B = 0, f, 0; C = 0, - i, - 1. Find the angle between the lines A B and B C. 89. Given a plane T whose traces coincide and make an angle of 45° with the G.L. Choose any point on the plane, not on either trace, and draw two lines lying in the plane and making an angle of 60° with each other. 90. Given a plane T with TO = 0, 0, 0. A point 1 on the V trace = 2i, — f , 0, and a point 2 on the H trace = 2i, 0, If. A line AB in it has A = 1, li, 0, and B = 2, 0, — . Draw a line in the plane making an angle of 90° with AB. 91. A line AB has A = 0, — f , f ; B = 2f , |, — i. Pass a plane 210 EXERCISES through the line and parallel to the G.L. At a point O, which divides the line AB in the ratio of 5 to 4, draw a line making 90° with it and lying in the plane passed through it. 92. O = 5, — I, — i. It is a point in a plane T, which is per- pendicular to H and makes an angle of 30° with V. Draw two lines through O making an angle of 60° with each other. 93. = 0, — 1,-2. It is a point in a plane containing the G.L. draw two lines through O and in the plane making an angle of 120° with each other. y 94. ^/ / "<^ 76 95. \^y T / 96. aa" 97. To Follow Section 66, Page 59. 94. Project the line AB on the plane T. 95. Project the line AB on the plan^ T. 96. Project the line AB on the plane T. 97. Project the line AB on the plane shown. of rrv TH h' 98. c/ 99. TH TV h lOO. 98. Project the line AB on the plane T determined by the point C. EXERCISES 99. Project the line AB on the plane T. lOO. Project the line AB on the plane T. lOl. f^/ / g t\ >^ \ N 102. 103. To Follow Section 68, Page 62. 101. Find the angle between the planes T and S. 102. Find the angle between the planes T and S. 103. Find the angle between the planes T and S. 104. Find the angle between the planes T and S. ./f \ r r / t v< \ 105. 106. \ / <^ \iS> 107. / 211 j^ P sV^ P 104. ^/^ y y 108. 105. Find the angle between the planes T and S, the latter be- ing determined by the point O. 106. Find the angle between the planes T and S. 107. Find the angle between the planes T and S. 108. Find the angle between the planes T and S. To Follow Section 77, Page 75. 109. A plane T has both traces coinciding and making an angle 212 EXERCISES of 45° with the G.L. toward the right. Draw the projections of a triangular pyramid of 21 inches altitude, and equilateral triangular base one inch on a side, an edge of the base in H and plane of base in T. no. A plane T has TV at 60° to the G.L. above and to the right. TH at 45° to the G.L., below and toward the right. A base of a cube lies in it with one corner in V and a diagonal of the base par- allel to V. The cube is H inches on edge. Draw its projections. HI. Find the locus of points equi-distant from the two points: A = 0, 1, f ; and B = If, H, H. 112. Find the locus of points in space equi-distant from t>he two following planes, T and S. Point 1 of T*= 0, i, 0, and 3 = 2i, i, 0. Point 2 = 0, 0, 1, and 4 = 2i, 0, 1. Point 1 of S = 0, If, 0, and 3 = 2i, If, 0. Point 2 = 0, 0, 1, and 4 = 2^, 0, 1. 113. Find the locus of points in space equi-distant from the points: A = 0, 1, |, and B = |, f , i^, and C = li^. If, 2. To Follow Section 83, Page 82. 114. A = 0, 1^, i; B = 1, 0, 0,; C = If, If, i. Find the locus of points in space equi-distant from the lines AB and BC. 115. N = 0, - i, - If; P = f, - U, - 1; O = If, - i, _ ^. Draw the locus of points in space equi-distant from the lines NO and OP. To Follow Section 102, Page 101. 116. MN is the axis of an oblique cylinder, with circular base H inches in diameter, in H. M = 0, 0, |; N = 2, If, li. The bases are parallel; O, P and Q are points on the surface; o' = i, f , 0; p' = 1, If, 0; g = 2i, 0, I; find their corresponding projections; draw a tangent plane to the cylinder at Q. 117. MN is the axis of an oblique cylinder, having a circular base in H, li inches in diameter. The bases are parallel; M = 0, *The odd numbers are points on the V trace, the even those on the H trace. EXERCISES 213 0, 1; N = H, H, H. P is a point on the cylinder at i, li, — ; draw a plane tangent to the cylinder at P; draw another plane normal to the cylinder at the point P. To Follow Section 114, Page 112. 118. Develop the cylinder of problem 117. To Follow Section 117, Page 116. 119. A righ circular cone of f inch diameter of base, If inches altitude, has its axis parallel to, and equi-distant from, V and H in the 3rd angle; center of base is at 0, — |, — f. A point P on its surface has its H projection at i^, 0, — 1. Draw a plane tangent to the cone at P. 120. An oblique cone, with circular base in H, H inches in diameter, and center on the G.L, at 0, 0, 0, has apex at li^, li, li^. A point P on its surface is at i^, — , f ; draw a tangent plane to the cone at P. To Follow Section 119, Page 119. 112. An oblique cone, with circular base in H, H inches in diameter, and center on the G.L. at 0, 0, 0, has apex at li\, H, 1^. A line MN has M = 1^, r«, — ts; N = 2^, 1,-1; draw a tangent plane to the cone and parallel to the line MN. 122. A right circular cone, of f inch diameter of base If inches altitude, has its axis parallel to, and equi-distant from, V and H, in the 3rd angle; center of base is at 0, — f , — |. A line BC has B = 2, — f , — 1^; C = 21, — i, — 1. Draw a plane tangent to the cone and parallel to the line BC. 123. A right circular cone has base in H, 2 inches in diameter, 2i inches altitude, and center at 2i, 0, Ij. MN is a line with M = 0, i, It, and N = li, li, i. Pass a plane tangent to the cone and parallel to the line MN. 124. Discuss the conditions of problem 123 when (1), the given line has the same inclination to the plane of the base of the cone that the elements of the cone do; (2), when the given line makes a 214 EXERCISES. less angle, with the plane of the base of the cone, than the elements do. To Follow Section 123, Page 122. 125. An oblique cone has apex at 0, If, 2. The base touches H at the point If, 0, If, is a circle in H projection, and its plane is perpendicular to V and makes an angle of 30° with H. A plane T, perpendicular to the axis of the cone, cuts the G.L. at f , 0, 0. Find the curve of intersection of the cone and the plane. To Follow Section 126, Page 125. 