a 5 SHADES, SHADOWS AND LINEAR PERSPECTIVE mnw iMi ^ i ^ w — I.O£:den HENRY NEEPrOGDENI Cornell university Library rfi^ <\ Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924015443579 DESCRIPTIVE GEOMETRY AND MECHANICAL DRAWING SERIES By Frederick Newton WiMson, C.E., A.M. The specialized treatises constituting this series are uniform in size (nine inches by twelve inches, with text-page six and three-quarters by nine), printed on plate paper, and elaborately illustrated with photo-engravings, wood-cuts, cero- graphic process blocks, half-tone plates, etc. They are adapted both to class-room use and self-instruction. 1. JStote^Taking, Dimensioning and Iiettcring. A text-book on Free-hand and Mechanical Lettering in general, and on the lettering and dimensioning of " working drawings ; " also con- taining full instructions as to the sketching of bridge and machine details, for inspection or design. Various conventional methods of represent- ing materials are also given. Roman and Gothic letters, vertical and inclined, together with the Soehnecken Round Writing, Reinhardt Gothic, and other alphabets much employed by engineers and architects, receive ample illustration ; while the total of sixty-five complete alphabets afFords an unusual range of choice among serviceable forms. Full instructions are given as to the proportioning of titles, spacing, mechanical "short-cuts," etc. ; also a large number of designs for fancy corners and borders. $1.25 nd. 2. The ThiPd Angle ^VTethod of ttiaking Working Dracaings. A practical treatise on the American draugh ting-office system of applying the principles of projection in the making of "shop " drawings. It contains a large number of geneial problems on projection, intersections, and the development of surfaces, together with such special problems as an upper-chord post-connection of a bridge ; standard screws, bolts and nuts, with tables of proportions ; helical springs ; rail and valve sections ; spur gear. Illustrated with eighty-five cerographic-process blocks, and eight wood engravings. $1.25 net. 3. Some ^VLathematieal CuPVes and theip Gpaphieal Constfuetion. This work presents in compact form the more interesting and important properties, methods of construction, and practical applications of the curves with which it is essential that the architect and engineer should be familiar. It is also adapted to class-room use in mathematical courses- Among its special features are sections on homologous plane and space figures, given in connection with the conic sections, and laying a sound foundation for work in projective geometry ; link-motion curves and centroids, as an introduction to kinematic geometry ; historic notes and problems. Among other topics treated are the Helix ; Common Cycloid and its Companion ; Curtate and Prolate Ortho-cycloids ; Hypo, Epi, and Peri-trochoids ; Special Trochoids, as the EUipse, Straight Line, Lima5on, Cardioid, Trisectrix, Involute, Spiral of Archimedes ; Parallel Cur\es ; Conchoid ; Quadratrix ; Cissoid ; Tractrix ; Witch of Agnesi ; Cartesian Ovals ; Cassian Ovals ; Catenary ; Logarithmic Spiral; Hyperbolic Spira*. Lituus, and the Ionic Volute, The work concludes with a chapter on the nomenclature and double generation of cycloidal curves. Illustrated with fifty-four ceiographic blocks and one half-tone. §1.50 net. 4. Pt^actieal Engineering Dpacaing and Third Hngle Pfojeetion. A practical course for students in scientific, technical and manual training schools, and for engineering or architectural draughtsmen. It includes not only the contents of the first three volumes of this series, but also full instructions as to the choice and use of drawing instruments and materials ; line tinting and shading ; conventional methods of representation; plane problems of frequent recurrence; blue-printing and other methods of graphic reproduction and illustration; isometric drawing; and cavalier perspective (oblique projection). One hundred and seventy-eight pages, two hundred and seventy illustrations, and sixty-five alphabets. 2.80 net. B, Shades, Shadocas and liinear Perspective. A short course for students of engineering or architecture, and for professional draughtsmen. For its reading a knowledge of elementary projection drawing is assumed. The methods employed in the best American practice receive especial emphasis. Illustrated by twenty-one cerographic blocks and three half-tones, 1,00 net. 6. Descriptive Geometry — Pure and Applied, caith a chapter on fligher Plane Curves and the fielijt. This work contains in logical sequence not only the matter constituting volumes 2, 3 and 5 of this series, but also a chapter on the pure descriptive geometry of Monge, with elaborate illustration of the mathematical surfaces of most importance to the graphicist, and with applica- tions to Trihedrals, Spherical Projections, Axonometric {including Isometric) Projection, One-plane Descriptive Geometry, Oblique Projection, etc. , the whole constituting a broad, educational course. One hundred and ninety pages, illustrated with two hundred and eighty-four cerographic blocks, six half-tones, and twenty-one wood engravings, $3,00 net. 7. Theoretical and Practical Graphics. This work embodies the entire contents of the six preceding treatises, in a volume of three hundred pages, with four hundred and seventy eight illustrations, sixty-five alphabets, and thirty-eight border designs. It constitutes a progressive course in graphical science. $4.00 net Published by THE MACMILLAN COMPANY, 66 Fifth Ave., New York. Loudon: niacmillan & CO.. Limited. SHADES, SHADOWS AND LINEAR PERSPECTIVE. SHADES, SHADOWS AND LiINEAR PERSPEGTIVE FOR STUDENTS OF ENCINEERINC OR ARCHITECTURE, PROFESSIONAL DRAUGHTSMEN, Etc. pfedcficfc l^emton UJillson, C.E., R.J/l., Professor of Deso-iptive Geometry^ Stereotomy and Technical Drawvhg in the John C. Green School of Science, Princeton University; Mem. Am. Soc. Mechanical Engineers ; Associate Am. Soc. Civil Engineers ,' Mem. Am. Mathematical Society ; Fellow American Association for the Advancement of Science. l^eca Vopk THE MACMILLAN COMPANY LONDON : MACMILLAN & CO., Ltd. 1898 RIGHTS RESERVED \J COPYRIGHT 1898 BY FREDERICK N. WIIvLSON. PREFACE , PINIONS differ as to what is necessarily comprised in a fundamental course in shadows and perspective — one of the essential preliminaries to successful work in art or architecture and to the critical examination of paintings and drawings. It is easy for a writer, after stating a principle, to multiply examples of its application, but it is assumed in the following pages that the student can be trusted to do some independent reasoning as to the employment of methods that have been clearly illustrated by one or two typical examples of their occurrence. Every principle is here furnished that would be required in dealing with even so elaborate a subject as a cathedral, whether in exterior or interior view; and only those methods are presented which have recommended them- selves in American practice and are in daily application. The course is offered in this form for the convenience of those who for any reason desire to use it apart from the other matter with which it originally appeared in the author's larger work entitled Theoretical and Practical Graphics. F. N. W. INDEX. Architectural perspective, exteriors, 612. Architectural perspective, interiors, 616. Brilliant points, 587. On surface of revolution, 588. On sphere, with curve of shade, 589. Centre of the picture, 599. Circle, perspective of, 611. Curve of shade, 571. On a sphere, 589. On a torus, 590. On a warped surface, 592. On a warped helicoid, 596. Cylindrical column and abacus, shadows, 584. Cylinder (half) and rectangular abacus, 585. Diagonals and their vanishing points, 604. Direction of light, 574. Doors and doorways in perspective, 616, 618. Groined arch, perspective, 6ig. Horizon, defined, 601 ; as locus, 602. Interiors, perspective of, by method of scales, 616, Inverted plan, how used in perspective, 610. Light, direction of, 575. Line of shade, 571. Lines of height, use, in perspective, 612. Niche, shadow on interior, 591. Perspective, linear, aerial, 598. Preliminary definitions, 599-604. Of vertical lines, 600. Of parallels to picture plane, 605. By trace and vanishing point, 606. By diagonals and perpendiculars, 607. Of a cube, by above methods, 6og. Of a cube, by inverted plan, 610. Of a circle, 611. As applied in architecture, 612, 616. Of shadows, 614-616 ; 619. By the method of scales, 616. Of a right lunette, 617. Of a groined arch, 619. General remarks, 620, Perpendiculars, vanishing point of, 603. Point of sight, 599. Shade versus shadow, 570. Shade, line of, 571, 589, 590, 592, 596. Shade lines, 573. Shadow, of point, 577. Equal to original line, 578. Of line perpendicular to plane, 579. Of parallel lines, 580, Of cube, 575, 581. Shadow of vertical pyramid, 582. Pier and steps, 583. Cylindrical abacus on similarly - shaped col- umn, 584. Rectangular block on vertical semi-cyhnder 58s. Vertical, inverted, hollow cone, 586. On the interior of a niche, 591. Of line on a warped surface, 593. Of helix on surface of screw, 595. Screw surface, shade and shadow, 594-6. Sphere, brilliant point and curve of shade, 589. Steps and pier, shadows, 583. In perspective, 616 (c). Surface of revolution, brilliant point on, 588. Torus, curve of shade on, 590. Vanishing points, 602 ; reduced, 616 (b). Vanishing point of perpendiculars, 603. Of diagonals, 604. Of lines inclined at various angles to H, 613. Of rays, 614. Visual ray and plane, 599. Warped surfaces, curve of shade on, 593. Shadows upon, three methods, 593. Shade and shadow on screw, 594-596. SHADES AND SHADOWS. 