'PRQPIfkl I Or a1M~i r2~~i~~a~~~5zr~~~~~~~a~~~~,817~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ R TE 9C 1 N7"I VERaFas ( CIIIIIY~CI4~C(L -— ~S~ IC~C- YC~ -~~~l 1J)Iv I THE DESCRIPTIVE GEOMETRY AND THE PERSPECTIVE OF THE STRAIGHT LINE WITH A BX-IEF INTARODUCTION TO THAT OF CURVES ACCOMPAN:ED BY MANY EXERCISES. BY.... WILLIAM J. MEYERS, PROFESSOR OF MNTHE'4ATICS, THE STATE AGRICULTURAL COLLEGE OF COLORADO. -.... The imago-ination --— it is mnatcrial ---to fortifiy and exalt it. " P.acoun- /lic Advancement of Learning. FORT COLLINS. COLORADO. WILLIAM J. MEYERS, PUBLISHER. 1896. PREFACE. The idea which has governed the outlining of the work indicated in the following pages has been that the value of the study of Descriptive Geometry lies, for the ordinary student, chiefly in the discipline which his imagination receives through his familiarizing himself with the fundamental operations of that portion of applied Geometry. My aim has therefore been to strike the happy mean between a treatment of the subject so abstract and difficult as to call forth no clear ideas in the mind of the student, and one so diffuse and easy as to call forth no real- mental exertion on his part. The method herein adopted is one which has been hammered out in the actual work of the class-room. Attention is particularly called to the use made of perspectives of figures embodying the fundamental relations as shown in Plate 1. In some respects, the use of perspective representations will be found to be equally efficient with that of models; in others the superiority of models is, of course, undeniable. In almost all cases, however, the student will find it iv PREFACE. advisable to construct for himself a rough model of the first quadrant (better yet of the four quadrants).. by nailing two pieces of thin board together so as to form a right diedral (two pieces of heavy card-board will serve fairly well), and then using threads to represent lines, and light cardboard or stiff paper cut to various angles to represent planes. He will find such a rough apparatus as this of considerable aid at the beginning of his study, but should early cast it aside and depend as much as possible on his imagination in tracing the relations of the different parts of the figu res. The chapter on perspective has been added out of deference to custom. In that, as in the preceding chapters, only such development of the subject as was thought to be of value as discipline has been attempted.- For any more extensive development of the subject the student is referred to books on the subject prepared especially for architects. Of these, Ware's Modern Perspective, published by James R. Osgood and Co., of Boston, is perhaps as good as any in the English language. The author will be glad to receive such criticisms and comments on the work herewith offered as may suggest themselves to instructors who may have occasion to use it with their classes.. WILLIAM J. MEYERS. Department of Mathematics, The State Agricultural College, Fort Collins, Colo. V TABLE OF CONTENTS. PAGE Preface.......................................... iii Table of Contents.................................. v N otation................................................ iii CHAPTER I. ARTICLE I Descriptive Geometry defined..................... 2 Projection, Center, Base, and Projecting Ray...... 2 3 Projection ani Projector of Figure................. 2 4 Centric Projection.............................. 3 5 Projectee, Parallel Projection..................... 3 6 O rthogonic Projection........................... 3 7 T race....................................... 3 8 Determ ination of Point........................... 4 9 The Problem of Descriptive Geometry............. 4 Io Horizontal and Vertical Bases of Projection........ 4 II Horizontal and Vertical Projections................ 5 12 Q uadrants...................................... 5 13 Ground Line................................... 5 I4 Mode of Denoting Projector....................... 5 15 Single planes for drawings........................ 6 15 Mode of distinguishing projectors. Symbols....... 6 17 Kinds of lines used.............................. 7 I8 Co-ordinates..................................... 8 19 O ctants......................................... Io 20 Side Plane, how shown........................... Io vi CONTENTS. CHAPTER II. FUNDAMENTAL PROPOSITIONS. A RTICLE PAG E 21-44 Propositions and Corollaries.................. CHAPTER llI. FUNDAMENTAL OPERATIONS. 46,48-62 Fundamental Problems with analyses............. I6 47 Digression upon the use of S...................... I6 63-64 Development................................... 22 65 Intersections of Surfaces....................... 23 CHAPTER IV. EXERCISES FOR DRAUGHTING PRACTICE. E xercises......................................... 25 A bbreviations.................................. 32 Notes on Problems, general....................... 34 Notes on Problems in Triedrals................. 37 Notes on Problems in Intersections................ 39 CHAPTER V. PERSPECTIVE. 66 Perspective defined............................. 4I 67 Projective Geometry defined..................... 41 68 The Point at Infinity............................ 42 69 The Line at Infinity.............................. 43 70 The Vanishing Point........................... 43 71 Ihe Vanishing Line.......................... 44 72 Picture Plane, Point of Sight, Center of Picture, Foot of Station........................... 44 73 Normal planes and lines, Axis, Distance.......... 45 74 Horizon Plane, Ground Plane, Meridian Plane, Horizon, Ground Line, and Meridian...............46 CONTENTS vii ARTICLE PAGE 75 Secondary Planes and Figures..................... 46 76 Principal Diagonals, Subordinate Diagonals........ 46 77-9I Fundamental Propositions in Perspective.......... 47 92 The Fundamental Problem in Perspective.......... 48 93 Modifications of Fundamental Method............. 49 94 Direct Practical Method........................... 50 95-Io I Indirect Practical Methods........................ 51 96 Two Problems involved in such Methods........... 5I 97 To Lay off the Perspective of a Given Length on the Perspective of a Given Line.'................ 52 98 To Find the Trace and the Vanishing Point of a Given Line under Certain Conditions.......... 53 99 To Divide the Given Perspective of a Sect into the Perspectives of Proportional Parts.............. 54 Ioo Vanishing Point off the Sheet. Centrolinead...... 55 IO1 V anishing Scales................................. 57 Io2-Io The Practical Problemm.................. 57 IO5 Parallel, or One-Point, Perspective................ 59 Io6 Angular, or Two-Po nt, Perspective.............. 59 107 Oblique, or Three-Point, Perspective.............. 59, 1os Shadow s..................................... 60 10) Order of Procedure.................................. 60 IIo Depressed Perspective Plans...................... 6r III Illustrative Problem............................. 62 Problems for Exercise........................... 66 viii NOTATION. a, b, c, d, e, f..straight lines or sects. g,.............ground line. gi,............first auxiliary ground line, the one I to H. g2,............second auxiiiary ground line, the on ' I to V. h,........... horizon. i, j,,,....... curves. m,............eridian. q,.... the normal through q, in perspective. o,............origin of co-ordinates. pq, -, r,, v, o, v,, points. s,............ point of sight. s'".......... center of picture. Si,............foot of station. x, y, z,..... co-ordinates. A, B, C,,D, F, F, I, J, K, planes. G,..........ground plane. in perspective. H,............horizon plane in perspective; otherwise the horizontal, or first, base of projection. L,,........ length of the sect joining p an q. L3a,...........length of portion of a between its traces on the firs and secozcd bases of projection. M,........... meridian plane in perspective. N,...........plane normal to PP in perspective. 0O. 07, etc,.... third octant, seventh octant, etc. R, U, [V, X, Y, Z, curved surfaces. PaH, or hPa,...projector of a on H, or horizontal projector of a. PP,...........in perspective, the picture plane. Q2, Qa, etc......second quadrant, third quadrant, etc. S,............ side, or third, base of projection. NOTATION. ix TaV, or vTa,...trace Of a on V, or vertical trace of a. V,............the principal vertical base of projection, also called the second base of projection. Vq,......... in perspective, the secondary plane (or plane parallel to PP) through the point q. c, c, c,....projections of c on H, V, and S, respectively. aa,............the angle that a makes with a'. Pa............the angle that a makes with a". ya,.........the angle that a makes with a"'. z ab..........the angle that a makes with b. AAB,........the diedral that A makes with B. ar,,........ a revolved into H about a' as axis. qrHr.......... revolved into H about a' as axis. ar!lHI,......... a revolved parallel to H about i as axis. PdIn,....... B developed into H about TBH as axis. r,............the angle between the traces of a plane. rBHS,.........the angle between the traces of B on H and S. Besides the symbols above noted there are to be used the ordinary geometric symbols for trigon, parallelogram, perpendicular, angle, etc. CHAPTER I. DEFINITIONS, NOTATION, ETC. i -DESCRIPTIVE GEOMETRY is that portion of applied geometry which is concerned with the investigation, by means of drawings in a single plane, of the properties of figures not lying in the plane of the drawings. It is the scientific basis of draughting. It finds application wherever the relations between the different parts of any solid are to be found without first constructing the figure, and in many cases the drawings used in it are far superior to the actual solid figure for purposes of investigation, to say nothing of the great saving of time and labor effected by their use; for instance, if a tin-smith wishes to make two intersecting tubes, he makes use of the principles of this science in laying out the patterns by which to cut his sheets of metal so that when fastened together the desired figure will result. Aside from its direct utility in the industrial arts, descriptive geometry also has great value as a means of mental discipline, its study being perhaps better adapted to the development and discipline of the imagination than that of any other science. 2 DEFINITIONS. 2.-If from any point q, (see fig. I,) we draw a line through any other point n to meet a given surface R, the point u in which the given line qn meets the surface R is the projection of n upon R from q as center of projection. The surface R is the base of pro. jection; the line from the center of projection q through the projected point n is the projecting ray of n. 3.The aggregate of the projections, upon any surface R, of the various points of any figure X from any center of projection q, is called the projection of Xupon R from q as center; and the aggregate of the projecting rays of the various points of X from q as center is called the projector of X from q as center. Q. and E. I.-What kind of figure is the projection of a straight line upon a plane? 2.-May the projection of a straight line upon a plane ever be merely a point? If so, under what conditions,-if not, why not? 3.-May a straight line ever be so placed as to have no projection on a given plane? If so, under what conditions? 4.-What kind of figure is the projection of a circumference upon a plane? 5.-May a circumference ever have such position that its projection upon a given plane will be a straight line?-a circumference?-an elongated closed curve?-an open curve?-two open curves?-two closed curves?-three open curves?-such that it will have no projection upon a given plane? If any of these questions receives an affirmative answer, state the conditions,if a negative answer, show why. 6. —What kind of figure is the projection of any plane curve upon a plane when the plane of the curve contains the center of DEFINITIONS. 3 projection? Are there any exceptions? What kind of figure when the plane of the curve does not contain the center of projection? 7.-What kind of figure is the projection of any non-plane curve upon a plane? 4.-The kind of projection just described is what is meant in the higher mathematics by the term "projection" when used without qualification. In this textbook, however, we shall call it centric projection, because we wish to use the term "projection" (unqualified) in a more restricted sense. 5.-If the center of projection be taken at an infinite distance from the projectee (i. e., the figure projected,) the projecting rays within any finite distance from the projectee will be indistinguishable from parallel lines. Under such circumstances we shall call the projection parallel. 8.-What kind of figure is the (parallel) projector of any nonplane curve? 6.-If, in the case of parallel projection, the base of projection is a plane perpendicular to the projecting rays, the projection is called orthogonic. Since orthogonic projection is the kind most frequently used, it is to be understood that that is the kind meant by the term "projection " when used without qualification in this text-book. 7.-If two figures, neither of which is contained entirely within the other, have any part common, the common part is called the trace of either figure upon the 4 DEFINITIONS. other; thus the trace of one line upon another is a point, of a line upon a plane is a point; of one plane upon another is a straight line, etc. 9.-What is the trace of a trigonic prism upon a plane to which4ts lateral edges are not parallel?-of a cylinder on a plane to which its elements are not parallel? —of a cylindric surface upon a plane?