126. Develop the cone of problem 125. To Follow Section 150, Page 161. 127. Draw the contour of an oblique helicoid of uniform pitch, with the following conditions: One helical directrix, H inches in diameter, second helical directrix, 4 inches in diameter; pitch of helices 4 inches; angle of elements with the axis 15°. Take the H plane perpendicular, and the V plane parallel to the axis of the helical directrices. Plot the curves of at least three points of the generatrix, in addition to those in which it touches the two given vdirectrices. 128. Draw the contour of a right helicoid of uniform pitch, with the following conditions: One helical directrix 1^ inches in diameter, second helical directrix 4 inches in diameter, pitch of helices 4 inches. Take the H plane parallel to the plane director, and the V plane parallel to the axis of the helical directrices. Plot the curves of at least three points of the generatrix, in addition to those in which it touches the two given directrices. To Follow Section 159, Page 168. 129. An oblate spheroid rests on H and touches V. The minor axis is vertical and is 2 inches long, touching H at 3, 0, H; the major axis is 2i inches long. P is a point on the upper surface, and its V projection is at 3f , li, 0. Pass a plane through P and the minor axis. Show the curve of intersection of the plane with the spheroid; draw a plane tangent to the surface at P. EXERCISES .215 To Follow Section 161, Page 170. 130. A sphere, 2^ inches in diameter, rests on H, with center li inches from V in the 1st angle. A point P, on the front surface of the sphere, has its V projection | inch above the center and | inch to the right. Draw a plane tangent to the sphere at the point P. 131. An ellipsoid of revolution has a major axis If inches long, minor, li inches. It lies in the 3rd angle, touching the H plane, the major axis is parallel to V, 1 inch from it, and is perpendicular to H. A point P, on the surface, has its V projection f inch to the left of the axis, and If inch below H. Draw a plane tangent to the surface at the point P. 132. A circle, 1 inch in diameter, revolves about a vertical axis which touches H at a point li, 0, 1|, thus generating a torus. The center of the circle is at a distance of 1 inch from the axis, and the torus rests on H. A point upon the surface is at a distance of f inch from the axis and li inches from V; it may have several positions, show them and pass planes tangent to the surface at each. To Follow Section 167, Page 180. 133. A line AB has A = 0, 0, 3i; B = 2f, 3, li; it revolves about an axis perpendicular to H, whose H projection is the point li, 0, 1|, thus generating a hyperboloid of revolution of one nappe. Draw the upper base generated by the point B, and the lower, generated by the point A; draw the circle of the gorge, and the principal meridian plane; draw through B an element of the second generation. Locate a point P, on the surface, at If, — , If ; a point R at I, I, — . Draw a plane tangent to the surface at P. Find the curve of intersection, with a plane S, cutting the G.L. at 31, 0, 0, and whose V and H traces make angles of 30° with the G.L. toward the right, above and below the G.L., respectively. To Follow Section 174, Page 187. 134. Two oblique cones, with circular bases in H, intersect each 216 EXERCISES other. One cone has base 2i inches in diameter, its center being at 0, 0, 2, and apex at 2i, 31, f ; the other cone has base 3 inches in diameter, its center being at 2|, 0, 2|, and apex at |, 2i, i. Draw the curve or curves of intersection of the two cones, using a sufficient number of points to obtain a good curve. 135. 136. 136. Prove that the point C does not lie on the line AB. 136. Prove that the two lines AB and CD do not lie in the plane T. 137. A = 0, 1, f ; B - I, U, I; P = If, |, |. A plane T has TO = If, 0. 0; TV makes an angle of 45" with the G.L. above and to the right; TH makes an angle of 45° with the G.L. below and to the right. Draw, through P, a plan© parallel to the line AB, and perpendicular to the plane T. 138. the line AB and distant 1 inch from P. Draw a plane through 139. A plane T has TO = 0, 0, 0; TV makes an angle of 30° with the G.L. above and to the right; TH makes an angle of 45° with with the G.L. below and toward the right. A point A, in the plane, is at 1, |, — . Through A draw a line making an angle of 30° with H. 140. A line AB makes an angle of 30° with H; its V trace, A, is at 0, I, 0; the end B has its H projection at If, 0, 1. Draw a line through the H trace of AB, lying in H, and making an angle of 45° with AB. EXERCISES 217 141. 141. Given the two pulleys as shown, and in 3rd angle projec- ion, and, assigning dimensional values to them, etc., find the position of the center, and the direction of the axis of an idler, of any given diameter, which will take the belt from the left hand pulley and deliver it to the right hand one without its slipping off. i \ €) 142. 143. 142. Assigning numerical values to the edges of the form shown, find: (1). The angles between the sides of the solid. (2) . The angles of the bevel of the edges between the different sides. 143. Find the curve of intersection between the stem and end of the connecting rod as shown. 218 EXERCISES 144. 144. Given the *two way' flue, as shown, and assigning dimen- sional values to the form, develop each piece as for sheet metal work. 145. 145. Given the form shown, which is a *cylindrical to square' offset, develop it by triangulation, as discussed in the appendix, namely, take four triangles, whose bases are the edges, respec- tively, of the square opening, and whose apexes are in the cylindrical portion below. Between these, divide the cylindrical EXERCISES 219 opening into a number of parts, the bases of another set of triangles whose apexes are in the corners of the square opening above. 146. 147. 