219 CMAPTER XIII. SHADES AND SHADOWS OP MISCELLANEOUS SURFACES. 569. The shadows cast by an object which is illumined by either the sun or some other source of light are, in the mathematical sense, projections, and the rays of light become the "projectors. 570. The shade of an object is that part of its own surface which receives no direct rays from the source of light, while the shadow is the darkened portion of some other surface from which the original object excludes the light. The rays through all points of a given line will determine either a plane of rays or a cylinder of rays, according as the line is straight or curved. 571. The line of shade on an object is the boundary between the illumined and the unillumined portions, and its shadow forms the boundary of the shadow cast by the object. If the object is curved, the line of shade is the line of contact of a tangent cylinder of rays, each element of which would be tangent to the object at a point at which the cyUnder and the E-ig-. 370. object would have a common tangent plane of rays. For convex f)lane-sided surfaces the line of shade is the warped polygon formed by the edges contained by non- secant planes of rays. ")72. It is the province of Descriptive Geometry, in its application to this topic, merely to determine the rigid outHnes of shadows and shades. The delicate effects of cross and reflected lights, which always exist in nature in greater or less degree, can only be theorized about in a general way and can be most suc- cessfully imitated in draughting by working from a niodel or by the aid of photographs. Figs. 370 and 371, which are half-tone reproductions of photo- graphs of plaster models, while illustrative Tnainly of shades as distin- guished from shadows, also show the absence on double curved surfaces of those rigid lines of demarcation to which theoretical constructions lead. Yet the ability to correctly locate the geometrical lines of shade and boundaries of shadows is as essential an element of the draughts- man's education as a knowledge of the laws of perspective; since, lack- ing either, he could neither make an intelligent visit to an art or architectural exhibition nor so Avork up an original design that it could bear critical examination. 573. A conventional rule, much employed for throwing machine drawings into sharp relief, is to S^lg-. 37±. 220 THEORETICAL AND PRACTICAL GRAPHICS. make the right-hand and lower boundaries of a flat surface shade (i. e., heavy) hnes, provided that they separate visible from invisible surfaces. A\'hen, however, they are located with reference to some source of light, as in architectural and other drawings, the shade lines are those which could cast shadows, and their determination usually requires more thought, as some of the succeeding problems will show. 574. Direction of Light. A direction quite often (though not necessarily) assumed for the light is that of the body - diagonal of a cube whose faces are parallel to the planes of projection; that diagonal, in particular, which descends from left to right in approaching the vertical plane. It is illustrated by the arrow in Fig. 372, the source of light being assumed to be the sun, whose rays may for all practical purposes be regarded as parallel. 575. The ray FR (Fig. 372), being the body -diagonal of the cube, will project in ER on the base, or in AR on the back. ER and AR, being diagonals of squares, make 45° with the edges of the cube, which has led to the expression "light at forty-five degrees," for the conventional direction. But the ray itself makes an angle (^) of 35° 16' with either E R or A R, as may be established thus: Taking the edge of the cube as unity we have ER = \/2; alwo tan (E R F) ^= 1 -T- ■v/2, which, in a table of natural tangents, corresponds to the value indicated. Shadows thus cast are seen to be true oblique or clinographic projections (Art. 14), although orthographic projections are usually employed as auxiliaries in their determination. 576. In pictorial illustration of a few general principles Fig. 372 is drawn in oblique projection. In so elementary a figure as the cube the edges to be drawn as shade lines are evident from the outset, since, for the given direction of rays, the back, right and lower faces are obviously in the shade. We proceed then directly to find the shadow of the warped polygon AB CD E HA, establishing at the same time some of the principles that are of most frequent use. 577. The shadow of a point upon a surface, being the intersection of the surface by a ray through the point, is found where the ray meets its projection on the surface. In Fig. 372 the point C is projected on the plane MP &t c. The ray Oc, meets its projection cCi at Ci, which is, therefore, the shadow sought. 578. Any line, straight or curved, is equal and parallel to its shadow, when the plane receiving the shadow is pafallel to the line casting it. Having c, we therefore draw c^h.^ and h^a^ parallel to CB and A B respectively. 679. A line that is perpendicular to a plcme will cast a shadow upon it whose direction is that of the projection of rays upon the plane. The trace, cCt, of the vertical plane of rays through CD, contains the shadow d^Ci, and the projections of both rays Gc^ and Dd^. SHADES AND SHADOWS. 221 580. Parallel lines cast parallel shadows on a plane, since they are the intersection of parallel planes of rays by a third plane. 581. In orthographic projection the construction of Fig. 372 is shown in Fig. 373. The rays are E-igr- 373. shown on V in e'x, f'y and c'z, and each projects to the proper plan to give the trace of a ray on H. 682. The shade and shadow of a vertical pyramid. As the point where a line meets a surface is the beginning of the shadow of the line on the surface, we have merely, in Fig. 374, to join t, the trace of the ray _ through the vertex, with b and d, where ^^^- ®''"*- the edges of shade (s b and s d) meet H, to include the shadow sought. Planes of rays through the other edges would in each case be secant to the pyramid, and therefore useless. 583. Shadows on vertical, horizontal and oblique planes are illustrated by Fig. 375, in which the pier flanking four steps receives on its inclined face the shadow of a vertical post, and in turn casts a shadow on the steps. The shadow of the post. To find the point y", y^, where the ray through the vertex y meets the face abed, regard y B as the trace of a vertical plane through the ray; note A and B, where it cuts bounding lines of the inclined sur- face; project these at A' and B', upon the elevations of the same edges, and draw A' B' for the V. p. of the intersection of these planes. This receives the ray from y' at y", which projects down to 2/i. A similar construction gives u", which joins with y" for the shadow of the edge y'u'. For the shadow of edge ry, r'y', we may regard the face a'b'c'd' as extended upward and to the left, sufficiently for a re -application of the method described, D' giving a direction for the line y"D', which is a real shadow only to the edge a'd'. The vertical edges of the post, whose plans are u and r, cast shadows on H in the direction of the projection of rays. The shadow of the -pier on the steps. The vertical edge at b casts a shadow beginning at the foot of the line, and running — in the direction of plans of rays — to b^, the h. t. of the ray b' N. At b^ the shadow of the line b' c' begins, and its direction — upon horizontal planes — is found by treating cc' as if it actually cast a shadow on the ground, t then being the h. t. of the ray from c', and thi the direction sought. At k, however, the shadow begins to fall on the front of the lowest step, and we project up to h' for its v. p. To get the direction of shadows on the fronts of the steps, assume on b' c' some point i, i'; imagine the front of step No. 1 extended to catch the ray through i' at I', then toward V the shadow M runs from h'. This direction once established, we have only to get one point of each shadow P and Q in order to draw them in; while R, S and T may be drawn parallel to b^t, when one point of each is known. The shadow M runs off the front of step 1 at 5, which projects down to 6 for the beginning of shadow R. The latter is parallel to b^t, and at 7 meets the lower edge of the front of the second step. It projects to 8, through which the shadow P is drawn parallel to h'l'. To determine where this process will terminate we may definitely locate the shadow of cc', either as a preliminary or at any stage of the work, thus: The ray c't' meets the level of the top step at s'; this projects upon ct at s, which is outside of the actual limits of step 4 and therefore 222 THEORETICAL AND PRACTICAL GRAPHICS. ■s^s- svs. unreal. The plan ct of the same ray meets the front of 4 at x, which projects at x', and this, being between the limits of the front of the step, is therefore a real shadow. At x' the shadow has an angle, corresponding to that between h' c' and c'f. Its direction above x' is most easily determined thus: Assume some point, as z' , whose shadow is likely to fall on the top step; find its shadow 2,, and through it draw the line o q par- allel to the line casting the shadow; then project o from the front edge of the upper step up to m ' n', the v. p. of the same edge, and there join with x'. 