m Student Wohk— State College of Washington. SHADES AND SHADOWS BY DAVID C. LANGE, M.S. INSTRUCTOR IN ARCHITECTURE IN THE WASHINGTON STATE COLLEGE NEW YORK JOHN WILEY & SONS, Inc. London: CHAPMAN & HALL, Limited 1921 :_3 Copyright, 1921 BY DAVID C. LANGE • PRESS OF «■ eRAUNWORTH & CO. eOOK MANUFACTURERS BROOKLYN, N. Y. PREFACE It has been the endeavor of the author to give in the following pages sufficient information to enable one to cast correctly the Shades and Shadows on any architectural object. The information contained herem was compiled with special attention for its use as a text book on Shades and Shadows. Architectural students in many colleges receive their early training under Engineering teachers. An at- tempt Vv'as therefore made to serve such students, by assuming the point of view of their engineering training at the beginning of the book, and lead them by a study of Shades and Shadows toward an appreciation of the artistic architectural point of view, so seldom developed in a strictly engineering course. The author wishes to acknowledge his indebtedness to the Faculty of the Architectural School of the University of Pennsylvania, where he received his first impressions and appreciation of Architectm-e, and especially to Professor Thomas Nolan for his able assistance in a review of the subject matter contained in the book. Also to those whose work is shown as illustrations. 4G8462 CONTENTS Chapter Page I. Elementary Prinxiples of Descriptive Geometry. Points, Lines, and Planes 1 Intersections of Solids with Planes 27 Intersection of Solids 34 II. Principles of Shades and Shadows. General Principles of Shades and Shadows 48 Shadows of Points 58 Shadows of Lines G6 Shadows of Planes 78 Shades and Shadows of Solids 82 Genera! Methods of Finding Shades and Shadows 90 Wash Rendering 130 NOTE Problems are referred to as plates. No illustrations accompany the following: Plates I to XIII Pages 36 to 44 Plates I and II Pages 63 and 64 Plates III, IV, V Pages 75 and 76 Plates XXVII, XXVIII, XXIX Page 135 SHADES AND SHADOWS CHAPTER I ELEMENTARY PRINCIPLES OF DESCRIPTIVE GEOMETRY POINTS, LINES AND PLANES 1. Descriptive Geometry is the art of representing a body in space upon two planes (called the Horizontal, H, and the Vertical, V, Coordinates, indefinite in extent and intersecting at right angles to each other in a line called the Ground Line), by projecting lines, perpendicular to the coordinates, from points of intersection of the contiguous sides of the body and from points of its contour, and thus solving graphically many geometrical problems involving three dimensions. (Figs. 1, 2, and 3.) NOTATION The'f oUowing notation is used : H Horizontal Coordinate Plane, V Vertical Coordinate Plane, P Profile Coordinate Plane, Q, P, R, S, etc., any other planes. HQ, HP, etc.. Horizontal Traces of any other planes, VQ, VP, etc.. Vertical Traces of any other planes, a, b, c, d, etc., any points in space, a", 6", etc., Vertical Projections of any points. PRINCIPLES OF DESCRIPTIVE GEOMETRY 2nJ Qua Irant Vertical Coordinate Horizontal Coordinate 16" \ch Fig. 3 Ii«. 4 G Fig. 5 POINTS, LINES AND PLANES 3 a!", h^, etc., Horizontal Projections of any points, a^, b'', etc., Profile- Projections of any points, A, B, C, D, etc., any lines in space, A", B", etc.. Horizontal Projections of any lines, A'', B' , etc.. Vertical Projections of any lines, A^', 5", etc., Profile Projections of any lines, GL, . Ground-Line. 2. The intersection of these Coordinate Planes form four angles or quadrants, called the First, Second, Third, and Fourth Angles, or Quadrants. (Fig. 1.) 3. In order to represent the two projections of an object on the same sheet of paper, the upper portion of the Ver- tical Plane is revolved backward about the Ground-Line as an axis, until it coincides with the Horizontal Plane. (Fig. 2.) 4. All points in the First Quadrant are vertically and horizontally projected, respectively above and below the Ground-Line. All points in the Third Quadrant are horizontally and vertically projected, respectively above and below the Ground-Line. All points in the Second Quadrant are projected above the Ground-Line. All points in the Fourth Quadrant are projected below the ^ Ground-Line. (Figs. 2 and 3.) 5. Two projections of a point are always on one and the same straight line, perpendicular to the Ground-Line. (Fig. 3.) 6. Two projections are always necessary definitely to locate a point or line in space. 7. The distance of a point in space above the Horizontal Plane is equal to the distance of the Vertical Projection of the point from the Ground-Line; and the distance of a point in space, in front of the Vertical Plane, is equal to the PRINCIPLES OF DESCRIPTIVE GEOMETRY .vjr I// Fig. 6 V .Fig. 7 1 1 II j 1 ^^ Jrf" Fig. 9 e\ ,f" I Jc'' Fig. 10 Fig. 11 Fig. 12 Fig. 13 Fig. 14 POINTS, LINES AND PLANES 5 distance of the Horizontal Projection of the point from the Ground-Line. (Figs. 2 and 3.) 8. A point situated on either Coordinate Plane is its own projection on that Plane, and its other projection is in the Ground-Line. (Fig. 6.) 9. A straight line is determined by the points at its extremities, and a solid is made up of lines; hence the pro- jections of a sufficient number of points of a line or solid determine their entire projections. (Fig. 7.) 10. Lines parallel in space have their projections par- allel. (Fig. 8.) 11. A straight line perpendicular to either Coordinate Plane has its projection on that plane as a point, and its other projection is perpendicular to the Ground-Line. (Fig. 9.) 12. A line parallel to either plane has its projection on that plane, parallel to itself, and equal to its true length; and its other projection is parallel to the Ground-Line. (Figs. 10 and 11.) 13. A line parallel to both coordinates, or to the Ground- Line, has both projections parallel to the Ground-Line. (Fig. 12.) 14. A point in a line has its projections on the pro- jections of the line. (Fig. 13.) 15. Projecting Planes are planes containing two or more projecting lines. 16. The Traces of a line or plane are the intersections of the line or the plane with the coordinates, and the two traces of the same plane must always meet in the same point in the Ground-Line, although sometimes at infinity, when the plane is parallel to a coordinate. The traces of a line lying in a plane must also be in the traces of that plane. (Figs. 14 and 15.) 6 PRi\riPLP:s OF descriptive ceometry POINTS, LINES AND PLANES 7 17. The Profile Plane is a plane perpendicular to the Ground-Line and hence to the Horizontal and A'ertical Coordinates, taken for convenience on the left side. Pro- jections of objects on the Profile Plane are shown by revolv- ing the Profile Plane about its Vertical Trace into the Vertical Plane. 18. If a straight line is perpendicular to a plane, its projectians are respectively perpendicular to the traces of the plane. For if an assumed plane is perpendicular to the given plane and a Coordinate Plane, and contains the straight line, it is perpendicular to their line of intersection; and any line in this plane is perpendicular to this line of intersection. But this assumed plane is also a Projecting Plane of the given line, and the lines of projection must lie in the traces of this assumed plane. Projections of the lines, therefore, are respectively perpendicular to the traces of the plane. (Fig. 16.) 19. To find the traces of a given line. Let A be the given line. (Figs. 17 and 18.) The traces of a line are the points in which the line pierce^ the Coordinate Planes. (Art. 16.) The projections of these traces must, therefore, lie in the projections of the line, and one projection oi each trace ^lies in the GL. (Arts. 14 and 8.) The other projection lies at the intersection of a line perpendicular to the GL through this point in the GL, and the projection of the line. (Art. 6.) To obtain the H Trace of the line, the line is continued to the H Plane, which is shown by the Vertical Projection intersecting the GL. This is the V Projection of the H Trace; and the H Projection is the intersection of a per- pendicular to the GL at this point with the H Projection of the line. The V Trace is found in the same way. 8 PHIXCIPLE!^ OF DESCRIPTIVE GEOMETRY Fig. 22 POINTS, LINES AND PLANES 9 20. To determine the true length of a hne joining two points in space. Let A be the given Hne. (Figs. 19 and 20.) A line is seen in its true length upon that coordinate to which it is parallel or in which it lies. By revolving the line, therefore, into or parallel to either coordinate, the true length of the line in space is determined. This is done in two ways: (1) By revolving the line parallel to a Coordinate Plane, by means of revolving the Pro- jecting Plane parallel to the opposite coordinate. (2) By i revolving the Projecting Plane about its intersection with i the same Coordinate Plane, as an axis, into that coordinate. ! 21. To pass a plane through two intersecting or par- allel lines. Let A and B be two intersecting lines. (Figs. 21 and 22.) The traces of a plane must contain the traces of the lines. The traces of the given lines, therefore, are deter- 1 mined, and like traces connected. These lines are the traces i of the required plane, and meet in the GL. (Arts. 16 and 19.) A plane may be passed through any three points by !^ passing two intersecting lines through the three points. 22. Given one projection of a line or point lying in a plane, to find the other projection. ' Let a^b'' be the H Projection of the line, and c^ be the ; H Projection of the point, lying in the plane P. (Figs. I 23 and 24.) The traces of the line in the given plane containing the given point must lie in the traces of the plane. The pro- jections of the traces are, therefore, determined, and from them the unknown projection. (Arts. 16 and 19.) 23. Given the projections of a point or line lying in a 10 l'inX(II'Li:S OF DESCRIPTIVE (lEOMETRY POINTS, LINES AND PLANES 11 plane, tc find its j)()8iti()n when the plane is revolved al)()ut its trace to coincide with either coordinate. Let ab be the line and c the point, lying in the plane P, and shown by their projections. (Fig. 25.) The axis about which the point or points of the line revolve must lie in the plane into which the point or points of the line is revolved. The revolving points describe a circle whose plane is perpendicular to the line of inter- section of the given plane with the Coordinate Plane. The intersections of this circle or circles with the coordinate are the required positions of the point or points in the line. Through the given point on the plane a plane is passed, perpendicular to the trace of the given plane, about which trace the given plane is to be revolved. The trace of the Auxiliary Plane is perpendicular to the axis of the revolving plane. On the trace of the Auxiliary Plane a point is laid ofT at a distance from the axis and trace of the given plane equal to the hypotenuse of a right triangle, one side of which is the distance from the projection of the point to the axis on the revolving plane, and the other side of which is equal to the distance of the point from the same projection. 24. To find the true size of an angle made by two inter- secting lines. (Fig. 26.) Let A and B be the tw^o intersecting lines. The angle betv/een intersecting lines may be measured when a plane containing the lines has been revolved to coincide with one of the Coordinate Planes. A plane is, therefore, passed through the two intersecting lines and its traces determined. This plane, with its lines, is then revolved about either of its traces as an axis, until it coin- cides with that Coordinate Plane in which the trace lies. The angle made by the intersecting lines, in the revolved position, is the required angle. (Art. 23.) 