-of a spheric surface upon a plane?-of a point upon a plane? Io.-What relation exists between the projection of any figure upon a given base of projection and the trace of its projector upon the same base? 8.-It is evident that the (orthogonic) projectors of any point upon two intersecting planes intersect at the point; and as two straight lines can intersect in but one point, the projectors of any point upon two intersecting planes determine that point. These projectors are themselves determined by their traces upon the planes to which they pertain, and these traces are the projections of the point under consideration upon the two planes. Any point then is completely determined by its projections on two intersecting planes. Any figure being determined whenever all its points are determined, it is completely determined by the projections of all its points upon two intersecting planes. 9.-It is exclusively with the projections of figures that Descriptive Geometry is concerned, and the problem of Descriptive Geometry may be defined to be -To determine from the projections of any figure the properties of that figure. rOi-For simtplificati tws on two p up which DEFINITIONS. 5 the projections of any figure are made'are usually so taken as to be perpendicular to each other, and are always to be understood to be so taken unless the contrary is'stated. One of the two planes is usually conceived to be horizontal, the other thus being vertical; for this reason they are called the H and V, bases of projection respectively, or, for brevity, the H and V planes, and are denoted by the letters H and V. ('See fig. 2.) II.-The projection of any figure upon V is called the vertical projection of the figure; that upon H, the horizontal projection. 12.-Two planes perpendicular to each other divide all space into four equal parts; these are called quadrants. With respect to H and V, the quadrant above H and in front of V (i. e., on the side of V next to the observer) is called the first quadrant, and is denoted by Q1; that above H and back of V, the second quadrant, denoted by Q,; that below H and back of V, the third quadrant, denoted by 0Q; and the remaining one, the fourth quadrant, denoted by Q4. I3.-The trace of V upon H is called the ground line, and is denoted by g. (See fig. 2.) 14.-The projector of any figure will be denoted by P followed by the symbol of the figure, and the base of projection will be denoted by the letter following the symbol of the figure'; thus, PaH will denote the projector of a upon H, PaY the projector of a upon V, etc-, 6 DEFINITIONS. 5.-In practice it would be very awkward for the draughtsman to be compelled to make part of his drawings on one plane, and part on another; so, in order to avoid this, H and V are conceived to be perpendicular to each other only for the purpose of conceiving the horizontal and vertical projections; and then, before executing the drawings, the two planes H and V are supposed to be revolved about g as an axis until the two faces of Q2 come together as likewise do those of Q4. That portion of the drawing-sheet which lies above g thus carries the projections appearing upon the upper part of V and the back part of H, while that portion lying below g carries the projections appearing upon the lower part of V and the front part of H. This must always be borne in mind by the student. ii.-In what quadrant is a point which has its vertical projection above g, and its horizontal projection below g?-both projections below g?-both above g?-horizontal above g and vertical below g?-vertical on g and horizontal below g?-above g?-horizontal on g and vertical below g?-above g? I6.-The confusion which might be expected to arise from the super-posing of the drawings of two projections upon the same portion of the drawingsheet is avoided in practice by the notation adopted, and by the use of special lines for special purposes. Throughout this text-book we shall use the notation which we shall now explain. Points in general (i. e., the points themselves in their actual positions) will be DEFINITIONS. 7 denoted by the lower-case Italic letters, m, j, p, q, s, t, u, v, and w. If any point be denoted by q, its horizontal projection will be denoted by q', and its vertical projection by q". Lines will be denoted by the lower-case Italic letters, a, b, c, e, and f; curves by i,j, k, and 1; the horizontal projections by a', b', c', etc., respectively, and the vertical projections by a", b", c", etc., respectively. The trace of the line a upon H will be denoted by TaH (read "trace of a upon Hi"), or by hTa (read "the horizontal trace of a"), as may be the more convenient; that upon V by Tav, or by vTa, etc. Planes will be denoted by the capital Italic letters, A, B, C, D, E, F, I, J and K. The trace of A upon H will be denoted by TAH or by hTA as may be the more convenient; that upon V by TAV or by vTA; that upon g by TAg. The complete trace of the plane A upon the two planes of projection will be denoted by TAHV or by hvTA. Curved surfaces will be denoted by the capital Italic letters, R, U, and W. The trace of the surface U upon H will be denoted by TUH or by hTu, etc. With such exceptions as may hereafter be noted, plane angles will be denoted by the lower-case Greek letters of the first part of the alphabet, diedrals by capital Greek letters of the first part, and polyedrals by capital Greek letters of the latter part of the alphabet. I7.-With regard to figures lying in Q1, the traces of given planes and the projections of given lines 8 DEFINITIONS. are drawn full,-thus, -------;those of required planes and lines are composed of long dashes,thus, -- -- Given points are denoted by small crosses, and required points have very small circles drawn about them. Planes and lines neither given nor required, but used to aid in the solution of any problem are called auxiliary; auxiliary planes have their traces composed of dashes of medium length with two dots or very short dashes in each interval,-thus, - ----—, while auxiliary lines have their projections composed of dashes of. medium length.alternating with single dots or very short dashes,-thus, - With regard to figures in Q2, Q3, and Q4, traces. of given planes and projections of given lines are drawn with dashes of medium length, and these are usually made fainter than the corresponding full lines for figures in Q,. Other lines are drawn as for figures in Q1,. but usually much fainter. The projections ofthe same point are usually connected by lines.composed of dots or of very short dashes. I8.-Co-ordinates. In locating the position of a point, we shall use the letter y to denote its distance from H, takingy to be algebraically positive when the point is above H, negative when it is below H; ' will be taken to denote its distance from V, positive when the point is in front, negative when it is back of V. It will be noticed that the position of a point is not completely ditermined by its distance from H and YV u but merely DEFINITIONS. 9 that of some line on which the point must be. In order to define the point completely, we must know its distance from a third plane not parallel to the intersection of the other two. Such a plane is, for simplicity, taken perpendicular to g and passing through some definite point of it. This plane is called, in Descriptive Geometry, a side plane and we shall denote it by S. The trace of S upon V is called the first auxiliary ground-line and is denoted by g,; that of S upon H, the second auxiliary ground-line, denoted by go. The distance of a point from S is denoted by x, and is called positive when the point is to the right-hand of S, and negative when to the lefthand. These three distances of the point, denoted by x, y, and z, taken with regard to their algebraic signs, are called the co-ordinates of the point. The planes from which they are measured, S, H, and V, are called the co-ordinate planes. Their intersections are called the co-ordinate axes; g, the x axis; gl, they axis; and g2, the z axis. The point common to the axes is called the origin of co-ordinates and is denoted by o. The side plane S is also used in Descriptive Geometry as a base of projection, but this use of it will be discussed later. In denoting the position of a point, the values of the co-ordinates are written in the order x,y, z, separated by commas, and enclosed in a parenthesis; thus, the point (2, -3, 5) is the point for which x=2, y= -3, and z=5. In the exercises called for in connection with co-ordinafto the student may lhcate the- pGint ovn g, but tned 10 I0 ~~~DEFINITIONS. not attempt to show the plane S, the auxiliary ground lines, or the side projections of. the points. i9.-Octants. The three planes, S, H, and V, divide all space into eight equal parts called octants. These are numbered similarly to the quadrants. The first four are those to the right-hand of S, and each one to theleft-hand comes fourth in order after the one immediately to the right of it. They are denoted 01, 02, etc. 12.-Give the octant to which each of the following-named pints belongs, and say how far each point is from each base of projection;- (I,-4,6), (3,7,-9), (-2,5,8), (-3,-5,7), (4,2,3), (o,-2,5), (3,0,6), (3,5,0), (-2,4,0), (4, —3,o), ( —2,o,-3), (-3, —6,o), (-6,o, —4), (5,o,-8), (-4.0,12), (3,0,0), (o,-7,o), (0,0,4), (o,6,o), (- 5,0,0), (o),O,- ), (o,o,o). 20.-When S is used as a base of projection, in.Descriptive Geometry, before drawings are executed upon it, it is revolved about g, so that the two vertical faces of 03 and of O8 comle together, as also do. those Of 02 and those of 03. CHAPTER 11. FUNDAMENTAL PROPOSITIONS. Certain fundamental propositions to which we shall have occasion to refer later are here stated and briefly elucidated where necessary. The student should note them carefully. 21. —Proposition I. If q be any point, PqH is parallel to V, and Pqv is parallel to H. (See fig. 2.) 22. —Cor. I. The distance of q' from g equals the distance of q from V, and the distance of q" from g equals the distance of q from H. (See fig. 2.) 23.-Cor. II. If q lies in V, q' lies in g, and conversely. So also if q lies in H, q" lies in g, and conversely. 24.-Cor. ll. If a line is parallel to either base of projection, its projection upon the other is parallel to g, and conversely. (See fig. 6.) 25.-Cor. IV. If any plane figure is parallel to either base of projection, its projection upon the other is a straight line parallel to g, and conversely. (See fig. 5.) 12 FUNDAMENTAL PROPOSITIONS. 13.-Say in which quadrant y7 is and at what distances from H and V if q' is 3 inches above g-and q" is 2 inches above g q' is 2 inches above g and q" is 22 inches above g q' is 2 inches below g and q" is 5 inches above g q' is 7 inches below g and q" is 4 inches below g q' is IYx inches above g and q" is 33 inches below g q' is on g and q" is 2 inches below g q' is on g and q" is 4 inches above g q' is 6 inches above g and q" is on g q' is 8 inches below g and q" is on g. 14.-Give the location on the drawing sheet of each projection of each point in ex. 13, taking first point I34 inches to left of S, next I inch to left, next X inch to left, etc. 26.-Proposition II. If q be any point, the line joining q' and q" is perpendicular to g. Conversely, if two given points, one in H and one in V, are on a line perpendicular to g, there is some point of which the given points are respectively the horizontal and vertical projections. First, —Pass a plane through PqH and PqV. It is I to H and to V,.'. to g. Therefore its traces on H and V are both I to g. But when H and V are revolved into their customary positions for drawing, these two traces form a single line I to g and passing through q' and q". Second,-If u be a point of H and w a point of V, and the line uw be I to g, intersecting it (when H and V are in position for drawing) at 1, the two lines tu and tw, when H and V are restored to their ideal positions, determine a plane I to g,.'. I to H'and to V. Lines erected at u and w I to H and to V respectively will lie in the plane utw, and being I to intersecting planes will be non II. They will.. meet, and their point of intersection is the point of which they are the projectors, and of which u and w are respectively the horizontal and vertical projections. (See fig. 2.) 27.-Proposition III. If the point q is on the line FUNDAMENTAL PROPOSITIONS. 13 a, q' is on a', and q" is on a". Conversely, for every point on any projection of a line, there is a corresponding point on the line. (See fig. 2.) 28.-Proposition IV. The projection of a straight line is always determined by the projections of any two of its points: for when a straight line is J to the base of projection, its projection is a point, and in all other cases it is a straight line. 29 —Proposition V. If both projections of any line are straight, the line is straight: for in this case the two projectors are both plane, and the intersection of the projectors of any line determines the line. The student may determine whether or not there is any case in which this proposition fails. 30.-Proposition VI. If one of the projections of a line is perpendicular to g, the line lies in a plane perpendicular to g, and its projections are therefore both perpendicular to g and meet it at the same point. (See fig. 3.) 3I.-Cor. If at two different points on g per. pendiculars be erected, one in H, the other in V, no line can be found of which they are the projections. 32.-Proposition VII. Projections of parallel lines upon the same plane or upon parallel planes are parallel.. (See fig. 4.) 33. —Proposition V111. If a plane is perpendicular to either base of projection, its trace upon the other is perpendicular to g, and conversely. (See fig. 4.) 