584. The shadow of a cylindri- cal abacus upon a similarly shaped column. Let MEN (Fig. 376) be the plan of the cylindrical column, and ah c that of the abacus. The vertical plane of rays y z gives, by its tangency at t, the element of shade t'r' on the abacus. A simi- lar tangent plane of rays 6 x gives the element of shade px' on the column. It contains 6 6', that point of the abacus which casts the last shadow, x', on the column. Any other vertical plane of rays, as d, will cut a point d d' from the abacus, and an element from the column, the latter catching the ray from the former at q. The point x'lg-. 373. last found is "' *' necessarily the highest in the shadow, as it lies in that plane of rays which contains the axis; in other words, the meridian plane of rays. 585. The shadow of a rectangular block resting on a vertical semi- cylinder. Let mnop (Fig. 377) be the plan of a block whose front coincides with the section -lined surfaces of the cylinder. The edge a't' will cast an elliptical shadow a"e"t', which is the intersec- tion of the inner cylindrical surface by the plane of rays through a't'. To find it regard ac, be, xy as traces of vertical planes of rays, as in the last problem; draw rays SHADES AND SHADOWS. 223 ray back from a^ a' a", b'e", etc., through the points casting shadows, and note where each ray meets the element lying in the same plane with it. The shadow a"ai is cast by an equal length of a'k', found by drawing The remainder of a'k' would cast the shadow ac upon any horizontal plane on which the object might be regarded as resting. 586. The shade and shadow of a vertical, inverted, hollow cone.- In Fig. 378 let a's'f, abf, represent the cone. The lines casting shadows will be the elements of shade and a portion of the base. The elements of shade will be the lines of contact of tangent planes of rays. Each of these planes will contain the ray st, s't', through the vertex, and will cut the plane of the base in tangents to the latter; hence from tt — the trace of the ray — draw ta and tb tangent to the base; then as and bs are the plans of the elements of shade. Of these the latter only is visible in elevation; and the surface b's'f on its right is the visible portion of the shade. The shadow on H. The ray t's' has its horizontal trace at s^, one point of the shadow on H. As the arc aedb must cast on H a shadow that is equal and parallel to itself (Art. 678) find Si, the centre of the latter, by the ray from 6'(s); then the arc a^kb,^, limited by tangents from s^, completes the shadow on H. The shadow on the interior. Any secant plane of rays through the vertex, as tsd, will cut a point c from the base on the side toward the light, and an element s d on the opposite side of the cone. The ray through c will then intersect the element at a point m which is on the limiting line aib of the interior shadow. Other points are analogously found, as i from n. The shadow obviously terminates at the tangent points, a and b. 587. Brilliant points. All are familiar with the marked contrast as to brilliance between the various portions of a highly polished surface. The point which seems brightest to the observer is, however, not actually the one receiving the light most directly, but is that from which the incident (direct) ray is reflected directly to the eye, in conformity with the well-known optical law that an incident ray and the same ray as reflected make equal angles with the normal to the reflecting surface. Since in orthographic projection the reflected ray is perpendicular to the paper, we proceed as follows to find a brilliant point: Obtain the bisector of the angle between a ray of light and a perpendicular to the paper; then pass a plane perj)endicular to such bisector and tangent to the surface. Its point of contact will be the brilliant point sought. 588. To find the brilliant point on a surface of revolution. In illustration of the principles stated in the last article Fig. 379 is presented, in which one -quarter of a surface of revolution appears, Ms"g' being its meridian curve. The direct ray through C" G is a'0",aC. The reflected ray from the same point is OD, G" . To find the bisector of the angle whose plan is a CD, carry the ray a' G" into H, when it appears at b.^G, b"G". Ge^ bisects angle BGD, and in space becomes e'G", 224 THEORETICAL AND PRACTICAL GRAPHICS. ^±S. STS. /p ■--/' eC. The vertical meridian plane through Ce may be rotated till parallel to V, when the bisector just mentioned will appear at /'C", and the meridian curve will project in Ms"g'. Tangent to the latter and at 90° to f'C" draw op, which represents an edge view of a tangent plane. Its point of contact, s", counter - revolves to s'(s), the brilliant point desired. 589. The brilliant point and the curve of shade on a sphere. In Fig. 380 the sphere is represented by one view only, its elevation, acjb; the vertical plane of projection being understood to contain the centre, B. The brilliant point. A B, parallel to the arrow, is the projection of the ray through the centre, the ray itself making an angle of 35° 16' with the plane of projection (Art. 576). The reflected ray from B is projected in that point. If the plane of the incident and reflected ray be rotated into V about Aj as an axis, the former will appear &i R B and the latter a.t h B. L B is the bisector of the angle RBb, and D therefore the rabatted brilliant point, E being its true position. Were JK a tangent mirror and R a luminous point, the incident ray RB would be reflected to , a point on Bb, a.t a distance from B :^i-s- 3so. equal to R B. The curve of shade. This will be the circle of contact of a tangent cylinder of rays. It may be found by points, thus: Take a series of auxiliary planes of rays perpendicular to the paper, as MN, P Q, etc. Each will cut the sphere in a circle which may be seen as such by a 90 "-rotation of the plane; a revolved ray can then be drawn 'tangent to the circle, and its true position found by counter revolution. In the auxiliary plane MN, for example, we have ikn for half the circle cut from the sphere, and a tangent thereto at y, by a ray parallel to R B, gives a point which, after counter-revolution, appears at d on the curve of shade c e bf. The point e is similarly derived from s, and g from x. Parallels to A B and tangent to the spherical contour at c and b, give the extremities of the major axis of the ellipse in which the curve of shade projects. 590. The curve of shade upon a torus. In illustration, the architectural torus (Fig. 381) is employed, although the same methods are applicable to the annular torus. The points on the "equator" of the surface, viz., m' and n', are the points of contact of two verti- cal planes of rays, each indicated by xy on the plan. Points on the apparent contour of the elevation are determined by the tangency of planes of rays perpendicular to V, as ts' and that through Q, each at 45° to the horizontal. The highest and lowest points, oo' and uu', which must lie in the meridian plane of rays, NN, are found by revolving the meridian section in that plane about the vertical axis through C, until par- allel to V, when it will be projected in the given elevation. The ray AC, A'C, being revolved at the same time, becomes A^C, A"C', the latter then making 35° 16' with H. Parallel to A" C draw tangents to the elevation (only one, P, drawn), and counter -revolve the contact points into the first position of the plane N N. They appear at o and u, from which the elevations are derived. SHADES AND SHADOWS. 