146. Given the form shown, which is a cylindrical chimney- above a square opening, and, assigning to it dimensional values, develop the surface, as for sheet metal work, by the method of triangulation described in the appendix, namely, take first, four triangles, whose bases are, respectively, in contact with the four lower edges, and apexes in the neck above; then a series of triangles whose bases are contiguous and form part of the neck, with sides touching and apexes at the corners where the four lower edges meet. 147. Develop the frustum of the conical figure shown, by the method of triangulation discussed in the appendix, namely, divide the surface into a number of triangles, whose bases are oppositely directed, and lie in the upper and lower bases of the form, respec- tively, and whose sides are touching one another. 220 EXERCISES 148. 148. Given the form shown, which is analogous to the slope sheet of a locomotive boiler, and assigning dimensional values to it, develop it, as for sheet metal work, by the method of triangula- tion, discussed in the appendix, namely, divide the upper and lower bases into a large number of equal parts; let the spaces between the divisions be the bases of a set of triangles, one set in the upper curve, the other set in the lower curve, and the apexes of each set in the opposite curve, the edges of the triangles coincident. 149. Given the form shown, which is a 'round to flat' flue, and assigning dimensional values to it, develop it as for sheet metal work, by the method of triangulation as described in the appendix namely, take two triangles, with apexes at a and d, respectively EXERCISES 221 149. and with bases 6c and e/ respectively; then, divide up the remain- der of each base into a large number of parts, each one into the same number. Regard the divisions in each base as the bases of a set of triangles with apexes in the opposite base, at the points of division on that base. INDEX PAGE Alphabet of the line 26 Alphabet of the plane 12 Alphabet of the point 6 Analysis defined 24 Angles, to draw a line through a given point at a given angle to an oblique plane 79 Angle of a plane with a co- ordinate plane, being given, and the trace with that plane, to find the other trace 68 Angle of a plane with a co- ordinate plane being given, and the trace with the corresponding coordi- nate plane, to find the re- maining trace 70 Angle, to draw a plane at a given angle to another and through a giveij line 65 Angles, dihedral 3 Angles between lines 58, 59 Angles, a line making given angles with the coordi- nate planes 45 Angles a line makes with both coordinate planes, to find its projections 73 Angles, lines lying in four dihe- dral angles 25 Angles a line makes with a plane 61 Angles between two planes 62 Angles between two planes, by change of coordinate planes 66 Angles, given the angles a plane makes with both coordinate planes to find both traces 72 Appendix 197 PAGE Approximate development 198 Archimedian spiral in classifi- cation, 86 Archimedian spiral defined.... 88 Archimedian spiral as the trace of an oblique helicoid and a plane 163 Asymptote defined 93 Auxiliary sphere, used to ob- tain traces of a plane with coordinate planes.. 72 Axes of reference 8 Axis of a cone defined 115 Axis of a hyperbolic paraboloid 154 Base of a cone defined 115 Base of a cylinder defined.... 99 Center of projection 3 Change of coordinate planes for angle between two planes 66 Change of coordinate planes for a line 31, 39 Change of a coordinate plane with respect to a plane. 40, 41 Change of coordinate planes with respect to a point. 17, 18 Characteristic name for a cone. 115 Circle through three given points 77 Circle as a locus 80 Circle, in classification 86 Circle, involute of, classified.. 86 Circle defined 87 Circular, right circular cone of revolution; cylinder of revolution 91 Circular, to develop a right circular cylinder 110 Circle as intersection of sur- faces of revolution hav- ing a common axis 171 224 INDEX PAGE Circle of the gorge of a surface of revolution 167 Circle of the gorge of a hyper- boloid of revolution .... 175 Classification of lines 86 Classification of surfaces 91 Cone, axis of, defined 115 Cone, base of, defined 115 Cone, in classification 91 Cone, defined 88, 155, 166 Cones, intersection of two 187 Cone, intersection of, and a convolute 191 Cone, intersection of, and a cyl- inder 189 Cone, intersection of any, by a plane 120 Cone, nappe of, defined 116 Cone, oblique, defined 115 Cone, to develop 122, 124, 125 Cone, to draw a plane normal to, at a point on the sur- face 119 Cone, to assume a point on 116 Cone, plane tangent to and par- allel to a line 119 Cone, plane tangent to, at a point on surface 116 Cone, plane tangent to, through a point outside 117 Cone of revolution defined 115 Cone, of revolution, right cir- cular 91 Conical helix classified 86 Conical helix defined 88 Conies defined 86, 87 Conoid, in classification ....91, 140 Conoid defined 155 Conoid, intersection of, and plane 161 Conoid, to assume a point on, and to draw a plane tan- gent at the point 156 Conoid, right elliptical, defined 158 Conoid, right helical defined... 155 PAGE Consecutive elements of a sin- gle curved surface lying in a tangent plane to the surface 98 Consecutive positions of a mov- ing point defined 83 Contour elements of a solid... 107 Conventions 13 Convolute, in classification 91 Convolute defined 89, 126 Convolute, helical, to represent 129 Convolute, helical defined 126 Convolute, helical, to develop . . 137 Convolute, helical, to draw a plane tangent to and par- allel to a line 135 Convolute, helical, intersection of, with a plane 132 Convolute, intersection of, with a cone 191 Convolute, intersection of, with a cylinder 191 Convolute, to assume a point on, and to draw a plane tangent to 131 Coordinate axes 8 Coordinate planes, line parallel to both 19 Coordinate plane, line perpen- dicular to 20 Coordinate planes of projection defined 1 Coordinate planes of projection, angles a line makes with both, to find its projec- tions 73 Coordinate planes of projection, angles a plane makes with both, to find its pro- jections 72 Coordinate planes of projection, change of, for angle be- tween two planes 66 Coordinate planes of projection, change of, for a line. .31, 39 Coordinate planes of projection, change of, with respect to a plane 40, 41 INDEX 225 PAGE Coordinate planes of projection, cliange of, with respect to a point 17. 