14 FUNDAMENTAL PROPOSITIONS. 34.-Proposition IX. The two traces of any plane upon H and V (called respectively the horizontal and vertical traces of the plane) meet g at the trace of the plane upon g. (See fig. 4.) 35. —Cor. I. If a plane is parallel to g, its traces are parallel to g, and conversely. 36.-Cor. II. If the line a lies in the plane A, the trace of the plane A upon any other plane B passes through the trace of the line a upon the plane B. (See fig. 6.) 37.-Proposition X. Traces of parallel planes upon the same plane are parallel. (See fig. 6.) 38.-Cor. I. If the horizontal traces of two planes are parallel and their vertical traces are parallel, the planes are parallel, provided the traces are not parallel to g. (See fig. 6.) 39.-Cor. II. If the horizontal projections of two lines are parallel and their vertical projections are parallel, the lines are parallel. (See fig. 4.) The student may determine whether or not there is any case in which this corollary fails. 40.-Proposition XI. If a plane and a line are perpendicular, the trace of the plane and the projection of the line upon any second plane are perpendicular. Call the plane A, and the line call b, and' the plane upon which the projection takes place call H. Then PbH.is I to A FUNDAMENTAL PROPOSITIONS. 15 and to H,.. to their intersection TAH. '. TAH is I to PbH, it is I to the trace of PbH upon H; but this trace is b',.'. b and TAH are I when b is I to A. 41.-Cor. If one of the two sides of a right angle is parallel to a plane, the projection of that right angle upon that plane is itself a right angle. Call the two lines a and b, and suppose them to meet at the point q and to include a right Z. Then if the line a is II to H, the lines a' and b' include a right Z. For, through q pass a plane I to a; this plane will contain the line b and will be I to H,.. it is I to a which is II to H. It will.'. be PbH, and its trace upon H will.'. be b', and this trace, by the preceding proposition will be I to a'. 42.-Proposition XII. Any plane figure is equal to its projection upon any plane to which the figure is parallel. (See fig. 5.) 43.-Proposition XIII. The projections (upon the same plane) of any two homothetic figures are homothetic, and their center of homothesy is the projection of the center of homothesy of the two given figures. (See fig. 8, left-hand portion.) 44.-Proposition XIV. If a straight line a is tangent to a curve k at the point q, then a' is tangent to k' at the point q'. (See fig. 8, right-hand portion.) 45.-Cor. If a curve k is tangent to another curve at the point q, then k' is tangent to ' at q'. CHAPTER III FUNDAMENTAL OPERATIONS. The various fundamental operations of Descriptive Geometry will be best understood by carefully considering their application in certain simple and elementary problems. These will now be stated and discussed. 46.-Problem I. Of a line a, given a' and a", and of q, a point on a, given q', to find q". Analysis q" is on a", and the line joining q' and q", is I to g..'. q" is where a I from q' upon g crosses a". Likewise when q" is given to find q'. (See fig 2.) 47.-Digression upon the use of S. It is evident that when a in problem I is I to g, the method there developed is inapplicable,.' a',and a" are both I to g. It becomes necessary, then, to make use of some other base of projection whose ground line is such that a is not I to it. For this purpose we usually employ S as the new plane, and use it in conjunction with both H and V. Projections upon S are denoted by three accents; thus, the projection of a upon S is denoted by a"'. Evidently q" is as far from g, as q is from V, and.'. as q' is from g, while q"' is as far from g, as q is from H, and.. as q" is from g. Given q' and q" FUNDAMENTAL OPERATIONS. I7 then, we may easily construct q"' when g, and g2 are known. The student may readily trace out by means of the perspective figure in the left-hand part of fig. 3, the use of the side plane in the solution of problem I. 48. —Problem II. Given a' and a" of a line a, to find TaH and TaV. Analysis. TaH, being a point of a, is vertically projected in a", and being a point of H is vertically projected in g..'. it is vertically projected where a" meets g. TaH, being a point in H, is its own horizontal projection, and must.'. be on a'. Since (TaH)" is known, TaH is readily found by means of the method developed under problem I. Likewise for Tav, and for TaS. In the case where a is I to g, S would be used. (See figs. 2 and 3.) 49.-Problem III. Of a line a and a point q not on a, given a', a", q', and q"; to draw through q a line b which shall be parallel to a. Analysis. b' must pass through q' and be II to a'; likewise for b", q", and a". 50.-Problem IV. Of a sect a, given a' and a", to find the length of a, and the angle,aa which a makes with a'. Analysis. If a' is II to g, a is 11 to V and a" shows the true length of a, and the Z which a" makes with g = aa. If a' is not II to g, let mz and n (see fig. 7) denote the ends of a. Suppose through one end of-a, m say, a line to be drawn II to a' and meeting PznH in q. Then the Znzq = aa, and the hypotenuse mn of the rt. A thus formed is a. Now suppose this rt. Amqn to be revolved about PnH as an axis until II to V. m will trace a circular,, centered at q, 11 to H, and terminating at a point (call it t) equidistant with q from V. (/mt)' will.'. be a circular., centered at q', having a radius = to a', and terminating at t', q't' being II to g; and (mnt)" will be a straight line II to g. t" may thus be readily found, and t"n" gives the true length of a FUNDAMENTAL OPERATIONS. while Zm"z'"n" = aa. Likewise to find /3a, the Z which a makes with a"; and 7a, the Z which a makes with a'". Or if desired the rt. A mzqn may be revolved on mzq as axis until 11 to H, and its horizontal projection then used. For this method it should be noticed that. mq is II to a', it is II to H, and.'. imz"g" is II to g; also that '. PzH is II to V, n = q"n". The method here indicated is usually less convenient in practice, however, than is the one discussed in the preceding paragraph. 51. Problem V. To draw through a given point q a horizontal line b to meet a given line a which does not contain q. Analysis. Suppose b meets a in r. Then. qr is II to H and contains q7, q"r" is II to g and meets a" at r"; r' being on a' is then readily determined and q'r' may readily be drawn. Likewise to draw through q a line which shall be II to V and shall meet a. 52.-Problem VI. To find the angle between two given lines. Analysis. If the two given lines do not meet, draw through any selected point lines II to the given lines. The lines so drawn will contain an Z - to the Z sought. Problem VI is thus reduced to the problem of finding the Z between two intersecting lines. Call the intersecting lines a and b, and let q denote their point of intersection. Through any point, i say, of a draw a line II to H to meet b at (say) the point t. Then revolve the Artq upon rt as axis until it is II to H. letting w denote the revolved position of q. Then (Z rwt)', which - Z rwt which = Zrqt, = the Z sought. But (Zrzet)' -= r'Ze't'. To determine w', notice that if from q a I be dropped upon rt meeting it at (say) /, q'I' will be I to r't'. q"/" may then readily be drawn and the length of ql, and.. of wl, be readily determined. w1'' is I to r't' and = zl. (See fig. 9.) It should also be noticed that w' is distant from r' by the true length of rq and from t' by the true length of tq; also that za' is on the 1I I'q' and distant from r' by the true length of rq. Of the three methods thus indicated, the last is usually the most convenient in practice. FUNDAMENTAL OPERATIONS. I9 It is to be particularly noticed that this construction also gives Z qrt and Z qtr, so that it is readily applicable to the case where one of the two intersecting lines is 11 to H. It thus enables us to find very readily TAHV. In this connection it may be well to note that if either trace of the plane A is I to g, -AHV is a rt Z. When more convenient, rt may be drawn II to V and the A rtq be revolved II to V about rt as axis. The method just discussed fails when both given lines are II to S. In this case, Zr"'q"' "'.= Z -qt. 53.-Problem VII. To find the distance from a given point q to a given line a. Analysis. From q draw a line II to H, to meet a in (say) r. About qr as axis, revolve a until it is II to H. he horizontal projection of the figure in its revolved position will show every thing in its full size, and the distance of q from a may then readily be determined. (See fig. 9.) The problem to find the distance between two parallel lines is simply a case under Problem VII, '.' all points on one of two 11, are equidistant from the other. 54.-Problem VIII. To find the plane C of two co-planar lines a and b. Analysis. TCH must contain TaH and TbH; similarly for TcV. If unable to find the traces of the given lines, use auxiliary lines lying in their plane. Usually the most convenient auxiliary lines are those II to H and those II to V. (See fig. io.) To find the plane of a line and a point outside the line, and to find that of three independent points, are easy variations of Problem V1II; so, too, to find one projection of a line lying in a given plane when the other projection is given, and to find one projection of a point lying in a given plane when the other projection of it is given. 55.-Problem IX. To pass a plane A through one given line b and parallel to another given line c. 20 FUNDAMENTAL OPERATIONS. Analysis. Through any point q of b draw an auxiliary line f 11 to c. The plane of b andf is A. Notice that if either b or c is II to H, it is II to TAH. To pass a plane through a point II to two given lines is an easy modification of Problem IX. So, also, is it to pass through a given point a plane 11 to a given plane for the traces of the given plane are lines to which the required plane must be II. Another simple modification is the problem to pass through a given point a plane I to a given line, for the traces of the required plane must be I to the corresponding projections of the given line. 56.-Problem X. To find the trace of one given plane A upon another given plane B. Analysis. '.' TAB is a straight line, it will be sufficient to find two points of it. One such point is evidently where TAH crosses TBH,and another where TAV crosses TBV. (See fig. I.) In the case where the traces of the given planes upon one of the bases of projection do not meet within the limits of the drawing, it becomes necessary to resort to other methods than the one above indicated. In this case we customarily use auxiliary planes 11 to the bases of projection. These of course cut out of the given planes lines II to their traces upon'the bases to which the auxiliary planes are II, and the intersections of these auxiliary lines give points upon the line sought. When no other method suffices, we resort to a "reduction of scale"; i. e.,to the construction of a miniature figure similar to the given one, and after performing the desired operations on this, enlarging the results in the proper ratio. Figures 12 and I3 will illustrate these methods. Figures I4-I6 illustrate two useful methods of drawing through a given point a line to concur with two given lines whose point of concurrence lies outside the limits of the drawing, a problem which sometimes arises in such problems as the one just discussed. 57.-Problem XI. To find the trace of a given line a upon a given plane B. Analysis. If through a any auxiliary plane C be passed, TCB FUNDAMENTAL OPERATIONS. 2I will intersect a and will lie wholly in B. Therefore the point where a crosses TcB is the TaB. Customarily we take for C one of the projectors of a. (See fig. 7.) 58.-Problem XII. To find the angle between a given line a and a given plane B. Analysis. Through any point of a pass an auxiliary line c I to B. Then Zac = go~ - ZBa. (See fig. 17.) 59.-Problem XIII. To find the projection of a given sect a upon a given plane B. Analysis. The projection of a upon B is the straight line joining the projections of its extremities on B. These may be found by drawing from its extremities I s to B, and finding their traces upon B. 6o.-Problem XIV. To find the plane angle of the diedral between two planes. Analysis. The Z sought is the z between the traces of the given planes upon any auxiliary plane I to their intersection. Or, two lines from any point within the diedral I to its two faces contain an Z supplementary to the Z sought. (See figs. I8 and I9.) 6I. Problem XV. To find the distance from a given point q to a given plane B. Analysis. The distance of q from B = the distance of q from ToB, C being any auxiliary plane containing q and I to B. In the application of the method here indicated it is customary to take C I to TBH or to TBV. (See fig. 20.) The distance sought also = the distance from q to its projection on B. (See fig. 21.) The problem to find the distance between II planes is a simple modification of Problem XV. 62.-Problem XVI. To draw the shortest line joining two given windschief lines, i. e., two non-planar lines. 22 FUNDAMENTAL OPERATIONS. Analysis. Call the two windschief lines a and b, and suppose that the shortest line joining them meets a in r and b in t. Then rt must be I both to ar and to b, for if it were oblique to one of them, b say, then a I from r upon b would be shorter than r-t. Suppose a plane H passed II to a and b and another plane V be taken I to H; then a' and b' cannot be II,.' a" and b" are II, both being 11 to g, '. a anJ b are both II to H and are not II to each other. At the point where a' meets b,' m say, erect a I to H. Since this i lies in both PaH and PbH, it will cross both a and b. Since it is I to a' which is II to a, and to b', which is II to b, the I to H erected at mz is I both to a and to b. Therefore that portion of it comprehended between a and b is the line sought.