12 PRINCIPLES OF DESCRIPTIVE GEOMETRY POINTS, LINES AND PLANES 13 25. To find the true size and shape of any plane surface. Let abed be the plane surface. (Fig. 27.) This plane surface appears in its true size and shape ^vhen the plane containing it is revolved to coincide with a Coordinate Plane. The plane, therefore, containing the given plane surface, is revolved into the coordinate plane about one of its traces as an axis, and the revolved position of the plane surface constructed. This is the true size and shape of the plane surface. (Arts. 23 and 24.) 26. To find the shortest distance from a point to a line, and to draw the projections of it in its position in space. Let A be the line and a the point. (Fig. 28.) The shortest distance from a point to the line is a per- pendicular from the point to the line. A plane, therefore, is passed through the line and point. (Art. 2L) This plane, containing the line and point, is revolved about its trace into the Coordinate Plane. This determines the revolved position of the line and point. (Art. 23.) A perpendicular line is now drawn from the revolved position of the given point to the revolved position of the given line. This is the true length. This perpendicular, revolved back, determines its projections. (r 27. To find the line of intersection of any two planes. Case L To find the intersection of two planes inter- secting within the limits of the drawing. Let X and P be the intersecting planes. (Figs. 29 to 32.) The intersection of the Horizontal Traces must be a point common to both planes, and therefore a point common to their line of intersection. The Vertical Projection of this point lies in the GL. Projections of another point in this line of intersection may be determined in the same way from the intersection of the Vertical Traces. (Art. 8.) 14 PRINCIPLES OF DESCRIPTIVE GEOMETRY POINTS, LINES AND PLANES 15 16 PRINCIPLES OF DESCRIPTIVE GEOMETRY Having given, therefore, the projections of two points in the hne of intersection, the projections of the Une itself are easily determined. (Art. 9.) Case II . To find the intersection of two planes when the traces do not intersect within the limits of the drawing. (Figs. 33 to 37.) A series of Auxiliary Planes are passed parallel to a Coordinate Plane. These Auxiliary Planes cut from the given intersecting planes straight lines which are parallel to the Coordinate Plane, and which intersect in points common to both the given planes, and lie, therefore, in their line of intersection. The following is another proof for the same problem: The line of intersection between the two planes is com- mon to each plane, and the traces of the line of intersection must, therefore, lie in the traces of each plane. (Art. 16.) Hence the point of intersection of the V Traces of the planes is the Vertical Trace of the required line of intersection, with its H Projection in the GL. Another point in the line of intersection is the intersection of the H Traces of the planes, and its Vertical Projection is in the GL. Having then the two projections of the two points in the line of intersection, the projections of the line of intersection itself is easily determined. (Art. 6.) Case III. To find the intersection of two planes when both intersecting planes are parallel to the GL, or when one of the intersecting planes contains the GL. Let P and be the two planes parallel to the GL. (Figs. 36 and 37.) On the Profile Plane the traces of the intersecting planes parallel to the GL are determined. A point common to both traces is a point in the intersection of the two planes. Since the intersecting planes are perpendicular to the Pro- POINTS, LINES AND PLANES 17 c) VP I vo A. A'' \ ^\ HO Fig. 36 18 PEINCIPLEvS OF DESCRIPTIVE GEOiMETRY POINTS, LINES AND PLANES 19 tile Plane, so also must the line of intersection be perpen- dicular. The V and H Projections, therefore, of this line are the projections of the line of intersection of the two planes parallel to the GL. 28. To find where a line pierces a plane. Let P be the given Plane and A the given line. (Figs. 3S to 41.) The given line must intersect the given plane in the line in which any Auxiliary Plane containing the given line intersects the given plane, at a point v>here the given line crosses the line of intersection of the Auxiliary Plane and the given plane. An Auxiliary Plane is, therefore, passed through the line to intersect the given plane. (Art. 16.) The Hne of intersection is then determined between the given plane and the Auxiliary Plane. (Art. 27.) The required point lies on this line of intersection and on the given line, or at the intersection of these two lines. There are several cases, as follows: Case I. WTien any Auxiliary Plane containing the line is used. (Fig. 39.) Case II. When the H or V Projecting Plane is used as the Auxiliary Plane. (Fig. 38.) Case III. ^Yhen the line is parallel to the Profile Plane, necessitating the use of the Profile Plane. (Fig. 40.) Case IV. WTien the plane is defined by two inter- secting lines. (Fig. 41.) 29. To find the shortest distance from a point to a plane. Let P be the given plane and a the given point. (Fig. 42.) The shortest distance must be measured along a line from the point perpendicular to the given plane, and the projections of this line are perpendicular to the traces of 20 I'RINCIPLKS OF DESCRIPTIVE GEOMETRY POINTS, LINES AND PLANES 21 22 1>K]N(1PLE.S OF DESCRIPTIVE GEOMETRY the given plane. (Art. 18.) The point in which this per- pendicuhir j^erces the gi\'cn plane is then determined. (Art. 28.) To find, therefore, the length of the perpendicular, a projecting plane containing the perpendicular line is revolved about its trace into the coordinate, where its true length is shown. 30. To pass a plane through a given point and parallel to a g'ven plane. Let P be the given plane and a the given point. (Fig. 43.) The traces of the required plane are parallel to the cor- responding traces of the given plane, and are fully known when one point in each trace of the required plane is determined. A straight line, therefore, through the given point, and parallel to either trace of the given plane, is a line in the required plane, and intersects a Coordinate Plane in a point in the trace of the required plane. Thus the required plane has its traces through this point and parallel to the traces of the given plane. (Art. 10.) 31. To pass a plane through a given point, perpen- dicular to a given line. Let a be the given point and A the given line. (Fig. 44.) The V and H Traces of the required plane are per- pendicular to the V and H Projections of the given line. (Art. 18.) The direction of each of the required traces is, there- fore, known; and if a straight line is drawn through the point, and ])aral!el to either of these traces, it is a line of the reciuired plane. Unless parallel to the GL, it inter- sects one of th(> i)lanes of projection at a point in the trace 24 PRINCIPLES OF DESCRIPTIVE GEOMETRY of the required plane. (Art. 16.) Therefore, a trace through the point thus found, and perpendicular to the corresponding projection of the given line, is one of the required traces of the required plane. The other trace meets it in the GL, and is perpendicular to the other pro- jection of the line. (Art. 16.) 32. To pass a plane through a given line, parallel to another given line. Let A and B be the given lines. (Fig. 45.) The required plane contains one of the given lines, and a line intersecting this given line which is parallel to the second given line. Through any point, therefore, in the first line, a line is passed parallel to the second given line. A plane con- taining these intersecting lines is the required plane. (Art. 21.) 33. To pass a plane through a given point parallel to two given lines. Let A and B be the given lines, and a the given point. (Fig. 46.) If through the given point lines are passed parallel respectively to the given lines, and a plane passed through these intersecting lines, this plane is the required plane. (Art. 31.) 34. To pass a plane through a given line, perpendicular to a given plane. Let A be the given line and P the given plane. (Fig. 47.) The required plane Q, contains two intersecting lines, namely, the given line, and an intersecting line B, per- pendicular to the given plane. (Arts. 18, 21, and 19.) 35. To construct the projections of the shortest line that can be drawn, terminating in two straight lines not in the same plane. POINTS. LINES AND PLANES 25 26 PRINCIPLES OF DESC^'UPTIVE GEOMETRY is the required angle which can be seen in its true size by revolving it and the plane O containing it about its traces into either coordinate. INTERSECTIONS OF SOLIDS WITH PLANES 27 Let A and B be two straight lines not in the same plane. (Fig. 48.) The shortest distance between two points not in the same plane is the perpendicular distance between them, and only one perpendicular can be drawn terminating in these two lines. Through one of the given lines a plane is passed parallel to the second line. (Art. 32.) The second given line is projected on this plane. (Arts. 12 and 29.) This pro- jection of the second given line on the Auxiliary Plane, intersects the first given line. At their intersection a line E, perpendicular to the Auxiliary Plane, is dra\\Ti. (Art. 18.) It intersects the second given line because it is a Projecting Line, and is the shortest distance between the lines. (Art. 26.) 36. To find the angle made by any two intersecting planes. Let P and R be two intersecting planes. (Fig. 49.) A plane is passed which is perpendicular to the line of intersection of the two planes. It is perpendicular to both 'of the intersecting planes, and cuts from each a line per- pendicular to the line of intersection. (Art. 18.) The I angle made by these lines cut out by the perpendicular i;^lane is the required angle, and can be seen in its true size Iby revolving the plane containing it, about its trace, into jpne of the Coordinates. (Art. 24.) INTERSECTIONS OF SOLIDS WITH PLANES 36. A solid is a magnitude that has length, breadth, and |thickness, as a Cylinder, Cone, or Sphere. A Cylinder is a solid generated by a straight line called the Generatrix, moving with all its positions parallel along curved lines called the Directrix, the two curved lines lie 28 PRINCIPLES OF DESCRIPTIVE GEOMETRY d" «"<•" b" Fig. 50 ab is an element Fig. 52 Fig. 53 INTERSECTIONS OF SOLIDS WITH PLANES 29 in planes forming the remainder of the boundary. The different positions of the Generatrix are called the Elements. (Fig. 50.) A Right Cylinder is one whose bases are parallel planes perpendicular to its axis; and a Right Section is a circular section cut out by a plane perpendicular to its axis. (Fig. 51 .) An Oblique Cylinder is one whose bases are parallel planes which are not perpendicular to its axis. A Cone is a solid generated in a manner similar to that of a Cylinder, except that its elements pass through a fixed point called the Vertex. (Fig. 52.) A Right Cone is one whose base is a plane, perpendicular to the axis of the Cone. (Fig. 53.) A Polyhedron is a solid bounded by plane surfaces. (Fig. 54.) The surface of a Cylinder or Cone are called single- i curved surfaces of revolution. I The surfaces of a Sphere, Ellipsoid, Torus, etc., arc ' called double-curved surfaces of revolution. They are generated by a curved line revolving about another line, called an Axis. The generating curved line is called the Meridian Line and the curved surface it generates, a Merid- :^an Plane. 38. To find the intersection of a solid with a Secant Plane. Let P be the Secant Plane intersecting different solids. (Figs. 54 to 57.) Auxiliary Planes are passed through the solid and the given plane. They cut straight lines from the plane and straight or curved lines from the solid. The intersection of one of these lines cut from the plane with the corre- sponding line cut from the solid by an Auxiliary Plane is a point of the required line of intersection. 