225 Points in any assumed meridian plane. Let D Ce be any meridian plane. Project the ray A C, A'C upon it at CB; rotate the plane to A^F, when x-is'- ssi. A goes from 5 to 6 and thence projects to the level of A', giving A'" C for the revolved trace upon DOB of a plane of rays perpendicular thereto. Tangents, as the one through R, parallel to A"'0', give the level of the points of shade in the meridian plane selected, and, after projection upon A^F, counter -revolve to DCB as at e, whence e'. Points in the meridian profile plane L X, as s" and Z, are at the same level as those on the apparent contour, owing to the equality of the angles A^CA and A OX. 591. The shadow on the interior of a niche, cast by its own outlines. The figure (382) shows the plan ABC, and elevation, A'R'C, of the surface, which may be defined as a vertical semi -cylinder, capped by a quarter sphere. The direction of the rays being given by sa, s'a', a series of vertical planes of rays are passed. One of these is ee^, containing point e', which casts a shadow, and cutting from the cylinder an element which catches the ray from e' at e". The plane of rays mn contains a point oo', whose shadow is received by the spherical interior, but which involves no difficulty in its determination, as the ray through o and the circle catching the ray may be shown in their true relation thus: Project upon mn at q; then gm is the radius of the small circle cut from the top, and n o^m is the revolved position of the circle. Make nr^ equal to n'r', and draw o^r^ for the revolved ray through o. It cuts the circle n o^m at a point which counter -revolves to b, and thence projects upon o'n' at b'. The tangency of a plane of rays {MM) perpendicular to V gives the point x at which the shadow begins. The point p", at which the curve leaves the spherical part, can be exactly determined by a special construction, but is located with sufiicient accuracy if enough points have been obtained on either side, by the previous method, for the drawing of a fair curve. 692. The curve of shade on a warped surface. This is most readily determined by connecting the points of contact of tangent planes of rays, applying the principle that any plane containing an ""~--VII_, ..-''' element of a warped surface is at some point a tangent plane to the surface, and if also a plane of rays its traces will contain those of any ray intersecting the element. wiV-,_ 226 THEORETICAL AND PRACTICAL GRAPHICS. 593. The shadow of any line upon a warped surface. Various methods may be employed, among them the following: (a) Pass a plane of rays so as to cut the line casting the shadow in a point; through the point draw a ray to meet the curve cut from the surface by the plane. (b) A plane of rays, containing an element on which the shadow falls, will cut the line casting the shadow at the point whose shadow falls on the element selected. (c) If two lines cast intersecting shadows upon a plane, a ray drawn back from such intersection will meet the line that is nearest the plane in the shadow cast upon it by the point where the same ray meets the more distant line. 694. The shades and shadows of warped helicoidal surfaces. Illustrating with the triangular -threaded E-igr- 3S3. screw, whose surfaces are helicoidal, let a'Q and a'e (Fig. 383) be the generatrices of the upper and lower surfaces, respectively, the point a' generating the outer helix a'd'P, while e and Q generate inner helices of the same pitch as the outer. (Art. 478). 695. The shadow of the outer helix, P D K', on the surface of the thread below. Assuming Rw, R' T', for the direction of light, a vertical plane of rays Rju' will cut the helix in a point R, R', and the heli- coid in a curve u' Z'&, the latter catching the ray from R' in the shadow, 7", cast by it. The curve u'Z?> is found by projecting u, j, w and 3, which are the plans of the intersections of the plane with various elements, up to the elevations of the same elements. For the elevation of j, which falls on element cd, the points d and j are carried to h and j, ; h then projects to k', whence D k' for the revolved elevation of element c d. Upon it j ^ appears at j', which in counter-revolution returns to Z. Vertical planes of rays parallel to R w are shown in ah, b g, If, and with each the process just described is repeated. 596. Points of a curve of shade, by means of a declivity cone. The direction of light with which we have been dealing so far in the case of this screw, puts the entire under -surface of the thread in the shade; but were any portion illuminated, as would occur with light as indicated by tc, t's',& curve of shade would have to be determined, point by point, one method for which is as foUows: Obtain a tangent plane to the helicoid at some point of any helix. This will be determined by an element and a tangent to the helix. (Art. 478). For the outer helix a'r' the tangent plane at a' a would be xea', as e is the h. t. of the element a's', while x is the h. t. of the tangent at a' (found by making a x equal to the rectified arc a d K). This plane cuts the axis at s', but is evidently not a plane of rays, since the H- trace, t, of the ray through s', does not fall on that of the plane. Since, however, all planes that are tangent to a helicoid at points on the same helix are equally inclined to H, we may find the point of contact of a tangent plane of rays by generating a cone with a line of declivity of the plane just found, passing a plane of rays tangent to said cone, and then finding the parallel plane of rays that is tangent to the helicoidal surface. SHADES AND SHADOWS. 227 The base of the "declivity cone" is onq, of radius oc, the plan of that line of declivity which lies in a plane with the axis. In aco we see the plan of the constant angle between elements and lines of declivity in the series of planes tangent along the particular helix in question. The ray through the vertex s' of this cone has t for its trace; tn is therefore the trace of a plane of rays tangent to the cone, and en the plan of its line of declivity. A parallel plane of rays, tangent to the outer helix, would then contain an element of the helicoid which would be projected as much to the left of en as ac is to the left of co; that is, angle mem is made equal to oca, and m(m') would be that point of the outer helix which belonged to the curve of shade. A similar process for the inner and any intermediate helices would give points of a curve of shade whose shadow could be found by either of the methods given in Art. 593. 597. When any two surfaces intersect, the shadows cast by either on the other may be found by applying the general principles of Art. 593, care being taken to so avail one's self of known properties of the surfaces as to simplify the construction as much as possible. 228 THEORETICAL AND PRACTICAL GRAPHICS, CHAPTER XIV. DEFINITIONS AND PEINCIPLES. - AECHITECTUEAL PEESPECTIVE FOE EXTEEIOES. — PEESPECTIVE OF SHADOWS.— PEESPECTIVE OF INTEEIOES BY THE METHOD OF SCALES. ,---|R 598. A drawing is said to be in perspective when its lines correctly represent those of a given object as it would appear from a point of view located at a given finite distance from both it and the plane upon which the drawing is made. If the representation is not only correct geometrically but is also shaded and colored, it is said to be in aerial perspective; otherwise it is simply a linear perspective. The construction of the latter is obviously a preliminary to all artistic work in oils or water colors. Perspective plane. The plane on which the drawing is made is called the picture plane or per- spective plane, and is always understood to be vertical; it will therefore be frequently denoted by the same letter (V) heretofore employed for the vertical plane of projection. It is usually taken between the eye and the object, in order that the perspective may be smaller than the object itself. 599. The general principles and definitions may be illustrated by Fig. 384, which is a pictorial representation of the various elements involved. The picture plane is the vertical surface BZRK, later transferred to X Z"R"Y. Point of sight. — Visual ray. — Visual plane. S is the supposed position of the eye, and is variously termed point of sight, perspective centre and centre of the pic- ture. Any line through S is called a visual ray, and any plane containing it a visual plane. 600. ABC DEC (Fig. 384) is a rectangular block whose perspective is to be constructed. It is so placed that one of its faces — A BCD — is in the perspective plane, making that face its own perspective. Visual rays, S F, S E, S H, inter- sect the plane V at points /, e, h, which are the perspectives of the original points. Joined with A, D and Q they give — with A D OB — the perspective of that part of the block which is visible from S. To find the trace, /, of ray S F, a vertical visual plane may be taken through the ray. (? s is the horizontal trace of such a plane, and om its vertical trace. The ray meets the latter at /. Similarly for other rays. Other methods are given in later articles. The figure illustrates the fact that the perspective of a vertical line is always vertical; for the verti- cal visual plane through EH must cut a vertical plane V in a vertical line, part of which is the perspective eh of the original. LINEAR PERSPECTIVE.— GENERAL DEFINITIONS. 229 601. Horizon. The point of sight, S, is projected upon the picture plane at s'. A horizontal visual plane will cut the perspective plane in a horizontal line through s', called the horizon. 