* Therefore, to solve Problem XVI, pass through one of the given lines an auxiliary plane parallel to the other, project that other upon it, and where its projection upon this plane crosses the first line erect a I to the plane and prolong it until it meets the second line. This will be the line sought. (See fig. 22.) 63. —Development. To develop means primarily to unwrap, or to unfold. The word is used in Decriptive Geometry to denote the spreading out flat of any surface capable of being so spread without distortion. It is evident that all surfaces whose parts are plane are capable of being developed after separation into their component plane parts, while those in which no straight line can be drawn are as evidently incapable of it; thus the surface of a prism and that of a pyramid are both developable while that of a sphere is not. Of those curved surfaces generated or generable by the motion of straight lines, those are developable in which consecu*That there is but one solution to this problem appears from the fact that if rt be prolonged it must be perpendicular to two lines drawn through its trace upon the plane H of the discussion 11 to a and b respectively, and.'. perpendicular to H,. lines drawn through any point of H 1 to a and b lie wholly in H. r't being perpendicular to H lies in both PaH and PbH, which being non-coincident can have but one commo'.- liie. FUNDAMENTAL OPERATIONS. 23 tive positions of the generator are either parallel or concurrent; the others are not; thus those of the cone and of the cylinder are developable, while the hyperboloid of revolution (that generated by revolving one windschief line about another as axis) is not. 64. — From the use of the term "develop" to denote the spreading or laying out flat of any surface, and the fact that this development is usually effected in H or V, the revolution of a plane into H or V about its horizontal or vertical trace as axis, is commonly called the development of the plane into H or V, and the position taken by any figure of the plane when the plane is so developed, we call the developed position of the figure. 65. —Intersection of Surfaces. The intersection of two planes has already been discussed in problem X. When it is desired to find the trace of a plane upon a curved surface, or of one curved surface upon another, we customarily proceed by passing auxiliary surfaces (usually planes) so as to cut both given surfaces. The points common to the traces of the auxiliary surface upon the given surfaces are points of the line sought. By the determination of a sufficient number of such points the line may be determined as accurately as may be desired. The problem of finding the trace of one surface upon another is thus reduced to that of finding the trace of a plane upon any surface. The mode of doing this depends entirely upon the nature of the surface. In the case of surfaces generated by straight lines the auxiliary planes 24 FUNDAMENTAL OPERATIONS. would usually best be taken so as to cut the surfaces along their rectilinear elements if possible; in the case of surfaces of revolution they would best be meridian planes, i. e., planes containing the axis of revolution; planes perpendicular to the axis are also frequently useful as they give circular traces which may readily be drawn, etc. A few instances of the general problem will be given among the exercises in the next chapter. CHAPTER IV. EXERCISES FOR DRAUGHTING PRACTICE. In the exercises below given, accuracy is the first essential, and neatness is the next. Without neatness there can be no great degree of accuracy. Great care should be taken to have each line show as nearly as possible by its character just what it is intended to represent, and literal symbols should be employed on the drawing only to denote what cannot be shown by the lines themselves. The student is particularly cautioned to bear continually in mind that all work in Descriptive Geometry is performed wholly by means of pro= jections. Whenever any figure, then, is said to be given, it is meant that the projections, traces, etc., of that figure are given; and whenever any figure is said to be required, it is meant that the projections, traces, etc., of that figure are required. I5.-Draw the three projections of each point given in ex. 12, using one-fourth of an inch as the unit of length, and denoting 26 EXERCISES. the different projections by the proper accents. i6,-(i) Given q' and q", find q"'I (iii) Given q" and q"', find q' I.Ofa line a oblique to g, given a' and a" and of a point q on a,given q' to find q"(i) When a crosses Q1 and q is in Q1; (ii) When a crosses Q2 and q is in Q3; (iii) When a crosses Q~, and q is in Q2; (iv) When a crosses Q, and q is in Q4. (V, vi, vii and viii,) same as above except a joins;1 and w (ii', li", w' and co" being given,) which are in a plane I to g and in (v) are in Q,, in (vi) are in Q2, etc. For these last four problems use S. 18.-Given it', n",5 andtf" to find LnAf. (I) ii in Q1,j in Q1; (i i) ii i nQ2,5 fi nQ2; (li) 77in Q3, fi in Q4; (iv) 77 in Q4,/5f in Q2. 19. -Given it', it", fi' and LlifiJ to find fi"; how many soluitions? 20 2IGIVEN FIND a ~alla, a"'., TaH, TaV, TaS, uaa, Otya, L~a, Lia, and L2a. 23 Remarks. In examples 20, 21, 22, and 23, a is oblique to S and crosses Qi, Q2, Q, and Q4, respectively. 24 ~ I GIVEN FIND 25 1 26 a',a",.. a"', TtaH, ToY, TaS,aa, 13a, ya, L~a, Li~a, and Lea. Remarks. In examples 24, 25, 26, and 27, a is parallel to S and crosses Qi, Q2, Q3, and Q4, respectively. 28..-Given a" and a,'", t6-find a,'. 29.-Given a', TaH, aa, find a". Get all possible solutions. 30.-Given a', TaH, ya, find a". Get all possible solutions. EXERCISES. 2 27 In examples 31 to 48 inclusive, get all possible solutions. GIVEN. FIND. REMARKS. 31.-Given a", TaV, LVa, find a'. 32.-a"' TaV,..a..... a', a". 33-61', TaS,/3a....a 34.-a"', TaS, L2a. -a', a" 36 —a"l, TaS, L2a.... a L.a',)a... a 38.-a", aa, L3a....a'. 39.-TaH, TaiS.... a', Cl 4o,-TaV, aa, O3a.... -a' a" 41.-TaH, aa, La.... a', a. 42.-TaH, /3a, L~a,... a', a 4 3 — aa, O3a L-3a, q' -a', a" 44.-aa, 3a, Lia, q".a', a" 46.-a' q",aa.....a 47.-a', q",y~a....a 4-aqLia....a. 49.-TAH, TAV....TAS, 50.-TAH, AAI-l....TAV. 5i.-TAV, AAS...I TAH. 52.-TAH, TFAHV... TAV. 53. TAtH, TAV, a'.. -" 54.-TAV, a', a". TAH. 5 5. -TAV, TAS, q. q.. 56. q', q", aa, Oa.... a', a". q is on a. q is on a. q is on a. q is on a. q is on a. AtH, etc., 7AHV, etc. [Solve also when A is I_ to S. TakeA acute in one instance, [obtuse in another. Take A acute in one instance, [obtuse in another. Take 7 acute in one instance, [obtuse in another. a is inA.- Take 7acute in one [instance, obtuse in another. a is in A.- Take a crossing Q2. q is inA. Take qinQ. q is in a. 57.-TAH, TAV, q'...qrHTAH1. q is in A. 58.-TAH, q', q"....TAV, qrHTAu-. q isin A. 28 28 ~~~EXERCISES. GIVEN. FIND. REMARKS. 59.-TV, q", AAV...TAH,qrVTAv. q isin A-,, 6o -TAH, q', qrHTA-1..TAV, q". qis in A. 6i.-TAV, q'I, qrVTAV..TAH, q". q is in A. 62.-TAV, q', TAHV...qrVTAV. q is in A. 63.-q', q", qrVTAV... JTAH, TAtV. q is in A. 64.-q", q1.HTA1t1, AAH.. AH, TAV. q is in A. 65.-q", qrV,.AAH....TAH, TAV. q is inA.4 66.-q', qr~HTAII, TAHV.TAH, TAV. q is in A. 67. -q', qrVTA V, AAV.. TAH, TAV. q is in A. 68.-TAH, TAV, a'.... arVTAV. a is in A. 69. —a' a",, b', b",...TCH, Tcv. a and b intersect, and [lie in C. 7o.-a', a", b', b".TcH, Tcv. a, and b are 11 and lie [in C. 71. -aaq', q".. T CH, T cv. a and q are in C, q is [not on a. 72.- a' a", b', 1".. TCH, TuV. a and b are in C, but TbHV are without the limi~ts of the drawing. 73.-q', q" 1'I", w', w"..TCH, TCV. q, I, and w are in C. 74.-q', q",1' "2'2" Find the true figure qtw without finding traces of plane of qiw. 75.-TAH, TAV, b', b".... ZbA. b pierces A. 76.-TAH, TAV, TBH-, TBV. TBA. 77.-TAH, TAV, b', b". TbA. b pierces A. 78.-TAH, TAV, q', q". -.distance from q to A. q is not in A. 79.-TAH, TAV, TBH, TBV.. AAB. 8o. -TAH, TAV, TAB, AAB. TBH, TBV. 8i.-TAH, TAV, TBH, TBV, q', Zal... -a', a". [a is in A, q is on a. 82.-TAH, TAV, TBH, TBV,q q', q" jaB. -a', a". q in neither A nor B; a goes through q and is IIto A. 83.-TAH, TAV, b', b", q', q"......TCH, Tc'v. q in neith*er A nor B; C contains q, is I to A, and IIto Ib. 84.-a', a", TBH, TBv, AcB..TCH, Tc'v.- C contains a. EXERCISES. 29 Triedrals. a, /, and y are face angles; A, B, and r are the diedrals opposite a, /3, and y respectively; and a, b, and c are the edges of the diedrals A, B, and r respectively. w is the vertex. In each exercise find the other parts, and where sums or differences are given, find the component parts also. The student should notice the analogy between the. relations holding among the parts of triedrals, and those among the corresponding parts of trigons. GIVEN. GIVEN. GIVEN. 85.-a, 3, y 89.-a, B, /3-+ 93.-a, /3, A+B 86.-a /3, r 9o.-a, B, / —y 94. —a, /3, A-B 87. —, /3, A 9I.-A, B, r 95 -a, B, A-r 88. —a, B, r 92.-a, A, B. 96.-Given an oblique prism with a scalene trigonic base, edges oblique to H and to V. Find the projections of the trace of the prism upon a plane oblique to H, to V, and to the edges of the prism, also the true figure of the trace. 97.-Given an irregular tetragonic pyramid. Find the projections, also the true figure, of the trace of the pyramid on a secant plane oblique to H and to V. 98.-Develop the lateral surface of the truncated prism between H and the secant plane in problem 96 above; also that of the truncated pyramid between H and the secant plane in problem 97. 99.-An oblique trigonic prism cuts into but does not pierce an oblique tetragonic prism. Find the trace of one on the other. Develop the surface of each prism, so as to find patterns by which the surfaces might be cut out of sheet-metal or card-board. Cut them out of card-board and see whether or not they can be made to fit properly. Ioo.-Having given the three lateral edges of a trigonic pyramid, no two of them being equal, construct the pyramid so that the face angles shall have given sizes. ioi.-Draw a trigonic pyramid whose:face angles at the vertex shall have given sizes, no two being equal, and whose lateral faces shall make equal diedrals with its base. io2.-Draw a trigonic pyramid whose lateral edges shall have 30 EXERCISES. given lengths, no two being equal, and whose lateral faces shall make equal diedrals with its base. How many solutions? 103.-Given a tetragonic pyramid having a trapezium for base, to cut it by such a plane that the section shall be a parallelogram. 104 — Having given three of the lateral edges, no two being equal, of a tetragonic pyramid, draw the pyramid so that the face angles at its vertex shall be equal and the diedrals between consecutive lateral faces shall be equal. 105.-Given a pentagonic pyramid with its base in the rear part of H, and its vertex in Q1. Find the shadow cast on H, also that cast on V, by rays of light emanating from a given point in Q1, nearer to V than is the vertex of the pyramid. and above and to the left of it. io6.-The same as in problem Io5, except that the rays of light are parallel, and their direction is given. Io7.-A cylinder whose base is an irregular curve in H, and whose elements are oblique to H, to V, and to S, is cut by a plane perpendicular to its elements. Find the projections, also the true figure, of the trace of the plane on the cylinder. Develop the curved surface of the truncated cylinder between H and the secant plane. o.08.-An oblique circular cone whose base is in H and whose vertex is nearer V than is the center of its base, is cut by a plane oblique to H and to V. Find the projections, also the true figure of the trace of the plane on the cone. Develop the curved surface of the truncated cone between H and the secant plane. lo9.-Find the shadow of a cone upon a cylinder. The base of the cone is irregular and lies in H; the cylinder has a circular right section and its axis is parallel to H and oblique to V; the rays of light are parallel, and the shadow of the apex of the cone falls on the curved surface of the cylinder. iio.-A cone of irregular base pierces a cylinder of irregular base. Find the projections of the trace of the cone on the cylinder. iii.-A given oblique plane cuts a given sphere. Find the projections of the trace of the plane on the sphere, also the true figure of the trace. Find also the shadow of the sphere on the plane, the rays of light emanating from a point on the same side EXERCISES. 31 of the plane as is the center of the sphere. 112.-Find the projections of the trace of a sphere upon a right circular cone. The apex of the cone is just without the sphere, and the trace of the conic surface upon the spheric surface consists of two closed non-circular curves. Develop the surface of the cone and find the developed trace. II3.-Two cylinders of circular right section are respectively two inches and three and one-half inches in diameter, and their axes are inclined to each other at an angle of 30~, and where nearest are one-half inch apart. Find the projections of the trace of one upon the other, and develop the surface of each, showing the developed trace on each. II4.-Find the projections and the true figure of the trace of an oblique plane upon an hyperboloid of revolution. The hyperboloid has its axis vertical and is obtained by revolving about the axis a straight line distant from it one inch and making with it an angle of 45~. 115.