18 Coordinate planes of projection, distance of a point from.6, IS Coordinate planes of projection, line making given angles with 45 Coordinate planes of projection, revolution of 4 Coordinate planes of projection, trace of, with a line. .21, 22 Curvature, curve of double, formed by the motion of a plane intersecting in successive elements of... 84 Curvature, development of sur- face of single 108 Curvature line moving tangent to a curve of double 126 Curvature, a right line tan- gent to a curve of single, defined 94 Concourse, point of, defined 143 Corresponding, use of word. ... 14 Corresponding projection of a point on a line 15 Corresponding trace of project- ing plane of a line 15 Curve defined as envelope of a moving line 84 Curve of double curvature, formed by the motion of a plane intersecting in successive elements of the curve 84 Curve of intersection of a plane and a surface 107 Curve, the meridian curve of a hyperboloid of revolution is a hyperbola 178 Curve, meridian curve of a sur- face of revolution 167 Curve, projection of, defined... 92 Curve, a right line tangent to a curve of single curvature defined 94 Curve, tangency of two, defined 94 PAGE Curve, to draw a tangent to an irregular plane, 95 Curved lines classified 86 Curved, double curved lines de- fined, also single 84, 88 Curved, double, surfaces class- ified, 91 Curved, double, surfaces defined 89 Curved, double, surfaces of rev- olution 91 Curved, single, surfaces, two consecutive elements ly- ing in a tangent plane to 98 Curved, single surface, the cyl- inder defined 98 Curved, single, surface, develop- ment 108 Curved, single, surface, inter- section with a surface of revolution 194 Curved, single, surface, to as- sume point on 99 Curved, single, surface, to as- sume an element on.... 100 Cycloid, in classification 86 Cylinder defined 88, 98, 166 Cylinder, to develop a 112 Cylinder, to develop a right cir- cular, 110 Cylinder, in classification 91 Cylinder, intersection of cone and 189 Cylinder, intersection of. and a convolute 191 Cylinder, intersection of two.. 186 Cylinder, intersection of a right circular, and a plane 108 Cylinder, normal plane to, at a point, 105 Cylinder, plane normal to, also parallel to a line 106 Cylinder, projecting, of a solid. 107 Cylinder, right circular, in classification 91 Cylinder, to draw a plane tan- gent to, and parallel to a line 103 226 INDEX PAGE Cylinder, to draw a plane tan- gent to, at a point on 101 Cylinder, to draw a tangent plane through a point outside 102 Declivity, line of greatest 57 Definition 86, 87 Degree of an equation, the maxi- mlum number of times a line may cut a curve 93 Descriptive geometry, defined . . 1 Descriptive geometry, proper study of 24 Develop, to, any cone in gen- eral 124 Develop, to, an oblique cone . . 125 Develop, to, an oblique cylinder 112 Develop, to, a helical convolute 137 Develop, to, a right circular cone 122 Develop, to, a right circular cyl- inder 110 Develop, to, a surface of single curvature lOS Development, approximate 198 Diametral planes of a hyper- bolic paraboloid 150 Dihedral angles 3 Dihedral angles, lines connect- ing points in different . . 25 Directrix, defined 87 Directrix, helical 123 Distance of point from coordi- nate planes 6, 18 Distance from a point to a line 70 Division, the projection of any division of an angle be- tween lines 59 Double curvature, line moving tangent to a curve of . . . 126 Double curvature, curve of, formed by the motion of a plane intersecting in successive elements of curve 84 Double curved lines classified 86 Double curved lines defined . . 84 PAGE Double curved surfaces classi- fied 91 Double curved surface de- fined 89, 165 Double curved surface, plane tangent to 165 Double curved surfaces of revo- lution in classification... 91 Eccentricity, defined 80 Element, to assumie, on a cone 116 Element, to assume, on single curved surface 100 Element of path traced by a moving point 83 Element of tangency defined... 98 Elements consecutive, lying in a plane tangent to a single curved surface. . . 98 Elements of contour defined... 107 Ellipse, defined 87 Ellipse in classification 86 Ellipse, as a locus 80 Ellipsoid of revolution in classi- fication 91 Ellipsoid of unequaled axes de- fined 92 Elliptical cylinder defined 99 Elliptical hyperboloid in classi- fication 140 Elliptical hyperboloid defined.. 140 Elliptical cone, oblique 117 Elliptical conoid, right, de- fined 158 End plane, line parallel to . . . 20 End plane, trace of a line with 23, 24 End vertical plane 1, 3 End vertical plane, ground line 13 Envelope, curve defined as the, of a moving line 84 Epicycloid in classification 86 Epicycloid defined 87 Equiangular spiral in classifica- tion 86 INDEX 227 PAGE First and third angle projection discussed 197 Focus defined 80 Front vertical plane 1, 3 Geometry, descriptive defined.. 1 Geometry, projective, defined.. 2 Generation of double curved surfaces 166 Generation, method of, of sur- faces 91 Generation of a cone defined . . 115 Generatrix of a cylinder. .\ 99 Generatrix defined 87 Generic name for cone 115 Gorge, circle of the, of a hyper- boloid of revolution .... 175 Gorge, circle of the, of a sur- face of revolution 167 Gorge plane, of hyperboloid of revolution 175 Gorge plane of a surface of revolution 167 Graphical locii 81 Ground line 3 Ground line with end vertical plane 13 Ground line as projection of V and H planes 20 Gussett, method of develop- ment 198 Hi plane, supplementary IS Helical conoid, intersection with, and a plane 161 Helical, conoid, oblique 156 Helical, conoid, right, defined.. 155 Helical convolute, intersection with a plane 132 Helical convolute defined 126 Helical convolute, to develop. . 