30 PRINCIPLES OF DESCRIPTIVE GEOMETRY INTERSECTIONS OF SOLIDS WITH PLANES 31 32 PRINCIPLES OF DESCRIPTIVE GEOMETRY While the AuxiHary Planes may be taken in any position, yet for simplicity they should be chosen so as to cut the simplest line or curve from the solid. In the case of a solid having rectilinear elements, as a Cylinder, Cone, Prism, Pyramid, etc., the proceeding described is prac- tically the same as determining where a certain number of elements or edges pierce the Secant Plane, since the inter- section of the Auxiliary Plane by these elements determines the lines cut out from the solid. (Fig. 54.) The true size of the section cut from the solid can always be found by revolving it about the trace of the Secant Plane into, or parallel to, a Coordinate Plane. (Art. 25.) 39. To find the intersection of a Cylinder with an oblique plane. Let a Cylinder be cut by the oblique plane P. (Fig. 55.) Auxiliary Planes are assumed which cut elements from the Cylinder. These plane 3 are parallel to the axis of the Cylinder, and in order to cut the simplest lines they should be perpendicular to one of the coordinates, as U, T, S, R, etc. Each Auxiliary Plane cuts two elements from the Cylinder, and a straight line from the Secant Plane. The intersections of these elements with the straight line cut from the Secant Plane are two points on the required curve of intersection, as a, b, c, d, etc. 40. To find the intersection of a Cone with an obHque plane. Let a Cone be cut by an oblique plane P. (Fig. 56.) Auxiliary Planes arc passed through the vertex of the Cone and p(M-pendicular to a coordinate, as R, S, T, etc. These Auxiliary Planes cut elements from the Cone which inter- sect the lines cut from the given plane by the Auxiliary Planes, as A, B, C, D. These intersect m the points a, b, c, and d in the curve of intersection. The additional INTERSECTIONS OF SOLIDS WITH PLANES 33 34 PRINCIPLES OF DESCRIPTIVE GEOMETRY points needed to determine the intersection are determined in the same way. 41. To determine the intersection of a double-curved surface of revolution with a plane. Let the double-curved surface of revolution be cut by the plane P. (Fig. 57.) Auxiliary Planes, perpendicular to the axis, are passed through the solid. These planes cut circles from the double-curved surface of revolution, and straight lines from the given Plane, as R, S, X, Y, Z. The intersection of these will locate points on the curve of intersection. Other points of the curve of intersection are determined in the same way. The points a, h, c, d, e,f, g, etc., determine the intersection of the double-curved surface of revolu- tion with the given plane. 42. To find the intersection of a Polyhedron with any oblique plane. Let the Polyhedron be intersected by the oblique plane P. (Fig. 54.) The intersection of the given plane with each of the bounding planes of the polyhedron is found (Art. 27), and this determines the required intersection. INTERSECTIONS OF SOLIDS 43. To find the intersection of any two solids. Auxiliary Planes are passed through the two solids. These Auxiliary Planes cut lines, either straight or curved, from each solid, and the intersection of these lines with each other determine points on the required line of intersection. (Fig. 58.) The lines cut from the solids by the Auxiliary Planes are C, D, etc., and 1, 2, 3, 4, 5, etc., are the points on the required line of intersection which are determined by the intersection of these lines. 36 PRINCIPLES OF DESCRIPTIVE GE(3METRY PROBLEMS All problems are to be worked out with accuracy and the draftsmanship is to be of the highest quality, as the plates are exercises in drawing as well as solutions of prob- lems in Shades and Shadows. The sheets are to be 15 by 22 inches in size and are to have a margin f inch in width. All plates are to be numbered at the middle of the top of the sheet, and each is to have the date in the lower left- hand corner and the name in the lower right-hand corner. Special attention should be paid to the lettering and notation. PLATE I Draw the projections of a point in each of the follow- ing positions: 1. In the First Angle. 2. In the V Plane, between the First and Second Quad- rants. 3. In the Second Angle, equally distant from the H and V Planes. 4. In the H Plane, betvveen the Second and Third Angles. 5. In the H Plane, in front of the V Plane. 6. In the Third Angle. 7. In the Fourth Angle. 8. In the GL. 9. In a Profile Plane. 10. In the First Quadrant, 2 inches from H and 1 inch from V. PROBLEMS 37 PLATE II Draw the projections of a point in each of the follow- ing positions : 1. In the Third Angle, 1 inch from H and l\ inch from V. 2. In T", and | inch below H. 3. In H, and 2| inches back of V. 4. In the Fourth Angle, 1| inches from H and \\ inches from V. 5. In the Second Quadrant, If inches from H and 2 inches from V. 6. In the Profile Plane, equidistant from H and V. 7. In the Profile Plane, and in the Second Quadrant. 8. Find the projection of any point whose projecting lines pass through a point 1 inch from the GL. 9. In the T^ Plane, and 1 inch from the H Plane. 10. In the GL lies one projection of a point. The point itself is 1 inch from the GL. Locate the point in space. PLATE III Draw the projections of a line in each of the following positions : 1 . In the First Angle, parallel to V and H. 2. In the Second Angle, parallel to T^ and oblique to H. 3. In H, behind V and inclined towards V. 4. In the Third Angle, perpendicular to H. 5. In the Third Angle, oblique to V and H. 6. In the First Angle, oblique to H and T^ and in a plane perpendicular to the GL. 7. In a plane bisecting the Fourth Angle. 8. Draw the projections of two intersecting lines in the Third Angle. 38 PRIXf'IPLES OF DESCRIPTIVE GEOMETRY 9. Draw the projections of two parallel lines, one in the First Angle and one in the Second Angle. 10. Draw the projections of a point in the line in the sixth position. PLATE IV Show the three projections of a point: 1. Two and one-half inches to the right of P, 1 inch in front of V , and 2 inches above H. 2. Two inches to the right of P, 2\ inches behind V, and 2 inches above H. 3. One-half inch from P, l\ inches in front of V, and 2 inches below H. 4. Three inches to the right of P, 2 inches behind V, and in H. 5. In P, H, and 7. 6. In the first Angle, and 1 inch from P, H, and V. 7. In P, and equidistant from H and V. 8. In the GL, and 2 inches from P. 9. In P, between the First and Second Angles. 10. In P and V, and 1 inch from H. PLATE V Find the projections of the lines in the following posi- tions: , 1. Inclined to V and H, in the Third Angle. 2. Parallel to P, inclined to H and V, and in the First Angle. 3. Inclined to V and H, in the First Angle. 4. Inclined to H, and parallel to V, in the Third Angle. 5. Perpendicular to V, and in the Third Angle. 0. Parallel to the GL, in the Fourth Angle. PROBLEMS 39 7. Inclined to H, and laying in V, between the Third and Fourth Quadrants. 8. Intersecting lines in the Third Angle, one parallel to H, and inclined to V, and one parallel to the GL. PLATE VI 1. Draw the projections of a line which intersects the GL at a point 3 inches to the right of P, and pierces P at a point If inches from H and 2\ inches from V. 2. The point a is 2 inches from H, 1 inch behind V, and \ inch to the right of P. The point 6 is 2 inches to ' the left of P, 1 inch in front of V, and lies in H. Draw the projections of a line through these points. 3. Draw the projections of a line passing through the First, Second, and Third Quadrants. 1 4. Draw two lines intersecting in the Second Quadrant, at a point 2 inches from H and V. 5. Draw the projection of any line passing through a point in H, and 2 inches behind V. Intersect this line at 1 this point with a line passing through the Second, Third, aiid Fourth Angles. 6. In any oblique line, find a point which is equidistant if from H and V. 7. The point a, of the triangle ohc, is 3 inches to the ■ right of P, 4 inches above H, and 5 inches in front of V. ' The point 6 is 7 inches to the right of P, 3 inches behind F, and 4 inches belo^v H. The point c is 2 inches below H. \ Draw the projections of the triangle ahc. PLATE Vn i 1. Draw the projections of a plane which is perpen- ' dicular to H, and makes an angle of 45° with V. ' 2. Given a plane which makes an angle of 30° with 40 PRINCIPLES OF DESCRIPTIVE GEOMETRY H, and whose H Trace is parallel to the GL, and 2 inches back of V . Draw the traces. 3. Draw the traces of a plane parallel to the plane, in VII, 2. 4. Given a line which is oblique to the GL, whose H Trace is back of V, and whose V Trace is above H. Draw the projections of a line in this position. 5. Given a plane which is parallel to the GL. Draw the projections of a point in this plane. 6. Draw an oblique plane and show its traces in all quadrants. 7. Draw a plane, perpendicular to H, but oblique to the other planes of projection. 8. Draw a plane, perpendicular to P, but oblique to H and F, and passing through the First and Fourth Quad- rants. 9. Draw the traces of a plane perpendicular to P, and parallel to F. 10. Draw the projection of lines in each Quadrant, and find its traces in the H, V, and P planes. PLATE VIII 1. Find the true length of a line ah, located in space as follows: One end of the line a, is 1 inch from H, and \ inch from T'. The other end 6 is 2 inches from H, and f inch from V. The line itself is 1^ inches long. 2. Pass a plane through tv\'o intersecting lines whose intersection is 1 inch from H, and 1^ inches from V. 3. Pass a plane through two parallel lines as follows: The line ah has a point a, \ inch from V, and f inch from //. The point 6, is 1 inch from H, and 1 inch from V. The line cd has the point c, f inch from 7, and 1§ inches PROBLEMS 41 from H. The point c? is 1| inches from H, and If inches from V. PLATE IX 1. Given a plane whose traces make an angle of 45° with the GL, and the H Projection of a point which lies in the plane and is 1 inch from the H Trace of the given plane, and 1 inch from the GL. Find the other projection of the point, and show its position when it is revolved into one of the coordinates. 2. Consider the triangular plane surface in Plate VI, 7, and find its true size and shape. 3. The point a, in space is 2 inches from H and F, and \ inch from P. The line ch has the point c at one end, 1 inch from P, and § inch from H and V. Its other end 6 is 2 inches from P, and 3 inches from H and 7, and it is 4 inches long. Find the shortest distance from the point to the line and draw its projections in space. 4. Two planes P and R have their traces 3 inches apart in the GL. The trace of P make an angle of 45°, the H Trace of R an angle of 30°, and the V Trace an angle of 60° with the GL. Find the line of intersection between ^ the two planes which intersect. 5. The traces of a plane S are parallel to the GL, and 2 inches from it. The H Trace of a plane T is parallel to the GL and 1 inch from it. Its V Trace is 3 inches from it and parallel to it. Find the line of intersection between the t\To planes. 6. The traces of the plane P make an angle of 45° \vith the GL. Consider a line in space which does not intersect the given plane, and determine where it intersects the plane. 7. A point a is 3 inches from H, 21 inches from V, and 42 PRINCIPLES OF DESCRIPTIVE GEOMETRY I inch from P. The H Trace of a plane makes an angle of 45°, and the Y Trace an angle of 60°, with the GL. The point of intersection of the traces with the GL is | inch from P. Find the shortest distance from the point to the plane and draw its projections. PLATE X 1. The point a is 2| inches from P, | inch from H, and f inch from V. Pass a plane through this point and parallel to another plane whose traces make angles of 60° and 30°, respectively, with H and V. 2. In the preceding problem pass a plane perpendicular to the given plane. 