-Given q', q", aAH and AAV, to draw A through q. 32 ABBREVIATIONS. ABBREVIATIONS. The student will find the following list of abbreviations useful in taking notes in geometry. They are used in the notes which follow. adj adjacent ctr center alt altitude cv apl apply cyl argt argument cylc aux auxiliary d bet between A bs bisect dg bsr bisector dm cdr consider dir cf compare o0- dnt '" with dp cmpt complement ds cmpv comparative dscs cmpn comparison dst cncd coincide cncr concur dt cncv conceive dvp cnst consist el cntg contiguous eql cntn contain es cnv convenient exp corn common f crs cross fig crsp correspond gm cstr construct curve cylinder cylindric draw diedral diagonal diameter direct denote drop desire discuss distant or distance determine develop element equilateral essential explement find figure g" given h horizontal hpt hypotenuse ht hight hyp hypothesis inc increment incd include incl incline int interior isn intersection isos isosceles ist intersect k know 1 long or length lc locate or locus lim limit It lateral max maximum nin minimum mntn mention mthd method geometry or mtl mutual geometric n notice ABBREVIATIONS. 33 nb notice well r revolve slv solve neg negative rad radius or sm small ny necessary radiate sn section o origin rd reduce sp suppose obl oblique rl relation or sph sphere ol original relative spr superior opp opposite rm remain sstn sustain p page rq require sup supple ment pb problem rsp respective surf surface pcd proceed rt right t take pgn polygon rvrs reverse tan tangent pj project ry readily thr through pl plane s side tr trace pm prism sec secant trnc truncated pos positive seg segment u usual prin principle sgst suggest v vertical ps pass sh show vx vertex psn position sim similar wh which, pt point sl slant whose or ptn portion slh slant hight where pyr pyramid sin solution Suffixes to these words will be indicated by the use of the last letter of the suffix as a superior affixed to the abbreviation; thus, d'1 indicates drawn, —pls, planes, etc. 34 ~~~EXERCISES. 34 NOTES ON PROBLEMS. 20-27.-a' and a" being gn~, the s pjfls of the ends of a may ry be d',.,. a"'. Pb II (art. 48, p 17) dsc8 the mode of ig TaH and TaV. A sim argt apis to Tas. Pb IV (art. 50, p 17) will sgst the mode of f9 aa, ~a, and ya,-also L~a, Lla, and L~a. 28.-Is sim to pbs' 20-27, two pjlls being g'5 to f the third. 29.-The ptn of a' bet TaH and g is the h leg of a rt A, wh v leg is the I fromn TaV to g. aa is the acute Z in this rt A adj to the h leg. 30.-The I from TaH to g2 is one leg of a rt A; ya is the acute Z opp this leg. The other lleg- shows the dst from (TaH)'" to Tas. 3i.-The ptn of a" bet g and g1 is one leg of a rt A of wh L-2a — the hpt, and the adj acute Z '- ~a. 32.-The dst of TaV from g1 -onie leg of a rt A and ya the opp acute Z. This dets the 1 of the other lego and.-. the psn of TaS. 33.-D any aux line ii to the rqd line, and then thr TaS ps the rqd line 11 to the aux line. 34.-The ptn of a'" bet TaS and g2 is one leg of a rt A of wh L~a is the hpt. The 1 of the other leg - the dst of TaH fromn g2. 35.-The I_ from TaV to g is one leg of a rt A of w5h L~a is the hpt and the other leg- =A he ptn of a' bet TaH ani g. 36. -L~a =the hpt and the, I from TaS to g2 is one le o rt A wh other leg is the ptn of a' bet g2 and TraH; nb that TaH is Vy pjd wh a" meets g. 37.-D any aux- linei I to rqct line and f aa. aa with L-2a Will enable TaH to be lCd. Pb 37 is thus rdct to pb 29. 38. - Pb 38 is esy sim to pb 3inaalysis. 39.-Two pts on the rq11 line are g'5. 40.-TaV with aa gs L~a. L~,a with O3a gs the 1 of the ptn of a" bet Tav and g. a" may then be d, then a'. See fig. 23. EXERCISES. 3 35 4i.-L~a and aa g 1 of ptn of a' bet Tall and g. a' may then be d". a' and L8a being k11, a" may ry be d". 42.-L2a is the hpt of a rt A and O3a the Z adj to the leg wh the ptn of a" bet g and gi. a" may then be d11. Tall being g'9 a' is ry d11. -See fig. 24. 43.-D b, an aux line, 11to a a'nd put TbH at q', so that a' will cncd with b'. bf and b" may be d'by pb 400or pb 4'. L~a and /3a dt the dst of Tall from g; Tall may then be ry lcd on b'..Then d a II to b thr Tall. See fig. 25. 44.-D b 11 to a putting TbV at q", thus getting a". Lia, and /3a dt the dst of TaS from gi. Lc TaS and thr it d a HI to b5. See fig. 26. 45. L~a and Oa, dt the dst of Tall from g. Lza and aa dt the 1 of the ptn of a' bet Tall and g while L~a and aa dt the I of that ptn of a' bet Tall and g2. 46.-Dt q' then r a on Pqll H to V and dt dst of TaV from g, then r a back to ol psn and d a". 47.-DtL q', then r a on PqS 11to H, then dt TaS and d a thr TaS and q. 48.-L~a and the ptn of a' bet g and g2 dt aa.. a may then be dn~ by pb 46. 49.-TAS and TAV meet on g1. AAll may be fd by using an aux PI BI to TAll and f9 the Z bet TBA and TBll. For 7AllV, etc., see pb VI, art 52, p i8. 50.-Rvrs the pcde of pb 49. 5i. -Rvrs the pcde of pb 49. 52.-Rvrs the pcde of pb 49. 53.-TaV is a pt of TAV. 54.-Rvrs the pcde of pb 53. 55.-First f TAV, then thr q d an aux line 1' lying in A. q' is onb. F b", then q". 56- so that ab =aa, and Pb = a; then d a HI to 6 thr q. 57.-Thr q pass an aux p1 B I to TAll. R the rt A wh ~s are T.B llVA into H on TBll as axis, ng that q is on T-BA and that Pqll will r II to TAll. The dst of q from TAll is thus g11 and the psn of q llTAIj may ry be dtd. See fig. 27. EXE~RCISES. 58.-F TAV by psP thr q any aux line b lyving in A. This will rd the ph to pb 57'. qHTAI{ may b e fdtwithout first f9 TAV, by dpg from q a Ion TAH, and fg its 1 by pb IV, art 50, p i9. See fig. 28. 59.-T~his pb is sirn to pbs 50 and 57. 6o.-This ph is sim to pbs 57 and 58. 6i.-q" is on a line thr q' I to g, and on one thr qr VTA\T to TAV. 62.-F TAtH by rvrsg pb 49. Then f q" and n that a line thr q II to TAH will, when rd' into V on TAV as axis make with TAV an Z =to CrAHV. 63.-TAV is I to the line thr q" and qrvTAVT. Sp I to dnt the pt wh TiV crss this line. Then the A qrytq is isos and has its vx at 1. Its base is the hpt of the rt Aqryqq". R this A into V on q~~vq' as axis, f 1, and then d TAV, after wh TAH may ry be d11. 64.-The Z q qr11 q' is Y2 A AH. 65.-The dir15 of TAv is kil (see ph 63). T PB an aux P1 IIto A TBV is II to TAV and A BH AAH. D TBV, then with that and ABH d TBH, then f ABV. Kg AAV, the ph becomes sim to pb 64. 66.-The dir15I of TAH is lk" (see ph 63). T B and ABH (wh - LXAH) as in ph 65 and then sly as in ph 64. 67.-I n the rt A qqrIq', the Z qqr q " is Y2 AAV and the leg qq"= the from q' to g. The leg- q"qr may then be dtt", after wh q" may he lc"1, and the ph rd"t to ph 63. 68.-F the rd psns of any two pts, on a. TaH and TaV are uy as cnv as any. 69.-TaH is a pt of TCH, etc. 7o.-Simn to ph 69. 71.-Thr q d an aux line 11 to a, or else one meeting- a. This will rd ph 71 to ph 70 Or ph 69). If the aux line meets a it is uy most cnv to t it I Ito H or to V. 72.-Use an aux line crsg a and b, or crsg one and 11 to the other. 3-Use any two CO-plr lines carrying the three Pt-,. EXERCISES 3 37 74.-Thr one of the three Pt' d an aux line 11 to H or to V and lying in the P1 of the fig. Use this line as axis and r the fig I Ito H or to V. 75. Thr any pt of b d an aux line c wh shall be I to A. The cmpt of Z bc is the Z bA. 76.-See pb X, art 56, p 20. 77. —See pb XI, art 57, p 20. 78. -See pb XV, art 61, p 21. 79.-See pb XIV', art 6o, P 21. 8o.-Rvrs pb 79. 8i. —F the dst of g from B. Kg this dst and the Z aB, the dst of q fromn Ta-B (wh, nb, is a pt of TAB) may ry he~ dt"1, then TaB, then a may be d'-1. 82. D any aux line b lying in A and making with B the ds' Z aB. Then thr qd a to b. 8 3.-Thr q d cl II to b, and c2 __ to A. C is the P1 of c1 and C2. 84.-From any pt of a (exce-pt TaB) dp a I to B and dt its I and its foot in B. Its I will with ACB dt the dst from its foot to TCB. TCB also goes thr TaB. TcB may then be- d11 and, it and a will be two lines of C. On the Sin of Triedrals. In sly5' triedrals we uy cnicv the tiriedral so Iccl that onie of the faces shall lie either in H or in V/, and one of the edges be Ito g, thereby much facilitating the sin. Thus sp (/ to lie in H, and b to be I to g, w being in the front part of H. Then the face y is I to V and T)-v makes with g the p1 Z of the A B, while A~JH I'F. A is then the only A to be f1 as the A bet tw plS, ~both oblique to g. See fig. 29 for a triedral thus g' and slv~'. The student may find a cdr11 of the crspg cases in sphcAs useful in aiding his imagination. 85.T anywh in front part of H. D b I to G, lay off y, on 38 EXERCISES. (say) the left-hand s of b, a on the opp, and P beyond a. A brief cdrn of the mode of slvg a triedral as sh" in fig. 29 will sgst the rmg pcde. 86.-Lay off p in H with c I to g and adHc. Since r is g", vTa may ry be d" and vTb be fd. The rest of the pb is ry slvd. 87.-Lay off P as in pb 86. a is hTy, and having a and AyH (= A), vTy may ry be dn. TbV is a pt of vTy and may ry be fd by the mthd shn in fig. 29. Dt in what cases there are two slns. 88.-Lay off a in H with b I to g; hT/3 and hTy being thus g", together with the values of B and F, the triedral may ry be dn. 89.-Lay off /3+y in H with b I to g; cstr B and put a in psn, thus fg vTc. F also the dpd psn of vTc when / is dpd into H. This pt and the true psn of vTc are eqdist from Tag. 9o. -EY sim to pb 89. 91.-Sp the edge a to be tn I to g, and /3 to lie in H. vTy may ry be dn. If a sph be dn about any pt of a as ctr and two cones of rn be cstrd, one with its base in y and its base Z = to B and the other with its base in p and its base Z = to r, a pl tan to these two cones will cntn a or be II to a. If the ctr of this sph be t" at vTa, the v trs of these aux cones may ry be d", also the h tr of the cone wh base is in P. The v tr of the tan pl psS thr the apices of these cones, the h tr is tan to the base lying in P. 92.-Lay off a in H with the edge b I to g, and cstr Tav so that when the triedral is completed y shall lie in H. F vTc, and kg thus one pt in vT/3 and one in hT/3, also A/3H (= A,) T/3 and T/H may ry be d'. 93.-Cncv the triedral in psn with y in H and a I to g. R a into the pl of P on c as axis so as to get a+/3, then r a+- into H on a as axis, rg so that a+/3 shall cover y. The 1 of c will then be shn and the dst from vTc to b may ry bedt'. Sp a I dpd from vTc upon b when the triedral is in psn, and rd into V about hP (vTc) as axis, and so as to lie on the opp s of this pjr from vT/3. It and vT/ will then form with g a A wh int base Zs on g will be rspY B and A, and whose ext Z at vTc will.'. = A+B. By cstrg this A, A and B may ry be fd and the cstrn pcdd with by the u mthd. See fig. 30. 94.-The first ptn of, the pcdg may be as in pb 93. Then nb that if of a A EFG two sS e and f be g" and the Z (E-F), EXERCISES. 39 the Zs E and F may be fd by cstrg a A in wh two ss shall be rspy = to e and f and the incdd Z -to (E-F). The ext Z adj tofwill then = E, and the int Z opp f will = F. A and B being fd by this mthd, the cstrn for pb 94 may pcd by the u mthd. 95.-If thr the edge c a pl be ps" making with P the AA, this pl, the face /, and the pl in wh y lies will form an isos triedral of wh p will be the base face. The bsr of the A bet the aux pl and the pl in wh y lies will be I to /, and the A bet the aux pi and the face a - A-r. The pb may be most ry slyv by tg a ifi H with c I to g. 96.-This is merely a pb in fg the trs of gl lines upon a g11 pl, and then fg the true fig of the pgn of wh these trs are the vxS. 97.-Sim to pb 96. See fig. 31. 98.-The dvpt will cnst in fg the true figs of the various It faces of the trnc figs and then unfolding them into a flat surf, keeping their proper edges cntg. See fig 3I for the dvpt of the It surface of the trnc pyr. 99.-Sim to pbs 96, 97, and 98. Ioo.-Lay out the It surf of the pyr in its dvpd form and thus find the sizes of the basal edges. F wh the _1s from the apex of the pyr upon the basal edges meet these edges, also the 1S of these I. Kg these and the psns their pjns upon the base will t, the alt of the pyr and the pjnS of its apex are ry dt". ioi.-The faces in order to make As with the base must have = slh", and the pyr must.' be circumscriptible about a cone of rn. Any It edge then must make - /s with the el of tanY of the two faces in which this edge lies. A brief cdr11 will sh how to put these els in psn in the dvld It surf, and then how to dt the 1 of the It edges after the slh has been chosen. 102.-Sim to pb ioI. Io3.-Cdr the rl wh opp ss of the 7 must bear to the com line of the two faces in wh they lie. Io4.-Cdr the rls sstnd by the dg pls of the pyr to each other. Io5.-The shadow will be dtd by the trs of pls dtd by the source of light and the two It edges of the pyr which will cause the A bet these pls to be a max. io6.-Sim to pb lo5. EXERCISES. I07.-A cyl being merely the lim of a pm, the prins made use of in pb 96 will apply here. See fig. 32. Io8.-Pb Io8 bears the same rl to pb 97 as does pb 107 to pb 96. Nb that in dvpg, if the sec pi had not been I to the els of the cyl, we should have had to use an aux pi I to the el and dvp the two trnc cyls thus obtained. Io9.-Same cdr"n as in pb Io5. See fig. 33. iio.-Aux pls cncrg in a line thr the apex of the cone 11 to the els of the cyl will have st trs on both cv'd surfs. The isns of these trs will be pt8 of the tr dsd. III.-From cdrns of elY geom the tr is k" to be a 0 wh ctr is the pjn of the ctr of the sp upon the sec pl, and wh rad is ry dt"' when the 1 of this pjr is cfd with the rad of the sph. By rg the sec pi into H upon its h tr as axis, this 0 may ry be dn and then rd back to its ol psn. Or aux pis cncrg in any cnv diam of the sph, say the v diam, my be used and the isns of their trs on sph and sec pi will be pts of the trace dsd. Or aux pls II to H or to V may be used. This last mntn' mthd is frequently very cnv. The shadow will be dt1 by the tr of the sph on the sec pi and that of the cone enveloping the sph and having its vx at the source of light. II2.