137 Helical convolute, plane tangent to and parallel to a line 135 Helical convolute, plane tangent to, through a point out- side 133 Helical convolute, to assume point on and draw a tan- gent plane to 131 PAGE Helical convolute, to represent 129 Helicoid, in classification 91 Helicoid, right and oblique, etc., defined 158 Helicoid, oblique, classified 140 Helicoid, oblique, to assume point on 161 Helicoid, oblique, the point of tangency of a plane with 161 Helicoid, oblique, trace of with a plane, an archimedian spiral 163 Helicoid, right, classified 140 Helicoid, right, illustrated 162 Helix, angular pitch 128 Helix, classified 86 Helix, conical, defined 88 Helix, defined 87,126 Helix, linear pitch 128 Horizontal plane 1, 3 Horizontal of a plane 37 Horizontal projecting line 5 Hyperbola in classification ... 86 Hyperbola, defined 87 Hyperbola, as a locus 80 Hyperboloid, elliptical classified 140 Hyperboloid of revolution, the mieridian curve of 178 Hyperbolic paraboloid, its axis. 154 Hyperbolic paraboloid, in class- ification 91, 140 Hyperbolic paraboloid defined 141, 146 Hyperbolic paraboloid, diame- tral planes of, 150 Hyperbolic paraboloid, a plane tangent to and through a line 155 Hyperbolic paraboloid, plane tangent to, at a point on the surface 149 Hyperbolic paraboloid, to assume a point on, .147, 154 228 INDEX PAGB Hyperbolic paraboloid, its pro- jections and sections by- coordinate planes, and tangent plane at any point 151 Hyperbolic paraboloid, theorem 12 143 Hyperbolic paraboloid, theorem 13 144 Hyperbolic paraboloid, the ver- tex of 154 Hyperbolic spiral, in classifica- tion 86 Hyperboloid of revolution, in classification 91, 140 Hyperboloid of revolution, de- fined 90, 173 Hyperboloid of revolution, in- tersection by a plane. . . . 178 Hyperboloid of revolution, me- ridian curve is a hyper- bola 178 Hyperboloid of revolution, to assume point on and draw tangent plane to . . . 180 Hypocycloid, in classification.. 86 Hypocycloid defined 87 Indeterminate projections of a line 15 Indeterminate position of a plane 11 Intersecting, angle between two, lines 58, 59 Intersection of two cones .... 187 Intersection of a cone and con- volute 191 Intersection of a cone and a cylinder 189 Intersection of a cylinder and convolute 191 Intersection of two cylinders.. 186 Intersection of a cone and plane 120 Intersection of an oblique heli- cal conoid with a plane. . 161 Intersection of a helical convo- lute and a plane 132 PAGE Intersection of a plane and hy- perboloid of revolution.. 182 Intersection, line as, of two planes 38 Intersection, line of, of two planes 47, 48 Intersection of two bodies bounded by plane faces. . 184 Intersection of two pyramids.. 185 Intersection of surfaces 183 Intersection of a surface and a plane 107 Intersection of a single curved surface and surface of revolution 194 Intersection, point of, of two lines 16 Intersection of a right circular cylinder and a plane . . . 108 Intersection of surfaces of revo- lution, general cases . . . 192 Intersection of surface of revo- lution by a plane 172 Intersection of two surfaces of revolution 171 Intersection of surfaces of revo- lution having a common axis 192 Intersection of surfaces of revo- lution, axes in the same plane 192 Intersection of surfaces of revo- lution, axes not in the same plane 194 Intersection, as a trace 11 Involute of a circle, in classifi- cation 86 Line, alphabet of 26 Line, making given angles with a coordinate plane 45 Line, angles made with both co- ordinate planes 73 Lines, angle between 58, 59 Line, angle it makes with a plane 61 Lines classified 86 Lines, double curved, defined. . 84 INDEX 229 PAGE Line, change of coordinate planes for 31 Lines as determining a plane. . 11 Line, distance from a point to. . 70 Line of greatest declivity 57 Lines, notation for 13 Lines, projecting 2 Lines, projecting planes of . . . 15 Lines generating a surface by- motion 88 Lines, horizontal projecting. .4, 5 Line, indeterminate projections of 15 Line, the intersection of two planes 47, 48 Lines, point of intersection of two IG Lines, length of, by change of coordinate planes 32 Line, length of by revolution. . 32 Line, path of moving point ... 83 Line, normal to a surface de- fined 97 Line, to draw, parallel to a plane through a point. . . 43 Line, parallel to both coordi- nate planes 19 Line parallel to a plane 20 Line perpendicular to a coordi- nate plane 20 Line perpendicular to a plane. 52, 53 Line, plane through, and a point 37 Line, plane through one, and parallel to another 44 Lines, the common perpendicu- lar of two non-intersect- ing lines, 67, 68 Line, to project on any oblique plane 59 Line connecting points in dif- ferent angles 25 Line, to designate a point on.. 15 Lines, perpendicular and one parallel to a coordinate plane 57 Line, projecting 5 PAGE Lines, projections of, fully de- termined 14 Lines, corresponding trace of the projecting plane of. . 15 Line, revolution of a point about 27, 31 Lines, right, defined 84 Lines, single curved, defined... 84 Line, tangent to a curve, de- fined 93, 94 Line, traces of 20, 21, 22 Line, trace of with any plane. 51 Line, trace of with an end plane 23, 24 Line, traces of, to contain a... 37 Line, the traces of its project- ing plane 14 Line, rule to find the traces of. with a surface 108 Lines, vertical projecting 4, 5 Linear pitch of a helix 128 Locii, defined 80 Locus, the circle, ellipse, hyper- bola, parabola as a 80 Locus of points equidistant from two planes 81 Locus of points equidistant from three points 81 Locus of projecting perpendic- ulars 14 Logarithmic spiral in classifi- cation 86 Meridian curve of a surface of revolution 167 Meridian curve of a hyperbo- loid of revolution is a hy- perbola 178 Meridian plane of a surface of revolution at a point is perpendicular to the tan- gent plane through the point 170 Nappe, of a cone defined 116 Non-intersecting lines, the com- mon perpendicular to . . . 68 Non-intersecting lines as ele- ments of a warped sur- face 141 230 INDEX PAGE Normal to a curve, defined 94 Normal to a surface, defined. .. 