602. Vanishing points. The convergence, in a drawing or photograph, of lines representing others known to be parallel on the original object, is a familiar phenomenon. To determine the point of convergence or vanishing point of any set of parallels, we have only to obtain the trace on V of a visual ray drawn parallel to the system of lines; for such trace on V may evidently be regarded as exactly covering the point at infinity at which we may conceive the set of parallels as meeting. The vanishing point of a line is one point of its perspective. The horizon is the locus of the vanishing points of all horizontal lines. 603. Vanishing point of perpendiculars. A perpendicular is a line at 90° to V. A visual ray parallel to a perpendicular must obviously be the projecting ray through the eye; and s', therefore, the vanishing point of perpendiculars. 604. Diagonals and their vanishing points. Horizontal lines making with V an angle of 45° are called diagonals. Sw' and Sw", the diagonals through the eye, meet the horizon at the vanishing points of diagonals, w' and w"-, also known as points of distance, since they are as far from the pro- jection of the eye as the space -position of the latter is from V. 605. Lines parallel to the perspective plane have their vanishing points at infinity; or, in other words, the lines and their perspective representations are parallel. This is illustrated hjEH and eh in Fig. 384. 606. Perspective by trace and vanishing point. Since any point in the perspective plane is its own perspective, we may obtain the indefinite perspective of any line, as FA, by joining A — ^its trace on V — with its vanishing point, the latter being s' in this case, as the line mentioned is a perpendicu- lar. The visual ray SF then intersects the indefinite perspective A s' at /, when Af is readily seen to be the definite perspective of A F. (For application see Art. 612). 607. Perspective by diagonals and perpendiculars. The student has already discovered, without formal statement of the principle, that a point is determined in perspective by its being the inter- section of the perspective of two lines passing through the original point. We saw in the last article that the perspective of F was obtained as the intersection of a ray SF with a line As', which might be regarded either as the trace of a visual plane or as found by joining trace of line with vanishing point. Obviously any pair of lines may be drawn through a point, and the inter- section of their perspectives noted; but the auxiliaries which, on account of their convenience, are most frequently used in perspectives of interiors, are the diagonals and perpendiculars already defined. In the figure, FD ia the diagonal of the square top of the block; Sw' is parallel to FD, and to' therefore the vanishing point of diagonals, to use in getting the perspective (Dw') of said diago- nal. This intersects A s' (perspective of perpendicular A F) at /. 608. Having illustrated pictorially the principles most employed in linear perspective, we have next to show how they are applied to the orthographic projections which are usually all that the draughtsman has, with which to start his constructions. Obviously, the perspective plane cannot be rotated backward into coincidence with the paper on which the object is represented in plan, without the latter drawing being in most cases overlapped by the perspective representation. The usual — probably because the most natural — way to avoid this difficulty, is to imagine the plane V trans- ferred forward to some position Z"XYR", where, if rotated into the paper about its trace XY, the perspective will clear the auxiliary views. 609. In Fig. 385 the method just described is illustrated as applied to the block of Fig. 384, and both figures may be referred to in the following description: 230 THEORETICAL AND PRACTICAL GRAPHICS. Fig. 3S5. Fig*- 3S©. Horizon B K is, the trace (and orthographic representation) of the entire perspective plane BZRK, and XY its transferred position, upon which the elevation of the object is drawn (A'B'C'D'). The plan ADEP is drawn back of PK, and in the same relation to it as the object to V. Ts is the same in each figure. The horizon is located in relation to the ground line XY in Fig. 885, at a distance from it equal to Ts' (or Ss) in Fig. 884. The vanishing 'point of diagonals, w', is at a dis- tance from s' equal to s T. We have now merely to apply either of the methods previously de- scribed, thus: (a) Es {D' s'^ is the visual ray through E (the latter being projected on V at D'). Erh is the vertical visual plane through this ray; rh is its vertical trace, and e is the trace of the visual ray and therefore the perspective sought. (b) D' is the trace of ED, and D' s' is its indefinite perspective, drawn to the vanishing point of perpendiculars. Ray sE meets the trace Bk at r, whence e for the perspective desired. (c) Wanting the perspective of F, we may draw through it the perpendicular FA and the diagonal FD. These, being on the top of the block, meet V at the level A'D'. A's' is therefore the perspective of the perpendicular, D'w' that of the diagonal, and their intersection / the point desired. 610. The method by inverted plan. In Fig. 386 the same perspective as in the preceding figures is obtained by assuming that the object has been rotated 180° about B K, so that it appears, inverted, in front of the perspective plane. Diagonals and perpendiculars then give the same result as before. Any diagonal, as G C, being inverted, is drawn in perspective in its true direction, D'lv'. ■Fxg. ssT". This method is quite convenient when ~~ >hs,., Z^ _.----" dealing with plane figures; but for large and complicated objects the conception of inversion is confusing, and renders it far inferior to the other. To show, however, its serviceability in the field indicated, as also the device of circumscribing polygons — usually resorted to in case of curves — Fig. 887 is given. 611. The perspective of a circle. In Fig. 387 the circle is circumscribed by an octagon. Through the various points of circle and octagon diagonals and perpendiculars are drawn. These vanish, in perspective, at the points iv', w" and s", and by their intersections either give points of the circle directly, or lines to which the perspective of the circle can be sketched in tangentially. ARCHITECTURAL-PERSPECTIVE METHODS. 231 The perspective of a circle will be a circle only when it is parallel to V, or when the visual cone to its points is cut by V in a sub -contrary section. In all other cases it is an ellipse, when the circle is on the opposite side of V from the eye. 612. Perspective by Trace and Vanishing Point, with special reference to its application to architectural constructions (exteriors'). In further illustration of the method of Case (b) of Art. 609, which is generally used in drawing the perspective of the exteriors of residences and other architectural constructions. Fig. 388 is pre- sented. The object dealt with has not only horizontal and vertical lines but also edges inclined at Flgr. 383. /* J13_.___B^ VI IV FRONT ELEVATION SIDE ELEVATION various angles to a horizontal plane, so as to illustrate the method of deaUng with any direction of line that can occur in a house perspective. The only reason for not presenting the plans and elevations of some actual building, is the impossibility of reducing such a series of views to the neces- sary limits of our illustration without sacrificing clearness as to the constructions made therewith; but if for the given elevations and plan there were substituted the elevations of a house, together with a general roof and wall plan of that part of the house which is visible from the point of view selected, the procedure from that point would be identical with that shown above. 232 THEORETICAL AND PRACTICAL GRAPHICS. The central view gives the best idea of what the object is like, a block devised simply with reference to compact illustration of the various principles involved. Lines of height. Vertical planes through the edges of the plan, as D^y, K-^Z, XY, are first drawn, and their trates shown on the (transferred) perspective plane in the verticals yx, 9-10, Zz, etc. These, in architect's parlance, are lines of height, since upon each is laid off the true height of lines in the plane it represents. These heights are most conveniently located by projecting over directly from one or the other of the elevations. Thus, the height of K^C^ is seen at C on the front elevation, and at C" on the other. Projecting from the latter, we have z as the height at which K^Li would meet the perspective plane; its trace, therefore, to use as in Art. 606. Similarly, 6£j produced gives OU for its line of height, and 5^ cuts it at the trace U. PiQi gives the trace 9-10, upon which PQ projects, giving 10 for the height of both P and Q. 613. The vanishing points. With s^ as the plan of the eye in its relation to the original position of V, draw the horizontal line s^I parallel to the longer lines of the plan, and project / to T" on the horizon, for the vanishing point of that set. Similarly draw s^t^, parallel to those hori- zontal lines of the object that are perpendicular to the first set, getting T' for the other vanishing point of horizontal lines. Were there horizontal lines in other directions on the object, their vanish- ing points would have to be determined by an analogous process. The vanishing point of lines making angle (see side elevation) with the horizontal is at i', found thus: Draw s,^I parallel to the plans of the lines whose inclination is 6. At I draw a perpendicu- lar to s,^I and prolong it to meet, at i, a line s-^i making 6 (at s^) with s^I. Make T"i' equal to li, when i' is recognized as the trace of a visual ray parallel to the lines whose inclination was given; for when the triangle s-^il is rotated upon its base s^I until vertical, and then placed with said base at the level of the eye, we would evidently find i at i'. The vanishing point of lines inclined ° to H is found by duplicating the last procedure in every detail, s^t^t.