-Use aux pis I to the axis of the cone. Had the cone been obl it would have been best to use pls cncrg in the line psg thr the apex of the cone and the ctr of the sph. II3.-Use aux pis 11 to the axes of the two cyls. See fig. 34. II4.-Use aux plS I to the axis of the hyperboloid. The hyperboloid is most cnvy t1 so as to have its axis I to H or to V. See fig. 35 in wh the axis is t1 I to H. II5.-The dsdDp A must be tan to two cones of rn having their apices at q, the one having its axis I to H and its base Z - AAH, the other having its axis I to V and its base Z = AAV. ' the apices of these two cones cncd, in order to d this tan pl some line must be fd tan to both cones. This may be done by cutting them both by some aux pi not psg thr the apex and then tg a line tan to the trs of this aux pi on the two aux cones. Dt how many sins. CHAPTER V. PERSPECTIVE. 66.-By the perspective * of any figure is meant the graphical representation of that figure upon any surface as it would appear when looked at through that surface treated as transparent. Theoretically it differs then only from the centric projection of the figure in that it is a combination of the two centric projections got from the two eyes of the observer as centers of projection instead of being a single centric projection. Practically, however, the distance between the two eyes of the observer is treated as negligible and a single centric projection is used.t A very little practice enables the observer to make this single projection serve equally as well as would a stereoscopic projection. 67.-Parallel projection, to which we have hitherto almost entirely confined our attention, being but a special and limited case arising under centric projection, it is Perspective is derived from the Latin, eier, through, + specio, look. ' An instrument called the stereoscope is frequently to be found in collections of optical instruments, and is usually to be tound discussed in text-Looks on optics. By its use the two slightly different central projections got from the two eyes of the observer as centers of projection are combined into one in such a way as to give a more complete appearance of solidity than a single perspective drawing can present. 42 PERSPECTIVE. necessary to consider somewhat in detail some of the complications of general centric projection which become eliminated in the change from that to the simpler case of parallel projection. A full consideration of the relations between figures and their centric projections constitutes what is known as projective, or modern, geometry. Into that, however, it is no part of our present plan to enter. We purpose merely to examine into a few of the most elementary facts such as we shall later find practical application for. 68.-If, of two concurrent lines, both passing within finite distances of the observer and one of which is considered fixed, the point of concurrence be conceived to recede along the fixed line, the two lines will become less and less divergent until as the point of concurrence recedes to an immeasurably great distance the divergence of the two lines will become immeasurably small, and we shall be totally unable to distinguish the two lines from parallel lines, so far, at least, as any characteristics they present within a finite distance of the observer are concerned. It is because of this that we say that parallel lines are lines meeting at infinity. Moreover, the point of concurrence having receded to an infinite distance it may continue receding indefinitely and we shall in no wise be able to distinguish any change in the appearance of the two straight lines within finite space. Therefore instead of speaking of the poinzts at infinity along any straight line which lies with PERSPECTIVE. 43 in a finite distance of the observer, we speak merely of the point at infinity, any one such point being, so far as any value it may have for us who remain in finite space is concerned, indistinguishable from any other. 69.-Likewise if the line of concurrence of any two planes be removed to an infinite distance from the observer, the two planes themselves continuing within finite distances of him, they will become indistinguishable from parallel planes. We may thus say that parallel planes are planes meeting at infinity, and just as in straight lines with regard to their point of concurrence when it is at infinity, so here with regard to planes meeting in a line at infinity, we are totally unable to distinguish any effect produced by varying the position of the line of concurrence of the movable plane with the fixed plane so long as that line remains at an infinite distance. We therefore look upon all points at infinity in a plane as being ranged in one straight line which we call the line at infinity for that plane. 70.-Now it is at once apparent that the centric projection of any point is the trace of its projecting ray upon the base of projection, and that the centric projection of the extremity of any line is the extremity of its centric projection. If, therefore, we take the projecting ray of the point at infinity of any system of parallel lines, the trace of this ray upon the base of projection is the projection of a point common to all of the lines and one to which all their projections converge. This projecting 44 PERSPECTIVE. ray is, moreover, parallel to the lines themselves, since it meets them at their point at infinity. Since it passes through the center of projection we shall call it the cen= tric line of the system. The centric projection of a system of parallel lines then is a system of concurrent lines whose center of concurrence is the trace of the centric line of the system upon the base of projection. The projections all terminating or vanishing at this point, it is called the vanishing point of the system. If the lines be parallel to the base of projection, their centric line will meet the base at infinity, and their projections will thus be parallel to the lines themselves, the vanishing point being the point at infinity for the system. 7I.-Similarly the vanishing points of all lines lying in any system of parallel planes will all be ranged in a straight line (the base of projection being here presumed to be a plane), for the centric lines of all the various systems of parallels will all lie in a plane through the center of projection parallel to the various parallel planes, and the traces of these centric lines on the base of projection will all lie in the trace of this plane on that base. This plane we shall call the centric plane of the system, and its trace on the base of projection is called the vanishing line of the planes of the system, since upon this vanishing line lie the vanishing points of all lines lying in such planes. 72.-In the execution of a drawing in centric projec PERSPECTIVE. 45 tion, or perspective as we shall call it throughout the remainder of this chapter the base of projection is commonly taken to be a vertical plane, and is called the picture plane (denoted hereinafter by PP). The center of projection is usually called the point of sight, sometimes the station=point, sometimes merely the station (denoted hereinafter by s); its orthogonic projection upon PP is called the center of the picture (denoted as usual by s"); and its orthogonic projection upon the ground-plane (see below) is called the foot of the station (denoted hereinafter by s1). 73.-Any plane perpendicular to PP is called a nor= mal plane, and will be denoted hereinafter by N with a subscript prefixed to denote to what system the plane belongs and one suffixed to denote through what point it passes; thus, 1Nq will denote a normal plane belonging to the first system and passing through q. What system shall be called the first system, what the second, etc., is of course a matter of choice. The centric normal plane of any system will have no subscript suffixed. Any line perpendicular to PP is called a normal line, and will be denoted hereinafter by n with a supscript suffixed to indicate through what point it passes; thus, nq will denote the normal line through q. The centric normal line is called the axis, and will be denoted by n without any suffix. That portion of the axis extending from s to s" is called the distance line because it gives the distance of s from PP. Its length is frequently called the distance. 46 PERSPECTIVE. 74.-Among the various systems of normal planes, two have preeminent importance. These are the horizontal normal planes, denoted hereinafter by hN, and the vertical normal planes, denoted hereinafter by vN. The centric horizontal normal plane is called the horizon plane (denoted hereinafter by H) and its trace on PP is called the horizon of the picture (denoted hereinafter by h); that one supposed to be at the general level of the ground is called the ground plane (denoted hereinafter by G), and its trace on PP is called the ground= line (denoted as before by g). The centric vertical normal plane will be called the meridian plane (denoted hereinafter by M), and its trace on PP the meridian line (denoted hereinafter by m). 75. —Planes parallel to PP will be called secondary planes, and will be denoted hereinafter by V followed by distinguishing subscripts; thus, Vq will denote a secondary plane through q. The centric secondary plane will be denoted by V without any suffix. Any line or figure parallel to PP will be called a secondary line or figure and will be denoted by a subscript 2 prefixed to the symbol for the line or figure. The ratio which the perspective of any secondary sect' bears to the sect itself is called the scale of the secondary plane in which that sect lies. 76.-Horizontal lines inclined at 45~ to PP we shall call principal diagonals. Their vanishing points are *A sect is any definite portion of a straight line. PERSPECTIVE. 47 evidently on the horizon line and at distances from s" equal to s"s; for this reason these vanishing points are known as distance points: the one to the left of s" will be denoted by d1, the other by d2. Lines inclined at 45~ to PP and lying in or parallel to M have their vanishing points on m, equally distant with d, and d2 from s"; these lines we shall call subordinate diagonals, their vanishing points being called lower and upper distance points, and denoted herein by d3 and d4 respectively. The vanishing point of normals is evidently s". The perspective of any figure we shall denote by the symbol for the figure followed by the subscript p,thus, ap denotes the perspective of the line a. A subscript v attached to the symbol for a line or for a plane will be used to denote the vanishing point or line (as the case may be) of the figure to whose symbol it is attached. 77.-Perspective drawing as practiced by the draughtsman is based on a few simple propositions some of which have already been stated. For convenience of reference we shall here state them in compact form. Their demonstration is left to the student. 78.-Proposition I. Every point in PP is its own perspective. 79.-Proposition II. If a is a straight line ap is either a straight line or merely a point. 80.-Cor. The perspective of a sect is determined by the perspectives of its extremities. 48 PERSPECTIVE. 8i.-Proposition III. If q is on a, qp is on a. 82.-Cor. 1. If a and b concur in q, ap and bp concur in p. 83. —Cor. II. Perspectives of concurrent lines whose center of concurrence is in V are parallel. 84. —Cor. III. Perspectives of secondary lines are parallel to the lines. 85.-Cor. IV. Perspectives of normals vanish at s". 86.-Cor. V. Perspectives of lines equally inclined to PP vanish at points equidistant from s". 87.-Proposition IV. If any figure is symmetric about the axis, its perspective is symmetric about s". 88.-Proposition V. If a lies in or is parallel to A, av lies in Av, and if A and B are parallel Av and Bv coincide, and conversely. 89.-Proposition VI. At* is parallel to Av. go.-Proposition VII. The perspective of any secondary figure is similar to the figure, and the ratio of similitude equals the scale of the secondary plane. I9.-Proposition VIII. If a straight line a is tangent to a curve k at the point q, ap is tangent to kp at qp. 92.-We now are prepared to consider the problem of finding the perspective of any given figure. Since * At will be used throughout the remainder of this chapter to denote the trace of A on PP. PERSPECTIVE. 49 every figure is composed of points, our problem when reduced to its elements consists merely in finding the perspective of a single point; this as has been so often said consists merely in finding the trace on PP of the ray determined by s and the point. It is then merely the problem of finding the trace of a given line on a given surface. Suppose (fig. 8 bis) s and q both given, and s, and q1 the traces made on any plane F by two lines through s and q parallel to each other and to PP. Also let F, be here denoted by a. Conceive a plane passed through the lines ss, and qq,. Its trace on F will be the straight line s1q, which crosses a at w, and its trace on PP will be a line through w parallel to ss,; where this trace is crossed by sq is the perspective of q. These considerations are entirely independent of the position of F, likewise of the direction of the lines ssl. qq,, and qpzv. It is readily apparent also that qp will remain unchanged in position if F revolves on a, carrying with it the points s, and q,, the lines ss, and qqI and the points s and q, provided these lines remain parallel to their original positions and retain their lengths unchanged. We may then evidently replace F by the revolved position of H as we use it in our ordinary work in Descriptive Geometry. 93.