97 Normal plane to a cone through a point on, 119 Normal plane to a cylinder, through a point on.. 105, 106 Normal plane to a surface de- fined 97 Notation 13 Oblique cone defined 115 Oblique cone, to develop 125 Oblique cone, elliptical 117 Oblique conoid, in classifica- tion 91 Oblique cylinder, defined 99 Oblique cylinder, to develop... 112 Oblique, conoid, helical, defined 156 Oblique helicoid classified 140 Oblique conoid, helical, inter- section with a plane.... 161 Oblique helicoid, defined 15S Oblique helicoid, illustrated... 162 Oblique helicoid of varying pitch, classified 140 Oblique helicoid, to assume a point on, 161 Oblique helicoid, point of tan- gency of any plane with. 164 Oblique helicoid. trace with a plane, an archimedian spiral 163 Oblique hyperbolic paraboloid.. 154 Oblique plane, change of coor- dinate planes for 40 Oblique plane, to draw a line through a point at a given angle to 79 Oblique plane, to draw a pyra- mid with base in 75 Parabola, in classification 86 Parabola, defined 87 Parabola, as a locus, 80 Paraboloid, hyperbolic, its axis 154 Paraboloid, hyperbolic, in class- ification 91, 140 PAGB Paraboloid, hyperbolic, de- fined 141, 14o Paraboloid, hyperbolic, diame- tral planes of 150 Paraboloid, hyperbolic, tangent plane to, through a line. 155 Paraboloid, hyperbolic, tangent plane to, at a point 149 Paraboloid, hyperbolic, to as- sume a point on 147, 154 Paraboloid, hyperbolic, projec- tions of, sections by coor- dinate planes and tan- gent plane to 151 Paraboloid, hyperbolic, the ver- tex 154 Paraboloid, hyperbolic, theoremi 12 143 Paraboloid, hyperbolic, theorem 13 144 Paraboloid of revolution, in classification 91 Paraboloid, variable, defined... 92 Paraboloid of revolution, de- fined 167 Parallel, line, to a plane 20, 43 Parallel, line, to both coordi- nate planes 19 Parallel, line, to an end plane,. 20 Parallel, plane tangent to a cone and parallel to a line 119 Parallel planes, traces of with a third plane 39 Parallel, plane normal to a cyl- inder and parallel to a line 106 Partial penetration of one form with another 185 Path, element of, traced by a moving point S3 Path as a locus 80 Penetration, complete or partial of one form with another 185 Perpendicular, line, to a plane 52, 56 Perpendicular, line, to coordi- nate plane 20 INDEX 231 PAGE Perpendicular, a plane through a line, perpendicular to a given plane 79 Perpendicular, the common, to two non-intersecting lines 67, 68 Perpendicular, a plane, through a point, to a line 53 Perpendicular, a plane tangent to surface of revolution is perpendicular to the meridian curve through the point 170 Perpendicular, projecting, as a locus 14 Perspective, defined 2 Pitch, angular, of a helix 128 Pitch, helicoid of uniform, de- fined 158 Plane the alphabet of, 12 Plane, angle between two . . 62, 66 Plane at given angle with an- other through a line in the latter 65 Plane, in classification 91 Plane, defined 88 Plane, the angles with both co- ordinate planes being given, to find its traces. 72 Plane of coordinate axes 8 Plane, end vertical, front verti- cal, horizontal 1^ 3 Plane, indeterminate position.. 11 Plane, notation for 13 Plane, given to draw a pyra- mid with base in 75 Planes, change of coordinate, with respect to a line. 31, 39 Planes, change of both coordi- nate planes with respect to a plane 40, 41 Planes, revolution of coordinate 4 Planes of projection, defined ... 1 Planes of projection for per- spective 2 Planes, diametral, of a hyperbo- loid of revolution 150 PAGE Plane, gorge of, hyperboloid of revolution 175 Plane, gorge, of surface of rev- olution 167 Plane, intersection of any, cone by a 120 Plane, intersection of, and a conoid, helical 161 Plane, intersection of, and a right circular cylinder. . 108 Plane, intersection of and a hy- perboloid of revolution.. 168 Plane, intersecting itself in suc- cessive elements of a curve of double curva- ture 84 Plane, intersecting any surface 107 Plane, intersecting any surface of revolution 172 Plane, line as intersection of two 38 Plane, line of intersection of two 47, 48 Plane, angle a line makes with, 61 Planes, angle a line makes with both coordinate, to find its projections 73 Planes, line making giving an- gles with coordinate.... 45 Planes, coordinate, line perpen- dicular to 20 Plane, the horizontal of 37 Plane, line through a point at a given angle to 79 Plane, through one line parallel to another line 44 Plane, line parallel to 20 Plane, to draw a line parallel to, through a point 43 Plane, through a line and point 37 Plane, line perpendicular to any 52 Plane, line perpendicular to, is .perpendicular to any line in it 56 Plane, through a line perpen- dicular to a plane 79 232 INDEX PAGE Plane, to project a line on any, 59 Plane, the vertical of 37 Plane, normal to a cone at a point 119 Plane, normal to a cylinder at a point 105, 106 Plane, normal to a cylinder through a point outside. 106 Plane, normal to a surface, de- fined 97 Plane, through three points ... 35 Plane, to assume a point in a . . 38 Planes, locus of points equi- distant from two 81 Plane through a point parallel to two lines 48 Plane through a point perpen- dicular to a line 53 Plane, as projected on coordi- nate planes 11 Planes, projecting, of a line... 15 Plane, revolution of supplemen- tary vertical 17 Plane, tangent to a cone 116 Plane, tangent to a cylinder and parallel to a line 103 Plane, tangent to a cylinder at a point on, point out- side 101, 102 Plane, tangent, defined 97 Plane, tangent to a cone and parallel to a line 119 Plane, tangent to a cone, through a point outside. 117 Plane, tangent to a conoid at a point on the surface.. 156 Plane, tangent to a double curved surface 165 Plane, tangent to a helical con- volute 131 Plane, tangent to a helical con- volute and parallel to a line 135 Plane, tangent to a helical con- volute through a point outside 133 PAGE Plane, tangent to a hyperbolic paraboloid and through a line 155 Plane, point of tangency with oblique helicoid 164 Plane, tangent to a hyperbolic paraboloid at a point on surface 149 Plane, tangent to hyperboloid of revolution at a point on surface 180 Plane, tangent to a sphere through a line 170 Plane, tangent to surface of revolution at a point on surface 167 Planes, tangent, to warped sur- faces 149 Plane, trace of a line with co- ordinate 21, 22 Plane, traces of 11 Plane, traces of not limited... 