3. A point a is 2 inches from P, 1 inch from V, and f inch from H. A line he has one end h, J inch from P, I inch from V, 1 inch from H, and is 2 inches long. The other end, c is 1 inch from V and 1^ inches from H. Another line de has one end f inch from P, 2 inches from V, and 1| inches from H, and is 2| inches long. The other end, e, is I inch from V and H. Pass a plane through the point and parallel to the two lines, 4. A line ab has its end a, \ inch from P, 1 inch from V, and H inch from H. Its other end, h, is If inches from V, and 2 inches from H. The line is 2| inches long. Another line cd has its end c f inch from P, 3 inches from H, 1 inch from V, and is 2| inches long. Its other end d is 1^ inches from V and If inches from H. Construct the projection of the shortest line that can ba drawn termi- nating in these two given lines. 5. The traces of a plane R make an angle of 45° with H and 30° with 7, and meet in a point in the GL, \ inch from P. The traces of another plane S make an angle of 60° with // and 45° with F, and meet in the GL, 3 inches PROBLEMS 43 from P. Find the true size of the angle between the planes. PLATE XI 1. A Right Cylinder, 2 inches in diameter, rests on H", and has its 4-inch axis 3 inches from P and 2\ inches from V. This Cylinder is intersected by a plane whose traces make an angle of 30° with H and 45° with V, and intersect in a point 4 inches from the GL. Determine the true size of the section cut out. 2. Replace the Cylinder in the above problem with a Cone of the same size and determine the true size of the section cut out. PLATE XII j 1. A Sphere, 2| inches in diameter, has its center 1§ • inches from H, V, and P. It is intersected by a plane i whose traces make angles of 30° and 45°, respectively, with , H and V, and meet in the GL at a point 2\ inches from P. Determine the line of intersection cut from the solid by the plane. 2. A Cone with a 2-inch base rests on H, with its center t;l| inches from V and P. Its axis is 3 inches long and makes an angle of 45° wdth the H and V Planes, sloping up f(/rward, to the right {ufr). Find the true size of a section cut out by a plane making with its H Trace an angle of 45°, and with its V Trace an angle of 30° with the GL, when its traces meet in the GL, 3| inches from P. PLATE XIII 1. Find the lines of intersection between two solids, as follows: A Cylinder, 2\ inches in diameter, with its base resting on H, its center 4 inches from H and Ij inches 44 PRINCIPLES OF DESCRIPTIVE GEOMETRY from P, has a 5-inch axis which slopes up back, to the right (ubr). A cone, with a 3-inch base resting on H, its center 4 inches from H and 5 inches from P, has its 5-inch axis sloping up back, to the left {uhl). The axis of the Cylinder makes, in V Projection, an angle of 60°, and in H Projection, an angle of 30° with the GL. The axis of the cone makes, in V Projection, an angle of 45°, and in H Projection, an angle of 45° with the GL. QUESTIONS IN DESCRIPTIVE GEOMETRY 45 QUESTIONS IN DESCRIPTIVE GEOMETRY 1. Define Descriptive Geometry. What are Coordinate Planes? Projecting Lines? Projecting Planes? Quad- rants or Angles? 2. What is done in order that two projections of an object may be shown on the same sheet of paper? What is the Ground-Line? What angles does the Horizontal Plane make with the Vertical Plane? 3. How are all points in the First Quadrant projected? In the Second Angle? In the Third and Fourth Quad- j rants? How are straight lines represented? How are I their projections determined? 4. How are the projections of a point shown with refer- ence to the Ground-Line? How many projections are necessary to locate definitely a point in space? Name the different projections a point may have. 5. What does the distance of the Vertical Projection from the Ground-Line show? How are the projections of a point lying in one of the Coordinates shown? 6. When the projections of two lines are parallel, define the position of the lines to each other in space? If a straight ^ine is perpendicular to either Coordinate Plane, what are the positions of its projections? 7. If a line is parallel to either coordinate, how will its projections appear? If the projections of a line are parallel to the Ground-Line, how is the position of the line in space determined with reference to the Ground-Line? 8. If a point is on a line, what are the positions of its projections with reference to the projections of the line? Wliat can be said of the projections of the intersection of any two lines in space? 9. Can the intersection which a line makes with the 46 PRINCIPLES OF DESCRIPTIVE GEOMETRY Coordinate Planes determine the direction of a line in space? 10. Define the Traces of a Plane. Name the different traces which a plane may have. Where do the traces of any plane meet? Explain why they meet where they do. 1 1 . Define Profile-Plane. Describe the traces of a plane parallel to either coordinate. How is a plane containing the Ground-Line determined? If a straight line is per- pendicular to a plane, what are the positions of its pro- jections in relation to the traces of the plane? 12. Explain the notation used in this book. If a plane is parallel to the Ground-Line, what are the positions of its traces? If only one trace is shown on the Vertical and Horizontal Coordinates, and if this is parallel to the GL, what is the position of the plane? 13. if a plane is perpendicular to either coordinate, what are the positions of its traces on the opposite coordi- nate. Where do the traces of a plane stop? Given the traces of a line, how are its projections determined? 14. Explain how the line of intersection of any two planes is determined. 15. Make the drawings and give the proof for finding the shortest distance from a given point to a line. 16. How is the angle which a given line makes with any oblique plane determined? How is the true size of any plane surface determined? 17. Given the projections of a point or line lying in a plane, determine the position of the point or line by draw- ing and explanation, when the plane is revolved to coincide with either coordinate. 18. Make' the drawing and give the explanation for finding the true size of an angle made by two intersecting lines in space. QUESTIONS IN DESCRIPTIVE GEOMETRY 47 19. Explain the relation of the traces of a line lying in a plane to the traces of the plane. Explain the method of passing a plane through three points in space. How many planes can be passed through a line in space? 20. Make the drawings for finding the angle which a given line makes with a given plane; a given plane with the planes of projections? 21. Make the drawings and give the explanation for finding the line of intersection of a plane with a polyhedron. 22. How is the intersection of any oblique plane with any solid determined? Of any solid with any other solid? Give the full explanation. 23. How is the. true size of a section, cut out by any oblique plane from any solid, determined? CHAPTER II I. GENERAL PRINCIPLES OF SHADES AND SHADOWS 1. Objects are visible to the eye because of the reflec- tions from their surfaces of Rays of Light of varying degrees of intensity. These reflected rays impinge on the retina of the eye, and result in visual sensation and perception of the illuminated object. The principal source of light is the sun, and its Rays of Light fall on objects at various angles. If the angle of incidence of all the Rays imping- ing on a surface are the same, the surface appears flat and IS a plane surface; while if the angles of incidence are dif- ferent, the surface is not a plane surface. 2. The importance of the subject of Shades and Shadows should be fully realized, especially by students of Archi- tecture. It is a matter of surprise that few students, even after finishmg the usual course of study, are able to deter- mine accurately the Shades and Shadows on architectural drawings. They often resort instead to mere copying to the use of Shades and Shadows merely guessed at, and to a lavish rendering to cover mistakes. The beauty and impressiveness of a drawing is more truthfully enhanced by the accurate casting of its shadows than by its rendering 1 he casting of shadows on architectural drawings is a part of architecture as a Fine Art, and an important part of it« function IS to give a detail its true shape and show its true character. r^r^J^'- ^T'""""" "^ """"'""^ "'''^'^""S *<' funda- mental principles cannot be impressed to ostrongly upon GENERAL PRINCIPLES 49 ?^r! Fig. 1. — An End Pavilion Student Work— Wm. Hough. University of Penna. 50 PRINCIPLES OP SHADES AND SHADOWS IMHH HHUBv AktsMSss^r-:, m Jl||(^=^ >- '.' ^^^^^^^K ' ,. m i 4 Fig .2. GENERAL PHLN'CIPLES 51 the student's mind, for when these are once mastered the rest of the work is relatively easy. The theory may be taught in a very short time, and when understood, enables one to solve easily any problem in the subject. It is, how- ever, necessary, in order to become thoroughly familiar with the character of the Shades and Shadows on various common architectural forms and shapes, that certain prob- lems be accurately solved. 4. The subject of Shades and Shadows is an appli- cation of Descriptive Geometry, and its purpose is to give a more realistic appearance to the representation of objects. 5. Architectural drawings may be divided into two general classes. Working Drawings and Rendered Draw- ings. It is by the use of Shades and Shadows in rendered drawings that a truthful and realistic representation is made of an object, which would otherwise appear flat. All elevations and plans (Coordinate Planes) are conven- tional, and so are Shades and Shadows. They should not be made to produce the effect of perspective when the drawing itself is not in perspective. 6. The casting of • Shades and Shadows enables one more readily to determine the shapes of various details. They give mass and proportion to objects, and with their aid a more intimate understanding of architectural com- positions is possible. Their use is important, both in the rendering of the drawing and in the study of the design. The designer models his building with the aid of Shades and Shadows, and gives it texture, color, relief, and pro- portion, so that cornices, colonnades, windows, and all architectural details, can be readily comprehended as they appear in reality. (Figs. 1 and 2.) 7. ShadoAvs greatly influence design and the character of a building. This becomes evident from a study of the 52 PRINCIPLES OF SHADES AND SHADOWS iM.i. 3. Fig. 4. GENERAL PRINCIPLES 53 developments of architectural styles in different latitudes. Greek architecture with its simple lines could not have reached its perfection in northern climates, and Gothic architecture, which was developed under the low-lying sun, could not have come into such prominence under the bril- liant southern sky. (Figs. 3 and 4.) 8. The beauty of massing and proportion, of the mold- ings and other details of a building, is greatly enhanced by contrasts of light and shade. Profiles of moldings, proportion, and massing, are of minor importance in themselves, if not considered in con- nection with Shades and Shadows. (Figs. 1 and 2.) 9. Since the sun is the assumed source of light, con- sidered at an indefinite distance away, all its Rays of Light are parallel. 10. A further conventional direction, also, for the Rays of Light which has been agreed upon, is the average of the different directions of the sun's rays. This Conventional Direction for the Rays of Light is assumed to be parallel to the diagonal of a cube, sloping from the upper, front, left- hand corner to the lower, back, right-hand corner, when the faces of the cube are parallel or perpendicular to the vertical or horizontal coordinates. (Figs. 5 and 6.) 11. The true angle \vhich the Conventional Ray of Light makes with the Coordinate Planes is 35°15' 22". The projections of the same Ray make angles of 45° with the GL, or Coordinate Planes. CFigs. 