-It is also evident that if through s1q, parallel lines be drawn meeting a in r and t respectively, and on these lines s3 and q, be taken so that rs3 _ -, then rS1 t71 sq3 will cross a at w; and'that if s3s, be taken parallel 50 PERSPECTIVE. and equal to sls and on the same side of sls3, and q3q, be taken parallel and equal to q,1 and on the same side of qlq3, then will sq3 pierce PP at qp. This last statement is equally true whether or not s3 is on the same side of a with sj, provided that, when s3 is on the opposite side of a from s1, q. shall be on the opposite side of a from q1. 94.-We are thus led to the following practical method of constructing the perspective of a point q when its projection on H (q') and its altitude from H (the length of q'q) are known. H being supposed revolved into PP about h, from q' draw.a line perpendicular to h and equal to q'q, laying it off above q' if q is above H, below q' if q is below H, and mark the other extremity of the line so drawn q. Then draw from s (i. e., from its revolved position as a point of H) a line to q' and where sq' crosses h erect a perpendicular to h. Where such perpendicular meets sq is qp (see fig. 41). It will be noticed that when sq is nearly perpendicular to h this method gives extremely unsatisfactory determinations of qp. In such cases it is better to lay off q'q parallel to h as is done in the righthand portion of fig. 41; that portion of h comprehended between the lines sq' and sqr evidently equals the distance of qp-from h. The lines shown as mere construction lines in the figure should of course in any practical perspective drawing be drawn merely in pencil and afterward erased. PERSPECTIVE. 5I 95.-Customarily the auxiliary projections, etc., are drawn on other sheets of paper than the one on which the finished perspective is to appear, and these are tacked to the draughting board in convenient positions (see figs. 41 and 42). Small scale plans and larger scale working drawings may be utilized in this way, and the results increased or diminished in the proper ratio to secure the perspective to the desired scale. Fig. 42 shows the mode of utilizing elevations when they are drawn to the same scale as the plans used as auxiliaries. In that figure s1t evidently bears the same ratio to s,q' as does sqp' to sq', so that if in the lower right-hand portion of the figure, showing the side elevation, a vertical line be erected at t, as far from q as t is from q' in the plan, and sl"' as far to the left of that line as si is from t, and at the same height above the ground line that s is above the ground plane, lines from s,"' to various points of.the elevation will cut the vertical line aforesaid at the heights of the perspectives of those various points above the ground line; thus, for instance t1q gives the height of qp above the ground line. Of course if the plan and the elevation are not drawn to the same scale the proper modifications in these distances to reduce all parts of the drawing to corresponding scales will have to be made. A careful consideration of similar figures will be all that is necessary in order to do this correctly. 96.-The method above indicated serves very well 52 PERSPECTIVE. when the perspective sought is that of a figure having few of its outlines and important points lying in the same plane or in systems of parallel planes. When, however, its bounding surfaces lie largely in systems of parallel planes as is the case in the greater number of practical applications of the principles of perspective in architectural drawing, we find it in most cases useful to find the vanishing lines and points of the various systems of parallel planes and parallel lines, and to locate the perspectives of points largely by the use of auxiliary lines through the points. This mode is particularly useful when the points whose perspectives are sought occur in ranges, or rows. In the application of the method here indicated, webneed to know how to lay off on a line whose perspective is given the perspective of a given length, also how to divide a line whose perspective is given into parts proportional to given sects, and to determine the perspectives of the points of division. 97. —lf, in fig. 37, q denotes any point on a and the trace and vanishing point of a are as indicated, we may proceed thus to find the distance of q from at:-From s draw a line to av; it will be parallel to a. The ray from s to q pierces PP at qp. The triangles savqp and qatqp are similar. Therefore the distance of q from at is to the distance of s from av as the distance of qp from at is to the distance of qp from av. But if through at and av two parallel lines be drawn in PP and any line be drawn in PERSPECTIVE. 53 PP through qp cutting the line through at at r and that through av at zu, aty and azv will be in the same ratio, and will therefore be equi-multiples of the distances of q and s from at and av respectively. Any such point as ze if taken so that avZ equals avs is called a measur= ing point for the system of parallel lines vanishing at av. If taken so that the distance avw equals n times the distance azvs we shall call it an n-measuring point for the system. Tie line drawn through at parallel to azv is called a measuring line for the line a. In case the line a is a secondary line, the perspective of any section of it bears the same ratio to that section as the distance of PP from V bears to the distance of a from V. 98. —Consider next the problem of finding the vanishing point and the trace of any line whose perspective is given, when the distances of two of its given points from a given centric normal plane are given. Let a denote the given line (see fig. 38), q and r be the given points, 4N the given centric normal plane, and dq and dr the distances of q and r respectively from 1N. Suppose 1Nq and 1Nr drawn; their traces will be parallel to that of 1N and at distances from it of dq and dr respectively. Suppose any plane A passed through a. A will cut out of 1Nq and iNr lines passing through q and r respectively and parallel to each other and to 1N. Their traces then will lie in the traces of 1Nq and iNr respectively, also in At, while their vanishing point is in the vanishing line of 1N, which it will be remembered is the 54 PERSPECTIVE. same as the trace of 1N. Suppose zv the vanishing point -of these two lines. Their perspectives pass through qp and rp respectively, and their traces are where the lines wqp and zwrp cross the traces of iNq and iNr respectively. Through these two points At passes and where At crosses ap is Zt. A, passes through w and is parallel to At, and where A, crosses ap is a,. Notice that sinceA is indefinite in position, zu may be taken anywhere on the vanishing line of 1N. 99. —The solution of the problem to divide a sect whose perspective is given into parts propor= tional to given sects, and to find the perspec=tives of the points of division depends upon the fact that parallel lines intercept proportional parts on traversers (or transversals as they are frequently called). Suppose ap (fig. 39) to be the perspective of the given line, qp and rp the perspectives of the extremities of the given sect, and a, as indicated. Any line through a, will be the vanishing line of some plane A containing the given line a, and any line through any point of ap parallel to A, will be the perspective of some line in the plane A parallel to A, and therefore parallel to PP; its perspective therefore will be similar to the line itself. Then through a, we may draw any line whatever and take it for the vanishing line of some plane A containing the line a, and through either extremity of the sect to be divided, q say, draw a line qx parallel to A, (and therefore parallel to PP); on the line so drawn if parts be PERSPECTIVE. 55 laid off proportional to the parts into which the given sect is to be divided and in the same order, lines through the points of division and the points sought on the given sect will be parallel in space, and in the perspective will converge to some point on Av. Ioo.-It frequently happens that lines are so slightly inclined to PP that their vanishing points are off the sheet. In that case, in order to utilize the vanishing point, we find it very convenient to make use of the fact that the bisector of any inscribed angle bisects the intercepted arc. The manner of doing this will now be explained. In fig. 36, let v denote a vanishing point, and let o denote the center of any circle struck through v. Then if n2+I denote the radius of this circle, any two points on its circumference symmetric about the diameter through v and distant 2n from such diameter will lie on a chord distant n2-i from o, and 2n2 from v (provided, of course, that o is between v and such chord). Suppose q and r to be two such points, and that their chord crosses the diameter vo at w, also that t is the other end of such diameter. The angle qtw is then tannrn, and the angle qtr is 2tai' n. Any ordinary To with fixed head may evidently be easily modified so as to make use of this principle. Pins should be driven into the draughting board so as to be symmetrically disposed about some line through the vanishing point and each distant from that line 2n, and the line joining the pins distant 2n2 form the vanishing point. The angle qtr PERSPECTIVE. should be cut into the TO head as shown in the figure, and should have a chord of 4n at a distance of 2 from the vertex of the angle. The distance of the vanishing point from the chord joining the pins is evidently 2n' n 2n, or -n times the distance between the pins, and n2 4n 2 times the distance zt in the figure. A slight examination of the various values which n may conveniently be given in practice may not be entirely worthless. Let it be assumed that the To head is Io" by 2 ", and that the blade is 3" wide; also that it is not desirable to have the distance wt more than I", or the distance between the pins less than Y" or more than 3 ", and that there is a vacant portion along the edge of the draughting board at least I " wide into which the pins may be driven without injury to the drawing. Then we shall have that vw. can not be less than Ix (=, or 2 2\ 4 / greater than 3'" x 2 (=1 — 8), or than (n') in case wt is not to exceed I". Then when n = 5, vw minimum I ", and vw maximum = 8X"; when n = 32, vw minimum = 8", and vw maximum 52"; and when n = 200,vw minimum = 50", and vw maximum = 325". It is readily apparent that a very small assortment of modified Tos will be sufficient for any practical work, since two lines each ten inches long and separated at their more remote ends by one inch would approach each other by only one-thirtieth of an inch if their point of convergence were distant three hundred inches, a rate of convergence too slight to be perceptible to any PERSPECTIVE. 57 except the most highly trained eyes. Instruments embodying the principle here discussed are to be had of dealers in draughting instruments. They are sold under the name of centrolineads. IOI.-Lines converging to a point off the sheet may also be drawn by making use of the fact that concurrent traversers (or traversals) make proportional intercepts on parallels, so that if the intercepts made on any pair of parallel lines by any pair of lines meeting at the vanishing point be divided proportionally, lines through the corresponding points of division will concur at the vanishing point. Strips of paper graduated to different scales maybe so tacked to the draughting board as to be parallel and to have equal numbers of graduations between any pair of given lines concurring at the vanishing point. By the exercise of a little judgment, lines may be drawn through given points so as to pass through corresponding points of the scales and thus be directed to the vanishing point. The points of division on the strips ought not to be more than one-eighth of an inch apart. Lines graduated to various scales and used as indicated in the discussion above given are called vanishing scales. See fig. 40. I62. We come now to the final and practical problem in perspective drawing,-viz., how best to dispose PP, s, etc., with regard to the object whose perspective is to be drawn. This object we shall hereafter call simply tle obyect. The disposition will be governed chiefly by 58 PERSPECTIVE. two considerations,-first, the production of a good pictorial effect; second, ease of making the drawing. I03.-With regard to the first it is to be noted that persons examining a picture will in the majority of instances direct their attention chiefly to the middle portion of the sheet on which the picture is. In examining it to any considerable extent, they will usually change their distances from it as well as shift their positions parallel to it; i. e., the positions of their eyes will only approximate to that of s. A brief consideration will suffice to show that the farther s is from PP in comparison with the distance of PP from the object, the less would the perspective be modified by a definite amount of shifting of s, and consequently the less distorted will the picture appear when viewed from any point within a given distance from s. We conclude then that s should be taken as far from PP, and s" as nearly midway from side to side in the picture, as convenience will permit. A good practical rule is to take s at a distance at least seven-tenths of the width of the picture. Io4.-In so disposing of PP with regard to the object as to facilitate the work of drawing, notice that if PP be parallel to any set of lines of the object their perspectives will be parallel to the lines themselves, and hence may very readily be drawn. The lines of architectural constructions usually have three principal directions,-one vertical, the other two horizontal and perpendicular to each other. PERSPECTIVE. 