16 Plane, one trace given and an- gle with corresponding plane to find remaining trace 70 Plane, one trace given and an- gle with plane of that trace to find the corre- sponding trace 68 Plane, traces of, to contain a line 37 Plane, corresponding traces of projecting plane of a line. .15 Plane, traces of parallel planes with any other plane... 89 Plane, trace of a line with any, 51 Point, the alphabet of 6 Point, change of coordinate planes with respect to. 17, ]8 Point, circle through three.... 77 Point of concourse, defined.... 143 Point, to assume, on a cone. . . 116 Point, plane tangent to, through a point outside 117 Point, on a conoid, to draw a tangent plane 156 Points, consecutive 83 INDEX 233 PAGE Point, to designate on a line. . . 15 Point, distance from, to a line. 70 Point, distance from coordi- nate planes 6, 18 Point, element of path traced by a moving 83 Point, to assume on a helical convolute 131 Point on line in different angles 25 Point, through given, to draw a line making given angles with any plane... 79 Point, to draw a line parallel to a plane through, 43 Point, locus of, equally distant from two planes 81 Point, to assume, on an oblique helicoid 161 Point, to assume, on a hyperbo- loid of revolution and draw tangent plane 180 Point, to assume, on hyperbolic paraboloid, and draw plane tangent to 149 Point, of intersection of two lines 16 Points, locus of, equidistant from three points 81 Point, notation for 13 Point, to assume in a plane. ... 38 Points, plane through three. . . 35 Point, plane through line and.. 37 Points, to pass a plane parallel to two lines through, ... 46 Point, plane through, perpen- dicular to a line 53 Point, the projections of 5, 6 Point, corresponding projections of, on a line 15 Point, the projections of, on a common perpendicular to the G. L 6 Point, position of, determined by its projections, theo- rem 1 7 Point, revolved about a line 27, 31 PAGE Point of sight 2 Point, to assume, on single curved surface 99 Point, to assume on surface of revolution 167 Point of tangency defined 94 Point, approximate, of tan- gency of a curve and its tangent 96 Principal meridian curve of a surface of revolution . . . 167 Prisms, intersection of two 184 Project, a line on any plane. .. 59 Projecting cylinder of any solid 107 Projecting lines, horizontal, ver- tical, etc 2, 4, 5 Projective perpendiculars, as lo- cus 14 Projecting plane of a line 15 Projection, center of 2, 3 Projection, corresponding pro- jection of a point on a line 15 Projection of forms 3, 6 Projections, to find the, of a line when angles with co- ordinate planes are given 73 Projections of a line fully deter- mined 14 Projections of a line indeter- minate 15 Projections of a point 5, 6 Projections of a point deter- mine its position 7 Projective geometry 2 Proof, form of 17 Pyramids, intersection of two 172 Pj^ramid, to draw, with base in given oblique plane . . 75 Reciprocal spiral 86 Rectangular hyperbolic parabo- loid 154 Rectified curve 95 Right lines defined 84 Right conoid, in classifica- tion 91 234 INDEX PAGE Right circular cone of revolu- tion 91 Right circular cylinder, in classification 91 Right line tangent to another, defined 94 Right line tangent to curve of single curvature 94 Right cylinder defined 99 Right circular cylinder, to de- velop 110 Right circular cylinder, inter- section by a plane 108 Right helicoid, classified 140 Right helical concoid, defined, right conoid 155 Right circular cone, to develop. 122 Right helicoid, defined 159 Right conoid, elliptical, defined 158 Revolution, cone of, defined... 115 Revolution, hyperboloid of, de- fined 173 Resolution, hyperboloid of, classified 140 Revolution, hyperboloid of, in- tersection by a plane 182 Revolution, hyperboloid of, me- ridian curve of, is a hyperbola 178 Revolution, hyperboloid of, to assume a point on and draw a tangent plane to. 182 Revolution, hyperboloid of, has two systems of genera- tion 176 Revolution, length of line by, 32 Revolution, paraboloid of, de- fined 167 Revolution of coordinate planes, 4 Revolution of point about a line 27, 31 Revolution, surface of 89 Revolution, surface of, classi- fed 91 Revolution, surface of, defined. 166 Revolution, surface of, intersec- tion of, general case 192 PAGE Revolution, surfaces of, inter- secting when having a common axis 171 Revolution, surfaces of, inter- secting, axes in same plane 192 Revolution, surfaces of, inter- secting, axes not in same plane 194 Revolution, surface of, intersec- tion by any plane 172 Revolution, surface of, intersec- tion with single curved surface 194 Revolution, surface of, me- ridian curve of, defined. 167 Revolution, surface of, plane tangent at a point is per- pendicular to the meri- dian plane through the point 170 Revolution, surface of, plane tangent to. at a point. . . . 167 Revolution, surface of, to as- sume a point on 167 Roulettes, classified, defined 86, 87 Rule, for intersection of plane and surface 107 Rule, piercing point of a line and surface 108 Rule, tangent plane to a sur- face 97 Ruled surfaces, classified 91 Ruled surfaces, defined 88 Ruled surface of revolution, in classification 91 Secants, curve of, in classifi- cation 86 Secant line, approaching a tan- gent as a limit, 93 Secant plane of a cylinder, as a base 99 Secant plane, approaching a tan- gent plane as a limit 97 Sight, point of 2 Single curved lines, classified. 86 Single curved lines, defined ... 84 INDEX 235 PAGE Single curvature, line tangent to a curve of 94 Single curved surfaces, classi- fied 91 Single curved surfaces, defined. 88 Single curved surface, develop- ment 108 Single curved surface, the cylin- der, defined 98 Single curved surface, two con- secutive elements of lying in a tangent plane 98 Single curved surface, intersect- ing a surface of revolu- tion 194 Single curved surface, to as- sume a point on 99 Sinusoid curve in classifica- tion 8C Solution of problems defined.. 