6 and 7.) 127 A Plane of Light is any plane containing a Ray of Light. (Fig. 8.) 13. Light falling upon any object may be Direct, Indirect, Diffused, or Artificial. Direct Rays of Light are those Rays which fall directly on any object. (Fig. 9.) 54 PRINCIPLES OF SHADES AND SHADOWS Fig. 8 GENERAL PRINCIPLES 55 Indirect Rays of Light are those Rays which are reflected back on an object from some other object. This is called, also, Reflected Light. (Fig. 1, a and b.) Diffused Light is that light which is widely scattered, and has its rays spread out in all directions from the sources of light. (Fig. 10, page 129.) Artificial Light is light produced by artificial illumi- nation as the source of light. (Fig. 11, page 129.) Direct and Indirect Rays of Light are nearly always used in rendering architectural drawings. Diffused Light is often used in rendering interior perspectives, while Arti- ficial Light is seldom employed in rendering. In this Chapter Direct Rays are the only Rays of Light considered. 14. Shade is that portion of the surface of a body which is turned away from the source of light, and which there- fore does not receive any of the Rays of Light. (Fig. 9.) 15. Shadow is that portion of any surface of an object from which light is obstructed by means of another object placed between it and the source of light. (Fig. 9.) IG. The Shade-Line is the boundary line of the Shade, and the Shadow of the Shade-Line of any object always determines and bounds the Shadow of the object casting the Shadow. From the Shadow of an object its Shade- Line can always be determined. (Fig. 9.) 17. The Umbra is that portion of space from which light is excluded. The Umbra of a point is a line. The Umbra of a straight line is a straight line or a plane; while the Umbra of a plane is a plane or a solid. (Fig. 9.) 18. Real Shades and Shadows are those which have actual existence. Imaginary Shades and Shadows are those which have no real existence, and which are used solely for the purpose of determining the real Shades and Shadows. Invisible Shades and Shadows are those which have actual 56 PRINCIPLES OF SHADES AND SHADOWS 1 Fie. 9 Shadow line Umbra Shade .Sliade Line Invisible shade Invisible shade line GENERAL PRINCIPLES 57 existence but cannot be seen in projection, and which are therefore dotted in on a drawing. (Figs. 9 and 13.) 19. The Shadow of any object is found by passing Rays of Light through the object, and determining where the Rays intersecting the object casting the Shadow inter- sect the surface of the object on which the shadow is cast. (Fig. 9.) 20. The subject of Shades and Shadows of objects may be conveniently divided into (1) Shadows of Points, (2) Shadows of Lines, (3) Shadows of Solids, and (4) Special Methods for Determining Shades and Shadows, c^ 21. Attention is again called to the fact that the Y Projection of an object represents the elevation of the same object, the H Projection the plan, and the P Pro- jection the side elevation or section of the same object. 22. The following additional notation to that given in Chapter I is used in this Chapter II : ct% b^% etc., the H Projections of the shadows of points in space on any surface, a", h'% etc., the V Projections of the shadows of points in space on any surface. R, a Ray of Light. R'', the H Projection of a Ray of Light. /?", the V Projection of a Ray of Light. R^, the P Projection of a Ray of Light. 58 SHADOWS OF POINTS n. SHADOWS OF POINTS 23. To find the shadow of any given point in space on any surface or plane. Let a be any point in space. (Figs. 13 to 19.) The point in which a Ray of Light, passed through a point in space, pierces the surface receiving the shadow, is the shadow of the point. The piercing-point of this Ray of Light with the plane receiving the shadow is the point in which a line having the direction of the Ray of Light, through the point, pierces the plane receiving the shadow. (Chap. I, Arts. 14 and 28.) Case I. When the shadow of a given point falls on the Vertical Coordinate, or is shown on the elevation of an object. (Fig. 12.) Let a be any given point, shown by its projections a* and a\ To find the shadow of a. The point in which a Ray of Light, passed through the point a (shown by the projections of the Ray of Light, B!" and R% through the projections of the point, a!" and a') pierces the V Plane, is the shadow of a on that plane; and it is graphically indicated by the point in which a per- pendicular, erected at the intersection of the H Projection of the Ray of Light with the GL, intersects the V Projection of the Ray of Light. Case II. When the shadow of a given point falls on the Horizontal Coordinate, or is shown on the plan of an object. (Fig. 13.) This case is the same as the preced ng one, except that the Ray of Light pierces the H Plane instead of the V Plane, and therefore the H Projection of the shadow of a is the point in which a perpendicular erected at the intersection of the V Projection of the Ray of Light PRINCIPLES OF SHADES AND SHADOWS 59 Fig. 12 Fig. 13 Fig. 14 Fig. 15 60 SHADOWS OF POINTS with the GL, intersects the H Projection of the Ray of Light. Case III. When the shadow of any point falls on the GL, or on both coordinates. (Fig. 14.) In this case the projections of the Rays of Light through the projections of the point, intersect the GL at the same point. This point is the projections of the shadow of the point. Case IV, To find the Imaginary Shadow of any point behind or below one Coordinate when it is on the other Coordinate Plane. (Fig. 15.) This is the problem of finding the point in which the Ray of Light through the given point, passed through the Coordinate Plane it first strikes, as though this plane were transparent, pierces the other Coordinate Plane, behind or below the first Coordinate Plane. This determines the position of the Imaginary Shadow of the point on that Coordinate Plane. This is shown graphically by continuing that projection of the Ray of Light (which first intersects the GL) through the like projection of the point, and determining the point in which the other projection of the Ray, through the other projection of the point, intersects the GL. The intersec- tion of a perpendicular, erected at the last mentioned point, with the first mentioned projection of the Ray of Light, is the required projection of the Imaginary Shadow of the given point. Case V. To find the shadow of a point, having given one Coordinate Projection, either plan or elevation, and the Profile Projection. (Figs. 16, 17, and 18.) The same proof applies to this case as to the preceding cases. The graphical method of finding the projections of the shadow of the given point is as follows : i PHIXCIPLES OF SHADES AND SHADOWS ()1 Fig. 16 ^ \.a> Fig. 17 Fig. 19 . IQ" Fig. 18 62 SHADOWS OF POINTS The two projections of the Ray of Light, R^ and R or R', are passed through the projections of the point. The projection of the shadow of the point is the point in which a Hne, erected at the intersection of the Profile Projection of the Ray with the H or V Coordinate, and perpendicular to the Profile Trace, intersects the other projection of the Ray, or intersects a perpendicular erected at the point in which the other projection of the Ray inter- sects the GL. Case VI . When the shadow of a given point falls on any oblique surface or plane. (Fig. 19.) Let a be a point in space and P any oblique plane on which the shadow of a falls. This problem is solved in the same way as the preced- ing problems in Cases I to V. The graphical solution is the same as that employed to determine the point of intersection of a line, representing a Ray of Light, with any plane. (Chap. I, Art. 28.) It is well to note here that the Ground-Line can be separated into two Ground-Lines, without affecting in any way the resulting shadows. In fact these are the usual conditions with which one has to contend in the practical casting of Shades and Shadows in architectural work, since the elevation is often on one sheet of paper, the plan on another, and the section on a third. Axiom I. A point in a plane is its own shadow on that plane. l\ PROBLEMS 63 PROBLEMS The same note applies to the problems in the following plates as to the problems in the plates in Chapter I, Descrip- tive Geometry. (See page 36.) PLATE I 1. Draw the true direction of the Ray of Light in a 1^-inch Cube. 2. Find graphically the true angle which the Ray of Light makes with either coordinate. Determine the shadows of the following points: 3. One inch above the H Plane and Ij inches in front of the V Plane. 4. Two inches above the H Plane and 1 inch in front of the V Plane. 5. One inch above the H Plane and 2 inches in front of the V Plane. 6. One and one-half inches above the H Plane and If inches in front of the V Plane. Determine the Visible and Imaginary Shadows on the coordinates of the following points : 7. One and one-half inches above the H Plane and 1 inch in front of the V Plane. 8. One and one-quarter inches above the H Plane and If inches in front of the V Plane. 9. Lying in the H Plane and 1 inch in front of the V Plane. 10. Lying in the 7 Plane and \\ inches above the H Plane. 64 SHADOWS OF POINTS PLATE II Making use of the Profile and Horizontal Projections, determine the shadows of the following points : 1. One inch from H and | inch from V. 2. One inch from H and 1| inches from V. 3. One and one-quarter inches from H and 1| inches from V. Making use of the Profile and Vertical Projections, determine the shadows of the following points : 4. One and one-eighth inches from H and f inch from V. 5. Three-quarter inch from H and 1| inches from V. 6. Find the shadow of a point on any line shown by its H and V Projections. 7. Find the shadow of a point on any oblique plane. 8. Find the shadow of a point 1 inch from H and V, on a. plane which is § inch to its right and perpendicular to V and H. 9. Find the shadow of a point on a plane whose traces make an angle of 45° with both coordinates, when the point is 1 inch from H, 1^ inches from V, and f inch from the oblique plane. 10. Find the shadow of a point lying in the Ground- Line paying special attention to the notation. QUESTIONS ON SHADES AND SHADOWS 65 QUESTIONS ON SHADES AND SHADOWS 1. What causes objects to be visible? Of what impor- tance is the Study of Shades and Shadows in connection with architectural drawings? 2. What and where is the assumed source of light? 3. \Miat is the conventional direction for the Rays of Light, and why so determined? 4. What angle does the conventional Ray of Light make with the coordinates? What angle do the pro- jections of the Rays of Light make with the coordinates? 5. Describe the difference between a Ray of Light and the projections of a Ray of Light. 6. \\Tiat is a Plane of Light? 7. Into what kinds of Light can Light be divided? Define each kind. 8. What is Shade on an object? 9. WTiat is Shadow on an object? 10. Define Shade-Line, Umbra, Real Shades and Shadows, Imaginary Shades and Shadows, and Invisible Shades and Shadows. IL How is the Shadow of any object determined? 12. Into what four subdivisions may the subject of Shades and Shadows of objects be conveniently divided? 13. Write all the notation used in discussing the problems of Shades and Shadows. 14. Explain how the shadow of any given point in space on any surface or plane may be determined. 15. Name the different cases for the finding of Shadows of Points on Planes. 