59 105.-PP may be placed parallel to two of these, one vertical and the other horizontal; the perspective is then called parallel, or one-point, perspective, only one of the three principal directions giving a finite vanishing point, which is evidently s". This disposition evidently gives the easiest possible perspective constructions. One-point perspective produces the best pictorial effect when the parts of the object are symmetrically disposed about some open space down which the observer is conceived to be looking, as in the case of the buildings on the two sides of the street; it may also be used satisfactorily when the object is entirely or almost entirely below H, so that some of the lines in the.perspective will be shown converging towards s". io6.-If PP is placed parallel to only one of the principal directions, the perspective is called angular, or two-point, perspective, the other two directions both giving finite vanishing points. In this case the drawing will be facilitated by so placing PP that the two directions oblique to it shall be equally inclined to it. Where but one object is to be represented the best pictorial effect is produced by two-point perspective, except in the case before mentioned where the object is entirely or almost entirely below H. 107.-If PP is placed parallel to.none of the three principal directions the perspective is called oblique, or three-point, this arrangement giving three finite vanishing points. If PP is equally inclined to these three 60 PERSPECTIVE. directions, their vanishing points will evidently be equidistant from s". The case of three-point perspective will not, however, need often to be considered in practical work. io8.-If shadows are to be shown, the work of obtaining them will be much facilitated, of course, by taking the rays of light parallel to PP if they are to be considered parallel as they may be when coming from a source at a very great distance. In case they come from a source so near as to make them essentially divergent, the placing of the source will have to be determined by the particular features of the problem. og09.-After the center of the picture, the- horizon,' and the ground line have been selected, the vanishing.points of the principal lines of the object are usually next determined. The vanishing point of any system of parallel lines, it will be remembered, is the trace on PP of the centric line of the system, i. e., of the line through s parallel to the lines of the system. The lines drawn from s to the vanishing points of any three systems form then the three edges of a trigonic pyramid whose base lies in PP, whose face angles are given (being the angles between the directions of the three systems), and the orthogonic projection of whose vertex s on its base PP is at s". A brief consideration will show tr:at two of the corners of its base may be chosen at random, and that the third is determined' by these two and s",' also that the altitude ss" is also determined. Then: having PERSPECTIVE. 6I chosen the vanishing points of two systems of principal lines determine that of the third, and find also the distance ss". Next determine the distance points, the measuring points of the principal systems of lines, the vanishing lines of the various oblique planes, the vanishing points of oblique lines, etc. The determination of these will consist in the application of simple'principles of Descriptive Geometry, such as have heretofore been amply discussed and illustrated and so need not here longer detain us, and of the fundamental principles of Perspective as heretofore laid down. These various points and lines having been determined, the next thing to. be drawn is the perspective plan. IIo.-The perspective of the ground plane occupies that portion of PP between g and h,. and of course the farther away the lines in it are from PP, the more is their perspective narrrowed and confined vertically, and the more oblique to each other dd the perspectives of intersecting lines become. For this reason it is often very convenient to make use of what are called de= pressed perspective plans, i. e., plans drawn in perspective on horizontal planes depressed to a convenient extent below the ground plane of the picture. These may in most cases easily be drawn at a sufficiently great distance below the ground plane to put the depressed plan entirely below g in the picture, thus permitting it to be drawn on a sheet separate from that on which the picture is to be drawn. This permits the depressed plan 62 PERSPECTIVE. to be saved, and used in drawing other perspectives if desired. Elevated perspective plans may of course equally well be used. The use of a depressed perspective plan is illustrated in fig. 43, which shows the construction of the perspective of a Greek cross on a square pedestal, the face of the cross being taken at an inclination of 30~ to PP, and the front vertical edge of the pedestal being taken in PP. The vertical dimension of the pedestal is taken equal to the thickness of the cross, the horizontal dimension three times as great, and the bevel on the upper portion of the pedestal has a vertical extension equal to one-fourth of the altitude of the pedestal. The shadow is cast by parallel rays, parallel to PP, the sun being taken at an altitude of 60~. I I I.-As this problem illustrates a considerable number of the principles already discussed it may not be amiss to examine it somewhat in detail. The horizon line having been chosen and s" thereon, s is located at as great a distance as convenience permits. From s lines are then drawn meeting h at 45' in the points d, and d2. The point d3 falls on the sheet at s and d4 falls diametrically opposite and equidistant with s from s". The vanishing line for the front face of the cross is a vertical line, since the face of the cross is vertical. To find it a horizontal line is drawn through s meeting h at an angle of 30~ at the point marked q. The vertical line qlq2 through this point is then the vanishing PERSPECTIVE. 63 line of the system of planes parallel to the face of the cross, since this face makes with PP the angle 30~. Likewise is found the vanishing line of the side of the cross, it being the vertical line through the point marked r. The front bevel of the pedestal cuts out of the side face of the pedestal a line elevated 45~ above a horizontal; to find the vanishing point of such line, s is revolved into h about r as center, and through the point zu thus obtained a line is drawn at an angle of 45~ to h to meet the vanishing line of the side face of the pedestal at the point r. Any horizontal line in the front bevel vanishes at q; the vanishing line of this bevel is therefore the straight line through r2 and q. Likewise the vanishing line of the left-hand bevel is the line through r and q, that of the right-hand bevel is rq2, and that of the rear bevel is r, q. Since the intersection of the front and the right-hand bevels lies in both of those planes, its vanishing point will be at z where their vanishing lines cross. Likewise the vanishing point of the intersection of the rear and the left-hand bevels is at x where the vanishing lines of those two surfaces cross; xz is therefore the vanishing line of that diagonal plane of the pedestal which contains the front vertical edge. This vanishing line might equally well have been found by drawing the bisector of the angle qsr, and through the point y where it meets h erecting a line perpendicular to h. At arw is a half-rneasuring point for lines vanishing at q, the distance qze, being one-half of qs; and in order to 64 PERSPECTIVE. minimize the work of drawing the depressed plan, since in its principal outlines the plan of the base is square, a half-measuring point (call it zt4) for lines vanishing at r was taken at a distance to the right of requal to onehalf the length ys. As this falls outside the plateborder it is not shown in the fiure, but it is the point to which the lines 4-8, 3-7, and 2-6 converge. The ground line 22-I5 was taken at the desired distance below h, and the auxiliary ground line I-5 was chosen at a sufficient distance below the ground line to permit the depressed plan to appear entirely below the ground line. The point 15 having been selected for the front corner of the base (it, of course, is entirely independent of the line sd, in the figure), from the corresponding point I on the auxiliary ground line lines are drawn toward q and r to serve as the front left-hand and front right-hand outlines of the depressed perspective plan of the base. Aiong the auxiliary ground line the distance I-5 is then laid off, that distance being one-half the horizontal dimension of the pedestal. A line from 5 to w1, the half-measurino point for lines vanishing at q, determines on the line iq the depressed plan of the left-hand corner of the base (at the point I2) while one from 5 to zu, determines on the line ir the depressed plan of the right-hand corner. Lines from these two points toward r and q respectively complete the remainder of the outline of the depressed plan of the pedestal. Likewise the remainder of the de PERSPECTIVE. 65 pressed plan of the figure is drawn, the pillar of the cross having for its horizontal dimension one-third of that of the pedestal, and the bevel one-fourth of that of the pillar. 11-I3 is the depressed plan of the upper edge of the front bevel, and 9-Io is that of the right-hand side of the pillar. In constructing the perspective of the cross itself, use is made of the perspectives of the traces made upon the plane of the right-hand siJe face of the pedestal by the various outlines vanishing at q. Thus the perspective of the trace made upon this plane by the front edge of the top of the pillar is at 21. This point is found by laying off on the vertical from i the distance of that horizontal line above the ground plane, thus obtaining the point 20, then drawing the line 2o-r, and taking the point 21 where the line 2o-0 is crossed by the vertical line erected at 7 in the depressed plan. Where the line 2I-q is met by the vertical from the point 9 in the depressed plan is the perspective of the front upper corner of the pillar. That this point falls on the line ur2 is, of course, merely a matter of luck, as also it is that the point 20 comes near the line qr2. In getting the perspective of the outlines of the shadow, use is made of the fact that since the rays are p'-rallel to PP the shadows of vertical lines on horizontal lanes are parallel to g and that their perspectives and those of the rays are thus parallel to the shadows and rays themselves. Thus, the perspective of the shadow of I6 falls 66 PERSPECTIVE. at 22, and that of the shadow of 23 falls at 24. Exercises. The short collection of exercises herewith given will serve as samples and may be readily added to at will by the instructor. As they embody only applications of the common principles and operations of Descriptive Geometry, together with those of Perspective already amply illustrated, explanatory notes are deemed unnecessary. In the exercises below presented, when a line is said to be given it will be understood to be given by its trace and its vanishing point; correspondingly a plane is to be understood to be given by its trace and its vanishing line; and a point may be taken to be given by the intersection of a line and a plane, or by that of two lines, or by the ray through s and the distance of the point from s. The student will find it a useful exercise to solve the proposed problems with the given points taken as given in theseseveral ways. When anything is said to be required, if position is concerned then tts perspective is to be understood to be required; if the size is called for, the true size is wanted. II6.-Given Av, At, Bv, and Bt; find TAB. 117.-Given Av, At, bV and bt; find TbA. i 8.-Given av, at, bv and bt; determine whether or not a and b intersect, and if so at what point. g19.-Given two windschief lines, to pass a plane through one parallel to the other. In the remaining problems, s" and the distance s"s are also to be understood to be given. 20. —Find the angle between two given lines. 12I.-Find the angle between a given line and PP. 122.-Draw through s a plane perpendicular to a given line. I23.-Draw through s a line perpendicular to a given plane. 124 —Find the diedral between two given planes. 125. Find the distance from s to a given plane-to a given line. 126.-Draw through a given point a line perpendicular to a given plane. 127. Draw through a given point a plane perpendicular to a PERSPECTIVE. 67 given line. 128.-Find the distance from a given point to a given plane. 129.-Find the distance from a given point to a given line. I3o -Find the projection of a given sect upon a given plane. I3I.-Find the angle a given line makes with a given plane. 132.-Draw through a given point a line parallel to a given plane to meet a given line. I33. —Find the distance between two windschief lines and draw the shortest line joining them. d-. A.. I. - e *smer air I ova& jSLD;X2 C ^ B /.*. ~..................... ALL^^- 4^ X N."\^/X ^'^ V ---- - -- S jg, al! \\\ g1 / | \ ^ ~,..\ —^ ^^ /F~<s^^~~~~~~~~~~~~~.~. II \ \ ' —" r l > A_ ---'' ^. /...-.-.. —.... — 49 /~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ x.,.. '. 1 ----/ - / K s 4 --- —-- — \ 4 A \ - -- -- - ii ^\ "" x. X X- - /./.-~... _ \ *^ s '''^^ \ & \ y ^ - \.4 / '" ~.) / r II L i 1 l l I Jil H il 'I I; I I il I II i i I - z / / - p/ I-l, 4e — -— (t F, Z -> -— n r \ I.I I I d~~~~~~~~~~~~T >N z2iiI~7\- __ '7,iii -I - ---- -- l t -— It- l~ — lr,.., ) r I;j-._ / /?- -- I I ( \ \ 1 _ — \ ci I I ---I m 4 -