24 Sphere, in classification, 91 Sphere, defined 16G Sphere, auxiliary, for locating plane, traces given 72 Sphere, plane tangent to, through a line 170 Spirals, defined 86 Spiral, archimedian, defined... 88 Spiral, archimedian, the trace of an oblique helicoid and a plane 163 Supplementary Hi plane 1 Supplementary vertical plane. . 17 Surfaces classified 91 Surfaces, double curved, , de- fined 165 Surfaces, intersection of 183 Surfaces, method of genera- tion 91 Surfaces, generated by motion of a line 88 Surfaces, curve of intersection of and a plane 107 Surface, normal plane to, de- fined 98 Surface, tangent plane to, de- fined 97 FAQE Surfaces of revolution. 89 Surfaces of revolution, classi- fied 91 Surfaces of revolution, with common axis, tangent or intersect in a circle .... 171 Surfaces of revolution defined, 166 Surfaces of revolution, intersec- tion of, general case 192 Surfaces of revolution, inter- section of, with common axis 192 Surfaces of revolution, inter- section of, with axes in same plane 192 Surfaces of revolution, inter- section by a plane 172 Surfaces of revolution, to assume a point on 167 Surfaces of revolution, tangent plane to, at a point 167 Surfaces of revolution, plane tangent to is perpendicu- lar to the meridian plane through point of contact 170 Surface, rule to find trace of a line with 108 Surfaces, ruled, defined 88 Surfaces, single curved classi- fied 91 Surfaces, single curved, the cyl- inder defined 88 Surface, single curved, devel- opment of 108 Surface, single curved, two con- secutive elements lying in tangent plane to ... . 98 Surfaces, single curved, inter- secting a surface of revo- lution 194 Surface, single curved, to as- sume point on 100 Surface, warded, defined 89 System of generation of a hy- perbolic paraboloid, . . . 145 Tangent, in classification 86 Tangent line to a curve, two curves tangent 94, 95 236 INDEX PAGE Tangent line, point of 96 Tangent line, to line, defined.. 93 Tangent line, the limit of a se- cant 93 Tangent lines, projected as tan- gents 94 Tangent line of surfaces of revolution 171 Tangent plane, to cone and parallel to a line 119 Tangent plane to cone at point on 116 Tangent plane to cone through point outside 117 Tangent plane to conoid 156 Tangent plane to helical con- volute 131 Tangent plane to helical convo- lute parallel to a line... 135 Tangent plane to helical convo- lute, through a point out- side 133 Tangent plane, to cylinder, par- allel to a line 103 Tangent plane to a cylinder at a point on 101 Tangent plane to a cylinder through a point outside. . 102 Tangent plane to double curved surface 165 Tangent plane, contains con- secutive elements of single curved surface ... 98 Tangent plane to hyperboloid of revolution 180 Tangent plane to hyperbolic paraboloid through a line 155 Tangent plane to hyperbolic paraboloid at a point on surface 149, 151 Tangent plane, to a sphere, through a line 170 Tangent plane to surface of revolution at a point on surface 167 Tangent plane, to locate traces of 105 PAGE Tangent planes to warped sur- faces 149 Tangency, point of, defined ... 94 Tangency, approximate point of, a curve and its tangent 96 Theorem 1. Projection of a point 7 Theorem 2. Line, as the trace of its projecting plane. . . 14 Theorem 3. Point of intersec- tion of two lines 16 Theorem 4, 5. Line in plane and line parallel to plane 34 Theorem 6. Traces of parallel planes 39 Theorem 7. Line perpendicu- lar to a plane 52 Theorem 8. Lines perpendicu- lar and one parallel to co- ordinate plane 57 Theorem 9. Projections of a curve 92 Theorem 10. Lines tangent in projection 94 Theorem 11. Plane tangent to single curved surface. ... 98 Theorem 12. Hyperbolic para- boloid projected on plane directer 143 Theorem- 13. Section of hyper- bolic paraboloid 144 Theorem 14. Trace of oblique helicoid an archimedian spiral 163 Theorem 15. Plane tangent to surface of revolution per- pendicular to meridian plane 170 Theorem 16. Surface of rev- olution tangent when having common axis 171 Theorem 17. Hyperboloid of revolution, two systems of generation 176 Theorem 18. Meridian curve of hyperboloid of revolu- tion is a hyperbola 178 Theorem, form of proof to 17 INDEX 237 PAGE Third and first angle discussed 197 Three points, plane through... 35 Torus, in classification 91 Traces of a line 20,21, 22 Traces of a line in different angles 25 Traces of a line with an end plane 23, 24 Trace, corresponding, of the projecting plane of a line 15 Trace of a line with any plane 51 Trace, rule to find, of a line with a plane 108 Trace of a plane 11 Trace of a plane to contain a line 37 Trace, to locate the, of a tan- gent plane 105 Trace of a plane given and angle with that plane to find its corresponding trace 68 Trace of plane given and angle with corresponding plane to find other trace 70 Traces of a plane are not lim- ited 16 Traces of parallel planes with third pliane 39 Traces of plane perpendicular to a line 52 PAGE Transcendental curves, in classification 86 Triangulation, a method of de- velopment 198 Trigonometric curves, in classi- fication 86 Trochoids, classified 86 Trochoids, defined 87 Uniform pitch of helicoid de- fined 158 Variable ellipsoid, in classifica- tion 91 Variable paraboloid, in classifi- cation 91 Variable paraboloid, defined ... 92 Vertex of hyperbolic parabo- loid 154 Vertical of a plane 37 Vertical projecting lines 4 Vertical plane, front, end....l, 3 Vertical projecting line 5 Vertical revolution of supple- mentary, plane 17 Warped surfaces, classified 91 Warped surfaces, defined ...89, 139 Warped surfaces, tangent planes to 149 Windschief lines defined 139 Windschief lines as directrices of a surface of revolution 173 Zone method of development. 198 Short-title Catalogue or THE PUBLICATIONS OF JOHN WILEY & SONS New York London: CHAPMAN & HALL, Limited ARRANGED UNDER SUBJECTS Descriptive circulars sent on application. 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