16. State Axiom I. 66 SHADOWS OF LINES III. SHADOWS OF LINES 24. To find the shadow of a given straight line on a given surface or plane. Let ab be any straight line in space. (Figs. 20 to 26.) A straight line is composed of an infinite number of points. If, therefore, the shadows of enough points are located, the shadow of the line can easily be determined. Generally it is only necessary to cast the shadows of the points at the extremities of the line, or of the points where the shadow of the line changes^direction. If these shadows are connected by straight lines, these lines will be the required shadow-lines. (Figs. 20 to 26, Art. 23.) The shadow of any straight line, also, may be obtained by finding the intersection of a plane of light, containing the given line, with the surface receiving the shadow. This intersection contains the required shadow. (Figs. 21 and 22.) Case I. When the shadow of a straight line falls on the Vertical Coordinate or elevation. Let a'b" and a''b'' be the projections of a line when its shadow falls on the V Plane. Rays of Light, shown by their projections Ri" and Ri^, and Rz" and R2', are passed through the points at the extremities a and h of the line. These Rays of Light determine the shadows of the two points, a and b. Like projections of the points are con- nected, and this determines the projections of the shadow of the line. (Fig. 20.) Case II. When the shadow of a straight line falls on the Horizontal Coordinate or plan. This case is the same as Case I, except that the shadow falls on the // Plane instead of the V Plane. (Fig. 23.) Case III. When the shadow of any straight line falls PHIXCIPI.KS OF SHADES AM) SHADOWS 07 1 1 II Mu \A Fig. 20 ^/^ 1 G j I Fig. 22 68 SHADOWS OF LINES Fig. 24 p. \ \ 6"^ Fig. 26 Section Elevation PRINCIPLES OF SHADES AND SHADOWS ()9 on both coordinates, or on the plan and elevation, or on any oblique plane. Let ab be the given straight line whose shadow falls on both the coordinates. (Figs. 24 and 22.) Imaginary Shadows are employed in this Case. The shadow of the entire line is determined on the H Plane, in front of and behind the V Plane. (Art. 23, Case IV.) That portion of the shadow behind V is imag- inary. The shadow of the entire line is then determined on • the V Plane, above and below the H Plane. That portion of the shadow below the H Plane is imaginary. The shadows on the V and H Planes meet in the GL. The shadow of a straigh t line which_ falls on more than one plane surface may;^sq be found by passing a plane of light'TTirough'the given line and determining where this plane intersects the plane surfaces receiving the shadow; and then passing Rays of Light through the extremities of the line to find where the shadow on the line of inter- section between the plane of light and the other given planes terminates the shadow of the line. (Fig. 22.) Case IV. To find the shadow of a straight line per- pendicular to a Coordinate Plane (either elevation or plan), across several adjoining surfaces r.nd planes or across a series of moldings on that coordinate. (Figs. 25 and 26.) Let ab and cd be straight lines, perpendicular to the H and V Planes respectively, upon which is a series of moldings. A plane of light is passed through the lines and the intersection of this plane with the moldings dct(>rmined. Since the shadow of the line is contained in the line of inter- section of the plane of light and the series of moldings, it is (Fig. 25), in Vertical Projection, a line making an angle of 45° with the GL; and (Fig. 26), in Horizontal Pro- 70 SHADOWS OF LINES Roman Doric Order. PRINCIPLES OF SHADES AND SHADOWS 71 CuKNlCB. 72 SHADOWS OF LINES jection, a line making an angle of 45° with the GL. (Chap. I, Art. 27.) Axiom II. The shadow of a straight line perpendicular ' to Oliy (inhe coordinates, is a line making an angle of 45° with the GL, in projection on that coordinate, or on any series of surfaces, forms, or moldings on that coordinate. 25. To find the shadow of parallel lines on a given plane to which they are parallel. Let ah and cd be parallel lines parallel to the plane on M hich their shadows fall. (Figs. 27 and 28.) The shadow of each line is determined on the given plane. (Art. 24.) This determines two lines of shadow on the given plane, jDarallel to each other, and parallel to the lines casting the Hnes of shadow. Axiom III. The shadow of any line on a plane to which it is parallel, is a line equal and parallel to the given line. Axiom IV. The shadows of parallel hues on any plane are parallel. 26. To find the shadow of any line, not a straight line, on a given plane. Let ahcd, etc., be any line, not a straight line, whose shadow falls on a given plane. (Figs. 29, 30, and 31.) .The line abed, etc., is made up of an infinite number of points. By casting the shadows of a sufficient number of these points, the shadow of the curve is readily deter- mined. (Art. 23.) Axiom V. The shadow of a line, straight or otherwise, is determined by casting the shadows of adjacent points of that line. PRINCIPLES OF SHADES AND SHADOWS 73 Fig. 27 Fig. 29 c'' cZ'' Fig. 30 74 SHADOWS OF LINES PROBLEMS ON CASTING OF SHADOWS OF LINES 75 PROBLEMS ON THE CASTING OF SHADOWS OF LINES Note. — In the succeeding problems the following abbre- viations are used: u for up; / for forward; r for right; d for down; h for back; and I for left. PLATE III Determine the shadows of the following lines. 1. A line whose H Projection is 1 inch long, makes an angle of 30° with the GL, slopes ujr, and has its lower end I 1 inch from H and 2 inches from V. Its V Projection I makes an angle of 30° with the GL. 2. A line whose H Projection makes an angle of 45° I with the GL, is 1^ inches long, slopes ufl, and has its lower j end 1 inch from V and 2\ inches from H. Its V Projection : makes an angle of 30° with the GL. 3. A line whose H Projection is 2 inches long, makes : an angle of 45° with the GL, ubl, and has its high end § i inch from H and 1| inches from V. Its V Projection makes I an angle of 30° with the GL. \ 4. A line whose length is 1| inches, has one end 2 inches ■ from V and § inch from H, and is perpendicular to V. 5. A line whose length is 2 inches, has one end j inch ,. from V and 2 inches from H, and is perpendicular to V. 6. A line whose length is If inches, has one end \ inch from V and 1 inch from H, and is perpendicular to V. 7. A line whose length is U inches, is parallel to H, has one end 1§ inches from H and | inch from V, and is not parallel to V. 8. A line whose length is 1^ inches, is i)aralk'l to V, has one end ^ inch from H and 2 inches from V, but is not parallel to H. 76 SHADOWS OF LINES 9. A line whose length is 1| inches is perpendicular to V. Use only the V and Profile-Projections. 10. A line whose length is 1| inches and is perpendicular to H. Use only the H and V Projections. PLATE IV 1. Find the shadow of a 2-inch line which is parallel to the GL, 1 inch from V, and 1^ inches from H. 2. Find the shadow of a l|-inch line which is parallel to V, makes an angle of 30° with H, and slopes ur. The lower corner is 1 inch from V and H. 3. Find the shadow of a 2-inch line which is parallel to V, makes an angle of 30° with H, and slopes ur. The lower corner is 2| inches from V and 1 inch from H. 4. Find the shadow of a 2-inch line which is parallel to V, makes an angle of 30° with H, and slopes ur. The lower corner is 1§ inches from V and 1 inch from H. 5. Find the shadow of a line perpendicular to the Profile Plane. 6. Find the shadow of a 2-inch line which is parallel to the Profile Plane, makes an angle of 45° with H and V, and has its ends touching both H and V. 7. Find the shadow of a line lying in the Profile Plane. " 8. Find the shadow of a 2-inch line whose ends lie in the H and V Planes, and whose projections are perpen- dicular to the GL. PLATE V L Find the shadow of two lines which intersect at a point 2 inches from V and 1^ inches from H. 2. Find the shadow of any two intersecting lines which lie in an Imaginary Plane perpendicular to the coordinates. 3. Find the shadow of a curved line, four points of PROBLEMS OX CASTING OF SHADOWS OF LINES 77 which are as follows: a, 1 inch from H and V, and | inch from P; ?>, Ij inches from H, 1 inch from V, and 1^ inches from P; c, 1| inches from H, 11 inches from V, and 2^ inches from P; (i, 2 inches from /f, Ij inches from V, and 3 inches from P. 4. Find the shadow of the same line when it is moved 1 inch farther from V. 5. Find the shadow of the same line when it is moved ^ inch farther from V. 6. Find the shadow of a 2-inch line which is perpen- dicular to the GL, and which makes an angle of 30° with the H Plane. Its lower end is ^ inch from the GL. 7. Find the shadow of a line and any point in that line. QUESTIONS ON THE SHADOWS OF LINES 17. How is the shadow of any straight line determined? Explain both methods. 18. What is the Imaginary Shadow of an object? How is it determined? 19. How is the shadow of any line which is not straight determined? 20. If a straight line is perpendicular to one of the Coordinate Planes, what is the character of its shadow on that plane, and on any series of surfaces, forms, or mold- ings on that plane? 21. What is the character of th(> shadows of parallel lines cast on a plane? 22. Describe the shadow of a lino on u plane to which it is parallel and on which its shadow falls. 78 SHADOWS OF PLANES IV. SHADOWS OF PLANES 27. To find the shadow of any plane surface on another plane surface. (Figs. 32 to 35.) Let ahc be any plane surface. Plane surfaces are bounded by straight or curved lines. Rays of Light, therefore, are passed through the points at the extremities of the straight lines or through a suf- ficient number of points of the curved lines to determine the curve. The intersections of these Rays with the sur- faces receiving the shadow are points of shadow in the bounding line of the shadow. These points are connected by straight and curved lines as required, and the figure thus bounded is the shadow of the plane surface. The shadows of plane surfaces may be classified, according to the location of the shadow, as follows : Case I. When the shadow of the plane surface falls on the H Plane, or plan. (Fig. 32.) Case II. When the shadow of the plane surface falls on the V Plane, or elevation. (Fig. 33.) Case III. When the shadow of the plane surface falls on both the H and V planes; or on the plan and elevation. (Fig. 34.) Case IV. When the plane surface is perpendicular to one of the coordinates and the shadow falls on that coordi- nnate. (Fig. 35.) Case V. When the shadow of any plane surface falls on any oblique plane. (Fig. 36.) 28. The shadow of a plane surface on another plane surface to which it is parallel, is equal in size and shape to the plane surface casting the shadow. Let abed be a plane surface casting a shadow on a parallel plane surface. (Figs. 37 to 39.) PRINCIPLES OF SHADES AND SHADOWS 79 Fig. 33 ir c"di Fig. 36 Fig. 35 80 SHADOWS OF PLANES o" 6" C d" Fig. 37 Fig. 39 QUESTIONS ON THE SHADOWS OF TLAMIS 81 Determine the shadows of the plane surfaces (Art. 27) and it will be seen that the shadows have the size and shaj^e of the plane surfaces casting them. Axiom VI. Any point or line lying on a plane is the Invisible Shadow of itself on that plane. 29. The shadow of a circular surface is determined by circumscribing a polygon about it. The shadow of the polygon is then determined and also the points of tangency of the polygon and the circle. These points, connected by a curve, determine the shadow of the circular surface or plane. (Fig. 40.) QUESTIONS ON THE SHADOWS OF PLANES 23. How is the shadow of any plane surface determined on another plane surface on which it is cast? 24. How does the shadow of any plane surface show- on another plane surface to which it is parallel? 25. How is the shadow of a circular plane surface deter- mined on an oblique plane? uljiTifil.wijiiUiii \ A Section throuoh a RriLniNr,. 