^ being then the triangle whose altitude t^t^ is laid off vertically from T', and toward which 3' 2' and 5' 6' converge. The remaining construction is as follows: Vertical visual planes are drawn through s^ and all points of the object. To avoid complicating the lines only a few of these are shown, SjGi, s^P^, Si-Fj, s^J. Taking K^P^s^ as illustrative of all, we draw its vertical trace Jclp. The line of heights for the point P being 9-10, project P upon the latter at 10; then 10- T' is the perspective of 2-P^, and its intersection p with the vertical kk is the perspective sought. Z",, in the same visual plane, has its height projected from K to z, upon the line of heights through Z; then z T' gives k. The perspectives of all the other points might be similarly found; but with two or three points thus obtained we may find the various edges by means of the vanishing points, thus: Starting with e, for example, prolong i'e to meet at / the trace of visual plane s^I\; then fT', stopping at g on trace s^G^. As gk and fe have the same vanishing point, we find k as the intersection of gi' and z T'. Then T'k prolonged gives I and c on traces of visual planes (not drawn) through L^ and Ci . The point d being found independently, we join it with c for the edge c d, for which we might also find a vanishing point thus: Obtain the base angle of a right triangle of base C^D^, and altitude equal to height of C above R; then use this angle (which we may call fi) and the direc- tion I>iCi exactly as 6 and F^J were used in the construction giving vanishing point i'. 614. Perspective of Shadows. These might be obtained from their orthographic projections in every case, but usually a shorter method is employed. Both ways are illustrated in Fig. 389. The object whose perspective and shadows are to be constructed is a hollow rectangular block, PERSPECTIVE OF SHADOWS. 233 The corner b being in the perspective whose plan is fbcd, and whose height is seen at A B. plane, we have in A B the perspective of the front edge. The vanishing points R and L having been found from s, as in the last problem, draw A R and B R, and terminate them ^tg. ses. on the trace mg of the visual plane sm. Similarly, terminate A L and B L upon the trace ' dk oi the vertical visual plane sf. Then FR and CL give the rear corner E, etc. The shadow. Let c'm' be the orthographic elevation of the edge whose plan is c. Then if a ray of light through c (c') has the projections ex,, c' X, we shall have x, for the shadow of cc'. In the same way the shadow might be completed in orthographic pro- jection, a portion only, being, however, actuallj' indicated. Then, treating x^ like any other point whose perspective is desired, we would find r — the vanishing point of hori- zontal lines parallel to mx^, and draw Dr for the perspec- tive of the plan of a ray; then X, the intersection of Dr with the trace ox of the vertical visual plane sx,, is the perspective of the shadow of 0. CE being horizontal, its shadow on H is in reality parallel to it, and, perspectively, has the same vanishing point; hence draw from x toward L to complete the visible portion of the shadow. CD being a vertical line, has its shadow Dx in the direction of the projection of rays on H. 615. The perspectives of shadows, without preliminary construction of their orthographic projections, are thus obtained: In Fig. 389, with c' X and ex, as the orthographic projections of a ray, draw s'r', St, for the parallel visual ray, when r' is seen to be the vanishing point of rays. Then r is obviously the vanishing point of horizontal projections of rays; and for shadows on horizontal planes the two points thus found are sufficient. For the shadow of C we have merely to take the direct ray Cr', and the plan Dr of the same ray, and note their intersection, x. For shadows on a set of parallel planes that are not horizontal, we would replace r by the vanishing point of projections of rays on the planes in question. 616. Perspective by the method of scales, (a) In Fig. 390 let s be the point of sight, mn the horizon, and m and n vanishing points of diagonals. Attention is called again, by way of review, to the fact that the real position of the eye in front of the perspective plane is shown by either ms or ns, and that all horizontal lines inclined 45° to V will converge to m or n. 234 THEORETICAL AND PRACTICAL GRAPHICS. We now apply the properties of the 45 "-triangle thus: To cut off any distance, perspectively, upon a perpendicular to V, lay off the same distance parallel to V and draw a diagonal. If the room is to be twenty -two feet deep, make Ch equal to that number of units and draw the diagonal h n, cutting off the perpendicular from C at i, making Ci the perspective of the given depth. The rectangle AB CD having been laid off in the perspective plane, from given dimensions and to the same scale, draw from A, D and B toward s, terminating these perpendiculars on a rectangle obtained by drawing id and ij, then jf and df parallel to the corresponding sides of the larger rectangle. (b) Reduced vanishing points. In case the point of sight has been taken at such a distance from the perspective plane as to throw m and n beyond convenient working limits, we may get the same E'ig-. sso. result by bisecting or trisecting sn and taking the same proportion of the distance to be laid off. Thus the point i might be obtained by bisecting Ch and drawing a line to the middle of sn. We might equally well lay off from C toward h any other fraction of Ch, and draw thence to a point on the horizon whose distance from s was the same fraction of sn. (c) The perspective of the steps. Let it be required to draw a flight of three steps leading to a doorway in the left wall. If the lowest step is to be six feet from the front of the room, make Da equal six feet and draw am, cutting Dd at a corner of the step in question. If the steps are to be three feet wide, make D b equal to three units, and draw b s for the trace of the vertical plane of the sides of the steps. Making ac three feet, draw cm, getting point 9, which should be even with the first corner found. PERSPECTIVE BY THE METHOD OF SCALES. 235 The widths of the steps being laid off from c at e and g, and their heights at 1, 2, 3 on D^, their perspective is completed by a process -which should need no further description. (d) The doorway in the left wall. Assuming this to be the same width as the landing, which — as seen at gr — ia evidently four feet; and also that the walls of the hallway are in the planes of the front and back of the landing, draw vertical lines from the left-hand corners of the latter, terminating them by a perpendicular Ps drawn from a point P whose height (ten units) is that of the top of the doorway. 3-4 shows the height of step from landing to hallway. The perspective of the door is obtained in this case on the supposition that it is open at an angle of 54°, for which a vanishing point (not shown) lies on sm prolonged, and from which a line J'v gives the direction vJ, and similarly H' H for the top of the door. To find / we may draw D K &t 54° to a vertical line (the 45°- angle indicated is an error, should be 36°) so as to represent a four-foot door swung through the proper arc, when by projects ing up DK to 4l-L, which is the level of the bottom of the door, a perpendicular Ls wiU cut J'v at /. Then a vertical line from / will cut the line H' at H. Were the door actually open 45°, the edge Jv would pass through m. The hallway on the right has its corner 7 at a distance of thirteen feet from C, and is seven feet high. The width of the passage may be ascertained by the student. The method of getting the perspective of a door by means of an auxiliary circle is shown in Fig. 391. (e) The location of the light, I. To locate the light five feet from the right wall, move five units from B, to E, when Es will be the trace, on the ceiling, of the vertical plane containing the light. If the light is to be five feet below the ceiling, mark off five units down from E, when Gs will be a horizontal line giving the level of I. Finally, to have the light a definite distance back, say eighteen feet, make Bo eighteen units on the front edge of the ceiling; draw on and get t, when txl^y will be a plane at the required depth, and its intersection I with Gs will be the position of the light. (f) The shadows. As in any other shadow construction, we have to note, in any case, where a direct ray through a point meets the projection of the same ray. All horizontal projections of rays will pass through Z,, which is the projection of the light on the floor. For the triangular block FM, we take a direct ray 1 8, through any point of the edge casting the shadow, and l^% for the projection of the same ray; then 8 is the shadow of the point selected. At F, where the edge meets the floor, the shadow begins, hence F-% is the direction and FN the extent of the shadow cast on the floor, and, obviously, N M that received by the side wall. (g) The shadow of the door. JHll^ is a vertical plane of rays containing JH. It cuts the left wall in a line Wz, found by continuing the trace from l^ to meet D d, erecting therefrom a vertical line and cutting it at TF by a ray from I through H. The shadow of Jv on the landing has the same vanishing point as Jv. When it meets the side wall it joins with v, since there the line casting the shadow meets the surface receiving it. The shadow of J is at the intersection of ray I J with the trace (not drawn) of the plane of rays JH I upon the top of the block. This would be found thus : Where the h. t. of said plane cuts the edge 9-10 draw a vertical line, and from the intersection of the latter with the top edge of the landing draw a line to the point below I on the line 14-s. This will give the direction of the shadow of HJ on the landing, since the line 14 -s is at the level of the top of the landing. (h) The shadows of the steps. The plane yEGl has the trace l^p on the floor; and if on the vertical pq we lay off distances equal to the heights of the steps and draw vanishing lines to s, these will cut 11^ at points which may be regarded as the projections of the light upon the planes 236 THEORETICAL AND PRACTICAL GRAPHICS. ■FLg. 3Si. of the tops of the steps, and should he used in getting the directions of the shadows of vertical lines upon said tops. The direction of the shadow of the vertical edge at 9 is given by ZjS, which is made definite as to length by a ray from I through its upper extremity. The shadow on the floor is then parallel to the front edge till it meets the side waU, where it joins with the end of the line casting the shadow. The vertical edges of the second and third steps would cast shadows whose directions would be found by means of the points above l^ on 12-s and 13-s, and which would run obliquely across the tops, instead of covering them entirely, as shown, incorrectly, by the engraver. 617. The perspective of a right lunette, the intersection of two semi- cylindrical arches of unequal heights. Let AMB be the front of one of the arches and DNC the opposite end, at a distance back which may be found by drawing from the vanishing point of diagonals, T, a line TD to meet A B pro- duced, giving a point whose distance from A is that sought. Let the smaller passage be at a distance A X back of A, and equal to X Y in width. Continue the vertical plane on AD to the level eS ot the highest element of the smaller arch, and in that plane construct Pmno — the perspective of half of a square whose sides equal X Y. In this draw the perspective of a semicircle Pic go. At ee', dd', etc., we see the amounts by which the elements of the side cylinder extend past the plane A enF to their intersection with the main arch, and these in perspective are ordinates of the curve o'e"g'. For any one, as hb', draw hS cutting the semicircle Pko at o and g. Horizontals through these points will be those elements of the smaller cylinder that lie in the hori- zontal plane hb'S; and the perpendicular h' S cuts them at the points o' and g' of the intersection. 618. The perspective of a door, found by means of an auxiliary circle in perspective. Let QP, Pig. 391, be, ijerspectivelj', the width of the given door. Construct PKO, the perspective of the circle that P would describe as the door opened. If Q P were to swing to Q 3, the prolongation of the latter would give 1 on the horizon for its vanishing point, which joins with J for the direction of the top edge, the latter being then limited at 2 by a vertical through 3. Similarly, K Q, prolonged to the horizon, gives a vanishing point from which a line through J gives the top edge J I. 619. The perspective of a groined arch. If the axes of two equal cylinders intersect, the cylinders themselves wiU intersect in plane curves, ellipses. In the case of arches intersecting under these con- ditions the curves are called groins, and the arches groined arches, when that part of each cylinder is constructed which is exterior to the other, as in Figs. 392 and 393. Were G joined with M in Fig. 392 and the line then moved up on the groin curves Go and Mg to A, it would generate between those curves one -quarter of the surface of a cloistered arch, but the curves would still be called groins. Fig. 392 represents in oblique projection that portion of the structure which is seen in perspec- tive above the piUars in Fig. 393. THE PERSPECTIVE OF A GROINED ARQH. 237 of the pedestals and abaci, make ah It, E'igr. 3S2. The pillars are supposed to be square, and to stand at the comers of a square floor set with square tiles. Each pillar rests on a square pedestal and is capped by an abacus of the same size except as to thickness. Taking the perspective plane coincident with the faces Fig. 393, of any assumed size; prolong hi until Iv equals the height assigned to the pillar; then complete the front of the abacus on qv as an edge, to given data. Locate on < Z a point whose distance from I equals that of the front face of pillar from the perspective plane, and draw therefrom the perpendicular hGs; also draw the diagonal I h d, giving h for a starting corner on the base of pillar. A parallel to tl through h is cut by the diagonal t Ed at E. The same diagonal gives G and two points on the diagonally- opposite pillar, corresponding to E and G. The vertical line through h is cut by a diagonal from the ■y- corner of the abacus at the point where that edge meets the abacus, and the completion of the perspective of the top is identical with that just described for its base. The prolongation of hv meets a diagonal from q at the point where the front semicircle begins on the top of the abacus. Joining it with the corresponding point on the other abacus and bisecting such line gives the centre of the front curve, which may be drawn with the compasses, as the circle is parallel to the perspective plane. Similarly for the back semicircle. The perspectives of the groins and side semicircles. As the cylinders on which these curves lie are either parallel to or perpendicular to the paper, we may refer to them as the parallel and perpen- dicular cylinders, respectively. The elements of the latter will converge in perspective to the point of sight, as m c and y Le, Fig. 393. On the other cylinder they will be parallel in perspective. A horizontal plane of section, as that through a h (Fig. 392) or B T (Fig. 393) will cut a square ah en from the outside of the structure, and two elements from each cylinder. Either diagonal of this square, as ac (Fig. 392) will cut the elements in points of the groin. In Pig. 393 the diago- nal B d cuts the elements m s and y s at c and e, two points of the groin. These points also belong to elements of the parallel cylinder, and the latter, if drawn, will meet the side walls in points of the side semicircles. This is shown in Pig. 392 by drawing the element fi to meet the trace he at i. In perspective this is seen in the horizontal element through e (Pig. 393) which meets the perspective perpendicular 5 s at point / of the side curve. A number of planes should be treated like B T to give enough points for the accurate drawing of the curves. The tiled floor is made of squares whose diagonal is seen in true size at gj. By laying off the latter on a and drawing diagonals to the vanishing points d, the floor is rapidly laid out. The shadow of the left-hand front pillar and of its abacus. Assume r, and r for the vanishing point of rays and of their horizontal projections, respectively, remembering that these points must be on the same vertical line, since a ray and its plan determine a vertical plane, which can intersect another vertical plane only in a vertical line. The ray from I to r^ meets its projection hr &t the shadow of I on the floor, whence a perpen- dicular to s would give the direction of the shadow of Is. Join h with r; it runs off the pedestal at s, whence a ray to rj will give the shadow s" on the floor, from which s"r is one boundary of 238 THEORETICAL AND PRACTICAL GRAPHICS. the shadow of the pillar. This meets the further pedestal, upon whose front the shadow runs up vertically, being the shadow of a vertical line. On the top of the pedestal the shadow continues toward r till it meets the face of the pillar, where it agaui runs vertically until merged in the shadow of the abacus qv. The shadow of the abacus qv on the diagonally - opposite pillar consists of a horizontal shadow cast by a small portion of the lower front edge; a vertical portion cast by qv; and a portion running SOME PRINCIPLES OF DESIGN AND CRITICISM. 