82 SHADES AND SHADOWS OF SOLIDS V. SHADES AND SHADOWS OF SOLIDS 30. To find the shadow of a Polyhedron of such shape that none of its faces is perpendicular or parallel to the coordinates or planes receiving its shadow. Let abed be any Polyhedron with none of its faces per- pendicular or parallel to the coordinates. (Figs. 41 and 42.) Polyhedrons are bounded by plane surfaces. The shadows, therefore, of the enclosing plane surfaces are first determined. These shadows together have the form of an enclosing polygon which is the shadow of the polyhedron. (Art. 27.) The sides of the enclosing polygon of shadow are the shadows of the shade-lines of the object casting the shadow. Hence the shade-lines are readily determined by finding what lines cast the bounding lines of the shadow. In this figure, it is at the outset impossible to determine the shade-line from which to determine the shadow. Many of "the Shades and Shadows in architectural drawings have to be determined by the methods used in this problem, namely, by first determining the shadow of the object, and then, from the bounding shadow-line, determine the shade- line of the object. 31. To find the shadow of a solid which has its faces par- allel with or perpendicular to the planes receiving the shadow. Let abcdefgh be a solid, whose bounding planes are parallel or perpendicular to the planes receiving the shadows. (Figs. 43 and 44.) This is a problem which is quite common in architec- tural drawings, and in which a sufficient number of planes of an object are perpendicular or parallel to the planes receiving the shadow, thus making it possible to determine the shadows by simple and direct methods. (Art. 27, Case IV, and Art. 28.) PRINCIPLES OF SHADES AXD SHADOWS 83 84 SHADES AND SHADOWS OF SOLIDS /" a^b'^ ■•''d" Fig. 43 PRINCIPLES f)F SHADES AM) SHADOWS 85 i^^T Fig. 40 « Fig. 40'' Fig. 40*1 SHADES AND SHADOWS OF SOLIDS Fig. 40« Fig. 40* PRINCIPLES OF SHADES AND SHADOWS 87 Fig. 40 e Fig. 40 k Fig. 401 SHADES AND SHADOWS OF SOLIDS PRINCIPLES OF .SHADES AND SHADOWS 89 In this problem, the shadow is found by first finding the shade-Une of the soUd and then the shadow of its shade- line. This determines the bounding line of the shadow. The shade-line of a simple object is determined by find- ing where the Rays of Light and the Planes of Light are tangent to it. The following are some of the problems frequently pre- sented for solution : Case I. A Prism with its faces parallel or perpen dicular to the planes receiving the shadows. (Fig. 40a.) A Plinth on a Prism. (Figs. 406 to 40^.) A Pedestal. (Fig. 40c.) A Chimney. (Fig. 40e.) A Dormer. (Fig. 40/.) A Hand-Rail on a Flight of Steps. (Fig. 47.) A Cylinder. (Figs. 40m, 40o, and 40p.) A Cone. (Figs. 40d, 40k, 401, and 40m.) A Plinth on a Cylinder. (Fig. 40r.) The Trim of a Window. (Fig. 40s.) QUESTIONS ON THE SHADES AND SHADOWS OF SOLIDS 26. How is the shadow of any solid determined? 27. Explain difference of methods of determining the Shade and Shadow of a Polyhedron, and a solid which has its bounding faces perpendicular or parallel to the planes receiving the shadow. 28. Explain how the principle of determining the shadow of any point on any plane enters into the determination of the shadow of any solid. 29. How is the shade-line of a polyhedron dotormined? How are the shade-lines of ordinary simple objects deter- mined? Case IL Case in. Case IV. Case V. Case VI. Case VII. Case VIIL Case IX. Case X. 90 GENERAL METHODS VI. GENERAL METHODS 32. There are four principal methods, of which one or more are generally employed in determining the Shades and Shadows of Architectural Forms. They are : (1) The Method of Oblique Projection. (2) The Method of Circumscribing Lines. (3) The Method of Circumscribing Surfaces. (4) The Method of Slicing. 33. The Method of Oblique Projection consists in passing Rays of Light or Planes of Light, tangent to a given object, to determine its shade-line, and in finding where these Rays of Light or Planes of Light intersect any other object on which the shadow of the given object falls. (Figs. 45, 46, and 47.) This method has been the one so far employed in determining the Shades and Shadows of various objects, and needs no further explanation. 34. The Method of Circumscribing Lines consists in drawing upon any solid, preferably upon any surface of revolution, a series of lines, straight or curved as the case may require, which are, if possible, parallel to the plane receiving the shadow. The required shadow of the surface includes the shadows of all the lines on that surface. The bounding line of the shadow is tangent to the shadow of those lines which cross the shade of the surface, at points which are the shadows of the points of crossing. The point of intersection of two shadow-lines is the shadow of the point of intersection of those lines casting the shadows, if the lines intersect on the surface; or the shadow of the point in which the shadow of one line crosses the other, when they do not intersect. It is evident from this that if the intersection of the shadows of two lines is considered, that is, the point of PRINCIPLES OF SHADES AM) SHADOWS 91 uW' :Kl" fV chfh gMi' Fig. 45 Fig. 46 a«b"a>^ \ c'-d'- v >N ro'- N. \ /<''A'' \^ \ /•• \,.r3 1 ; ; 1 1 1 a'' 1 1 1 o>' 1 rh,l>' i 1 1^- 1 ,./,,-/, 1 ,jn. X 1 a"h'' «:-. 1 /.''/'' t y / >■ V' Fig. 47 92 GENERAL METHODS PRINCIPLES OF SHADES AND SHADOWS 93 tangency of the. shadow of one line with the hounding Hne of the shadow, the points in the shade-hne on the surface of the object which casts these points of shadow may be determined by passing back along each Ray of Light thi'ough the point of tangency. A curved hne connecting these points is the shade-hne. Let a Sphere (Fig. 49) be a sohd, whose surface is a surface of revolut on, which casts its shadow on the obhque plane P, or on one of the coordinates. (Fig. 4S.) In each case a series of lines is drawn on the surface of the Sphere parallel to the plane receiving the shadow in Fig. 48 and not parallel in Fig. 49. These lines may be determined by passing planes through the Sphere, parallel to the given plane. The lines of intersection of the assumed planes with the Sphere are the lines required. The shadows of these lines may now be cast on a plane to which they are parallel. (Arts. 25 and 28.) These shadows are now cir- cumscribed bj^ a curved line, which is the bounding line of the required shadow. The shade-line is found by passing back along the Rays of Light, Ri, Ro, etc., through the points of tangency of the bounding line of the shadow with the shadows of the assumed lines 1, 2, 3, etc., and determining where these rays of light, Ri, Ro, etc., are tangent to the surface of the Sphere. These points, which are points in the shade-line, are connected b}' a curved line, which is the reciuired shade- line. The procedure in finding the Shades and Shadows in Fig. 49, is the same, except that the circumscribed lines are not parallel to the surface receiving the shadow. 35. The Method of Circumscribing Surfaces consists in circumscribing the surface casting the shadow, witli the surface of an object whose shade-line is readily determined. 94 GENERAL METHODS Tonic Cap. PRINCIPLES OF SHADES AND SHADOWS 95 Corinthian Cap and Base. 96 . GENERAL METHODS Then at a point of tangency of the two surfaces, whatever is true of one surface is also true of the other surface, for such a point is in common. This method is only used in determining the shade-line on double-curved surfaces of revolution. Let it be required to find the shade-line on a Sphere. (Fig. 50.) The shade-line of a Cone is readily determined. The Sphere is circumscribed with a Cone, or with Cones in different positions. The shade-points on the lines of shade of the Cone where the Cone is tangent to the Sphere, are points of shade on the shade-line of the Sphere. These points are connected by a curved line, which is the shade- line required. 36. The Method of Slicing consists in cutting through the object casting the shadow and the object receiving the shadow with planes of light perpendicular to one of the coordinates, and in determining points of Shade and Shadow by passing rays of light through the shade-points in the slices casting the shadow, and determining where they intersect the slices receiving the shadow. (Figs. 51, 52, and 53.) By the use of this method the shadow of any object can be found. It is, however, often difficult of appli- cation, as the process of constructing the slices is relatively slow and often complicated. Let it be required to find the Shades and Shadows of a Sphere, Torus, and Scotia by the Slicing Method. (Figs. 51, 52, and 53.) Planes of light are passed through each object, per- pendicular to the H Coordinate, SET, etc. The slices cut out from the object by these planes are first determined. (Chap. I, Art. 4L) The rays of light, Ri R2, -etc., are passed through the points of shade or the projecting por- PRINCIPLES OF SHADES AND SHADOW; 97 PRINCIPLES OF SHADES AND SHADOWS 99 tions of the slices, and the points where these rays intersect the sUces receiving the shadow are determined. These points are connected by a curved Hne which is the ro(iiiired bounding Hne of the shadow. 37. The SUcing Method is used to great advantage in determining which faces of a Polyhedron are in light and which in shade, since it is sometimes not possible to deter- mine this by the method of Oblique Projection. The sides in shade are determined by passing a plane of light, per- pendicular to one coordinate, through the polyhedron (Fig. 54), and drawing the slice cut out by this plane of light. The surface in shade are then readily determined by passing rays of light, and they are the planes containing the bound- ing lines of the slice which do not receive the Rays of Light. 38. Another method of slicing, called the Slicing Method by Auxiliary Planes, is sometimes used. In this method the object receiving the shadow is cut by Auxiliary Pianos, parallel to either coordinate. A portion of the shadow of the object is cast on each Auxiliary Plane. The point of intersection of the shadow-line on the Auxiliary Plane with the line of intersection of the given surface with the Auxili- ary Plane, (which is a point in common on both planes), is the shadow of a point of the object casting the shadow on the object receiving the shadow. Hence it is a i)oint m the shadow-line. A sufficient number of points of shadow may be deter- mined in this way; and if they are connected by a line, straight or curved as required, they determine the nHjuired line of shadow. Let a Niche be as shown (Fig. 5o), and determine its shadow by this method. Auxiliary Planes, RST, etc., are passed through the Xichc, parallel to the V Plane. The shadow of the shade-line of the 100 GENERAL METHODS Fig. 54 Fig. 55 /o«« /' /- ■S' .^, Shadow of ah on pianes R-S-T-V Plan Planes of light 90° with H plane Shadow of ab en- planes R-T-S-V Fig. 56 Section on line^ A A 102 GENERAL METHODS niche is cast on each of these planes by casting, the shadow of the center of the Shade4ine on the AuxiUary Planes, and the points in which these shadows intersect the shces (cut from the Niche by the Auxihary Planes), will be points a, b, c, d, e,f, shown by their projections a^% b^% c^\ d!"', etc., and a''% h'% c", d'% etc., in the required shadow-line. These points are connected by a curve line, which is the bound- ing line of the required shadow. Let the Cornice-Moldings (composed of a Crown-Mold- ing or Cavetto, and an Ogee-Molding) at the top of a Pedi- ment be as shown. (Fig. 56.) It is required to find the shadows of the moldings by this method. Auxiliary Slicing Planes, SRT, etc., are passed, parallel to the V Plane, through the moldings, and the slices cut out are determined. The shadows of the shade-line a b, and c d then cast, on each Auxiliary Plane in order to deter- mine w^here each shadow intersects the slices respectively. As these are points in common, on the surface of the Pedi- ment and the Auxiliary Planes, they are points in the required shade-line. These points are connected by a curved line, which is the boundary-line of the required shadow. 