239 from u parallel to sr^ and cast by a part of the edge running from q toward s. The plane of rays through the edge last mentioned would be perpendicular to the paper, and its trace necessarily parallel to that of the projecting plane of the ray of light through the eye. The front semicircle casts a shadow on the rear pillar, whose centre in found by extending the plane of the front of the pillar sufficiently to catch a ray drawn through the centre of the original curve. The radius of the shadow would be the (imaginary) shadow of the radius of the front semicircle. The shadow of the arch- curve on the interior of the perpendicular cylinder is found thus: Pass planes of rays parallel to the axis of the cylinder. R A^ is the trace of one such plane. It is parallel to SJ-j. It cuts the element xS from the cylinder, and the point n from the curve casting the shadow. The ray nr, meets the element x S In a point of the curve of shadow. The shadow begins at the point where a plane of rays, parallel to R N, is tangent to the face curve of the arch. The shadow, P Q, of the side semicircle, is found by taking planes of rays parallel to the axis of the parallel cylinder. The traces of such planes upon the side face of the structure, (which is per- pendicular to V), involve the location of the vanishing point of projections of rays on profile planes. The ray of light through the eye (which meets the perspective plane at rj, projected on the profile plane through the eye, appears at So; hence o, at the level of rj) and on the vertical line through s, is the vanishing point sought. Lines radiating from o, as o w, are perspective traces of planes of rays. Each cuts the arch curve in a point as w, and the cylinder in an element as ik. The ray wr, through the point then meets the element in a point of the shadow. As k is not on the real part of the cylinder, it is useful only in connection with other points similarly found, to determine the shape of the curve PQ. 620. Some hints as to planning a drawing, and on intelligent criticism of works of art. At this point, though we grant acquaintance on the part of the student with the principles of this and the preceding chapter, upon whose correct application the success of the architect or artist so largely depends, there is no certainty that he could make a drawing which should not only be mathematically correct but also pleasing to the eye, or that he could pass just criticism on the work of others. The artistic sense, to be cultivated, must be innate; and originalitj^ or inventive- ness can only in small degree be inculcated by either precept or example. Yet one who is neither the "born artist" or "natural architect" can, by the mastery of a few cardinal principles, be not only guarded against the making of glaring errors, but also have his interest in works of art materially enhanced. In bringing this chapter to a close it seems advisable, therefore, to give a few hints with regard to the more important points upon which successful work depends. In the first place, the location of the f)oint of view is by no means immaterial. It is not well to attempt to include too much in the angle subtended by the visual rays to the extreme outlines of the object drawn, and the frequently - recommended angle of 60° may safely be taken, as, in most cases, the maximum for pleasing efi'ect. Nor should the eye be taken too near the perspective plane, since this involves a degree of convergence amounting to positive distortion, as all must have noticed in photographs of architectural subjects taken at too short range. In case of an error in first loca- tion of view -point, one can diminish the convergence of the lines without reduction in the size of the perspective result, by a simultaneous increase of the distance of the eye from the perspective plane and of the latter from the object. Usually, and, in particular, in an architectural perspective, the eye should not be opposite the centre of the structure, a more agreeable effect resulting from a lack of rigid geometrical symmetry and balance. Nor should the lines of the structure make equal angles with the perspective plane. 240 THEORETICAL AND PRACTICAL GRAPHICS. When viewing a picture, the endeavor should be made to put one's self at the point of sight selected by the artist. In fact, one will instinctively make the attempt so to do; but, to succeed, it is well to bear in mind the principles previously set forth as to the location of vanishing points. Among other essential preliminaries to which careful thought must be given is the quality called ''Balance'''' by artists. It is the adjustment of the various elements of a picture, so that while leaving no doubt as to its purpose there shall not be over - emphasis of its main feature, but a general interest maintained in the various accessories, the office of the latter to be, evidently, how- ever, contributory to the central idea or object. Probably no better example of balance can be found, to say nothing of its illustration of the other requisites of a good picture, than Hofman's well-known painting of Christ preaching on the shore of Galilee. (National Gallery, Berlin). When a drawing has been well planned as to its geometrical character and arrangement, and its main lines pencilled in, the next point to be considered is the style of finish. For architectural work there may be all degrees, from the barest outline drawing in black and white, to the most highly finished water color work, with sky effects, and foliage, water and figure "incident." To secure the sketchy effect which is so desirable, and avoid the harsh exactness of geometrical diagrams, all inked lines should be drawn free-hand except in work upon which considerable free- hand shading is intended. And even when the inked lines are ruled, they may preferably be in a succession of dashes of varying length, rather than in continuous lines. Whatever guide lines are required for the boundaries of surfaces that are to be "rendered" (i. e., brush-tinted) in water colors, should be ruled in pencil only. Probably the most practical as well as pleasing style of work is that in which each stroke of pen or brush suggests, by its location or its weight, quite as much as it actually represents, — impres- sionist work, in technical language. Scarcely second to correct planning and outlining is the chiaroscuro, or light and shade effect due to the values or intensities of the tones given to the various surfaces. Not exactly synonymous with it, yet dependent upon it, is the quality termed atmosphere, upon which the effect of distance largely depends. When well rendered, the foreground, background and middle distance are harmo- niously treated, and the idea conveyed is the same as by a view in nature, changing from the clearness and sharp definition of that which is nearest, to the hazy air and general indistinctness of detail of the remote. The architect ordinarily has considerably less to do with these qualities than the artist, the element of time usually being, for him, of too great importance to permit of the highest finish of which he may be capable; but the fundamental principles upon which they depend, so far as they apply to plane surfaces, with which he is mainly concerned, are the following: Illuminated surfaces, parallel to the perspective plane, and at different distances, are lightest at the front, and get darker in tone as they recede. When unilluminated, the exact opposite is the rule. On an illuminated surface seen obliquely, the lightest part is nearest the observer. This rule is also reversed, like the preceding one, for a surface in the shade when viewed obliquely. When the surface receiving a shadow ■ is of the same nature (material) as that casting it, the shade should be darker than the shadow. The intensity of a shadow diminishes as the shadow lengthens. Should the student wish to go thoroughly into the technique of free-hand sketching in black and white, he should obtain Linfoot's Picture Making with Pen and Ink; while for the beginner in color work as applied to architectural subjects, F. F. Frederick's Rendering in Sepia is admirably adapted. With these, and Delamotte's Art of Sketching from Nature he will find himself fully advised on every point that can arise in connection with the free-hand part of his professional work. Slili