39. To find the highest and low^est points in the shade- line or shadow-line of a double-curved surface of revolution. Let a Sphere and Scotia' be given. (Fig. 57.) It is required to find the highest and lowest points in the shade- lines and shadow-lines. This problem is a special application of the Slicing Method. A Slicing Plane of light, perpendicular to either coordinate, is passed through the axis and center of each object. The slice cut out is then determined, and the Rays of Light applied, in order to determine the points of Shade PRINCIPLES OF SHADES AND SlI AHOW: 103 .c and V are the highest and lowest points of shade Fig. 57 x'^J-'" is the highest point of !tliade RPINCIPLES OF SHADES AXl) SHADOWS 105 and Shadow. (Art. 37.) These points are the highest and lowest points in the shade-hne or shadow-Hnes of the Sphere or Scotia, because the sUce cut from the object by the plane of light is a great circle on which are the h ghest and lowest points of Shade and Shadow on the shade-line and shadow- line. 40. The forms used in architectural designs are such that it is very often advantageous to find the Shades and Shadows of an object by more than one of the methods given. Let it be required to find the Shades and Shadows of a Light-Standard. (Fig. 58.) The shade on the Sphere may be determined by cir- cumscribing surfaces. (Art. 35.) The Slicing IVIethod deter- mines the shadow on the Scotia. (Art. 36.) The shade on the Cylinder may be determined by the jNIethod of Oblique Projection. The circumscribing lines are used on the Torus to determine its shade. In order to find, therefore, the Shades and Shadows of an object, the problem must be analyzed, and those methods used which give the quickest and easiest results, "and which do not introduce complicated methods of con- struction. 106 GENERAL METHODS FOKVM •/Avcvsri ROME Corinthian Order QUESTIONS 107 QUESTIONS 1. Name the different general methods used for casting the Shades and Shadows of sohds. 2. Explain the Slicing Method. 3. What is the Method of Oblique Projection, used to determine the Shades and Shadows of an object'.' 4. Explain the other methods sometimes used to deter- mine the Shades and Shadows of an object. 5. \\'Tiat method is used to determine the Shades and Shadows of complex architectural forms? Explain th(' usual procedure followed in determining the Shades and Shadows of such forms. 108 GENERAL METHODS PLATE VI Find the Shades and Shadows for the following Plates using complete notation on all problems ^ Ifs" ' ', IV V n III ( > - i ll- VI X VII vm XI a IX ^■r^^ PROIU.KMS 109 no GENERAL METHODS PLATE Vin II K.I ■'• 1 VI PROBLEMS 111 PLATE IX Find only shadows of the lines in these problemi Section or Profile ni IV 112 GENERAL METHODS PLATE X ./ / / 1 ' / / y^ t j I T; 1^ ^^ C\) / CO I \ ^^^ / / ' 2 i V>^/ ! : , i I I I ^ : I I I I II I 12 ' ' ! ' M I I I j-J I I I'i^^ I J--"^^^ I / I -^^ "'' I / -^^^--< I A, I I I I I « I ' 1 I ' ni ,t. ^ ^'^" . s '2 L. •J v^ ' — ] -J - p> I V PROBLEMS li:"{ PLATE XI f ' 'i y ^ 2 ^ ^ 'a ^ -V ~f ' i Ix — : 1" 2" -■ eg -J -) ^ L L L-— -- t p^ r "■ -L :i? ?i" ^,' '»■ ,r 2' '^ CJ ' 1 *;.t F- '^^ * !■■ '4--; ' ) Vi' ■?f •"■I — I III IV 114 GENERAL METHODS PLATE Xn Use Oblique Projection Use Circumscribing Lines Use Slicins Method III Use Circumscribing Surfaces IV PRC^BLEMS 115 PLATE Xm Plan — 4' Find the Shades and Shadows by three different methods. 116 GENERAL METHODS ' PLA'l'iii XIV ) K (^ 1 '^\ 1 I? t !'" . y y- 1 ( \ 1 \ Elevation Plan PHOBLKMS 118 GENERAL METHODS PLATE XVI •^h Profile T^rr .n r 2Vi" Elevation II PROBLEMS 119 PLATE XVII j^==^ Profile Elevation 120 GENERAL METHODS PLATE XVIII Elevation Plan Section PROBLEMS 121 PLATE XIX C*=S(|n I cW8=3(!Q 122 GENERAL METHODS PROBLEMS 12:^ PLATE XXI n Elevation Section These Problems are to be done at twice this scale 124 GENERAL METHODS PROBLEMS 12: PLATE XXIII This Problem to be worked out at twice this »culc 126 GENERAL METHODS PLATE XXIV - : 1", 4'// 1\ 4H" < l"> Light Wash '« Medium Wash - Dark Wash 00 Light to Medium to Dark Wash I Date - Name ' PROHLEMS 127 128 GENERAL METHODS PLATE XXVI /Dark to\ / Light \ 1^ . ^ 2" ^ ^c Other washes and dimensions same as Plate II .. 1 1 lU Vi" , 1>4" . m" Light tor )ark '^ ■ \e< - Medium „ Light to Dark/ „ / Light / /^^Dark to / / Light ^-^ PRINCIPLES OF SIIADKS AND SHADOWS \2\) Fig. 10. fmMii^^^i^ Fig. 11. 130 WASH-RENDERING VII. WASH-RENDERING 41. An architect's drawing is a representation of a structure as it will appear in reality. The natural laws of light and shade are followed in the conventional manner explained in the foregoing pages of this book to the extent that it will advance this represen- tation. One medium used in the wash-renderings of draw- ings is India ink, especially in the presentation of drawings submitted in competitions. The materials needed for wash-rendering are Chinese, Japanese, or India stick-ink, an ink-slab, a small and a large sable-hair brush, a nest of saucers, a sponge, and a bowl or pan. Whatman's paper (rough) is a good paper for wash-drawings. To stretch the paper, the entire back of the sheet is wet, except a 1-inch margin around the edge, which is kept dry for the paste. After the paste is smeared on the edges, the surplus water is removed, and the sheet turned over. The right side, except the edges, is then wet and the edges pasted down starting from the middle of the sides. Gently stretch it, and work towards the corners, which are pasted down last. All the water should then be taken up before leaving the sheet to dry. Care should be exercised to pre- vent the water from wetting the edges. When they are wet the paper dries before the paste adheres to the board, and loosens the edges. Boards which, when wet, stain paper should not be used. Soft pencils should not be used in making these draw- ings. The HB, H, and F ''Kohinoor" or similar grade of pencil give excellent results. Care should be taken in making the drawings themselves, as a good pencil-drawing with careless washes generally presents a better final appear- PRINCIPLES UP .SHADES AM) SHADOWS 131 Student Work— State College of Washington. ^s^ *=s A Museum. Student Work— Clcinsoii College. WASH-RENDERING Hr^-^^^"^^^ A Museum, Student Work. n /:::._ _n'_ A Museum. Student Work. PRINCIPLES OF SHADES AND SHADOWS 133 ance than a poor drawing with more carefully made washes. Good washes never disguise poor drawings. A long, round point (not a chisel-point) should be kept on the pencil at all times. A point of this kind can be made with a sand- paper sharpener. When used to draw a line, the pencil should be pulled across the sheet and twisted at the same time, in order to keep the point sharp. A line should be of uniform width, should have a definite beginning and end of a firm touch, and should not taper off and tlisappear gradually. All construction-lines can run past their inter- sections. It is not necessary to rub out construction-lines, but they should be drawn very light. For large washes the stick India ink should be freshly ground on a clean slab each time it is required. Ink, after standing several hours, should be thrown away. The ink is rubbed on a slab until the solution is very black, and allowed to settle in the covered slab. Ink-wash evaporates rapidly, and dust settles in an uncovered slab. After the ink has settled, enough for the required wash is taken from the top with the point of the brush and put with clean water in a saucer. The ink should always be taken from the top, with no disturbance of the sediment, as the latter streaks and spots the wash. In laying on a wash, plenty of the color-solution used is kept on the paper and in the brush, and the wash is ''floated" on. In bringing the color up to the boundary-lines, it is squeezed from the brush and brought to a sharp point. The brush is then used to bring the wash to \ho l^oundary- line. This is done, also, as the bottom of the wash is approached, and all the surplus color is then taken up just before reaching the bottom. Where only a small surface is to be rendered, the tint may be mixed on a \nrrr of jiapcr in the same manner as in th(^ saucer. 134 WASH-RENDERING Beginners have difficulty in keeping a wash from ''streaking." Much skill is necessary to prevent this and is acquired only by practice. Some other cause of streaks are, dust on the paper, grease on the paper through contact with the hands, water-streaks or marks, and a disturbed surface from the use of a hard rubber. Before applying a wash the drawing should be sponged off. If a rubber is used, it should be soft and used before this washing. A skillful handling of the brush is acquired through constant practice. It is held like a pencil, with the arm free and not resting on the drawing, but with the little finger in contact with the drawing. Only the point of the brush should touch the paper. In laying on the wash the board is slightly tilted in the direction in which the wash is being applied. The wash should be applied so that all portions of the paper are throughly and evenly wet. A portion partly dried should not be again wet in that condition, and the draughtsman should wait until it has thoroughly dried. Washes which grade gradually are made by the addition of water or color each time there is an advance. These additions must be carefully managed so that the change in color is even. It is difficult to lay an evenly 'graded wash in one application. It is better to first lay a light, flat, or slightly graded wash, and then the graded wash, in one or several operations. The ''washing out" of a wash should be avoided; but if it is necessary, a sponge and a great deal of water should be used; or the drawing should be placed under a faucet, the water turned on, and the sponge used on the parts to be lightened. Sometimes gradations are obtained by laying on successive flat washes, each wash beginning a little lower in value than the previous cnc, or vice versa. PRINCIPLES OF SHADES AND SHADOWS 135 In laying on a wash, a good-sized "puddle" should be kept on the sheet. The bottom of the wash must be kept wet or it will streak The tint should be carried down evenly across the board, and the brush moved rapidly from side to side. The center of the wash is kept lower than the sides. Before putting on a wash it is a good plan to dampen the paper with a soft sponge so as to have the wash applied more evenly. Washes which are put on after the space has l)een sponged and before the paper is entirely dry, that is, while the paper is still damp but not wet enough to make the wash run, usually give good results. In the following plates there are three grades or values. Medium is supposed to be half way in value between light and dark. The spaces are to be drawn in with a soft pencil, on Whatman's cold-pressed paper. PLATE 27 1. Draw a corner of Vignola's Doric Order, with its Entablature, making the Column 12 inches high. Cast all the Shades and Shadows. Wash in the Shades and Shadows, background, etc., with graded ink-washes. PLATE 28 1. Do the same as in Plate IV, for the Ionic Order. PLATE 29 1. Do the same as in Plate IV, for the Corinthian Order. Wiley Special Subject Catalogues For convenience a list of ihe Wiley Special Subject Catalogues, envelope size, has been printed. These are arranged in groups — each catalogue having a key symbol. (See special Subject List Below). To obtain any of these catalogues, send a postal using the key symbols of the Catalogues desired. 1 — ^Agriculture. Animal Husbandrj'. Dairying. Industrial Canning and Preserving. 2 — Architecture. Building. Concrete and Masonry. 3 — Business Administration and Management. Law. Industrial Processes: Canning and Preserving; Oil and Ga« Production; Paint; Printing; Sugar Manufacture; Textile. CHEMISTRY 4a General; Analj-tical, Qualitative and Quantitative; Inorganic; Organic. 4b Electro- and Physical; Food and Water; Industrial; Medical and Pharmaceutical; Sujar. 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