I 



PERSPECTIVES 



PERSPECTIVE. 



IM 



113. By one or other of these principle*, the image* of any definite 

 right line*, aixl therefore of any rectilinear figure, may be obtained. 

 For one or more original line* may be alway* assumed a* pairing 

 through one or more points, the image* of which are required ; so that 

 the indefinite image* of the assumed line* will give those of the |x>int8 

 ought, by it* intersection* with the image* of other line*, in which 

 thoM point* lie : and these assumed line* may be ao taken a* to define 

 the image* with more precision, or to obviate the necessity of drawing 

 radial* of line* but little inclined to the intersecting line* of the plane* 

 in which they lie. I R is the perpendicular distance of the point in 

 which the auxiliary radial cut* the original plane from it* intersecting 

 line ; R therefore i the centre of the circular section of the conical 

 surface before alluded to (102). Make z n' in z v' equal to z R : draw 

 v B to make at v, with r n, the complement of the angle at which the 

 face* of the tetrahedron are inclined to each other. From R', a* a 

 centre, with RS for a radius, describe a circle. Draw lines to touch 

 thi* circle, parallel; respectively to A n, B u, A u. Through the point p, 

 in which the tangent parallel to A B cut* Y z, and through p,, the 

 vanishing point of AH, draw p, p,, the vanishing line of the face of the 

 solid meeting the face A B D in A B : and on the same principle* p, !, 

 p, p,, the vanishing lines of the two remaining faces are found ; then 

 ''. '' '' > n which these vanishing lines intersect each other, will be the 

 vanishing points (94) of the edge* of the solid, and lines accordingly 

 drawn from a,b,d to these points will complete the image of the 

 tetrahedron. 



114. Simple as is the construction above described, for finding the 

 vanishing lines of planes making any proposed angle with a given plane, 

 it may frequently be avoided by availing ourselves properly of the 

 symmetry of the solid to be delineated. Thus, in the example before 

 us, after finding the image, a, b, d, of one face of the tetrahedron, we 

 might have determined the image of the centre of that face by drawing 

 those of the perpendiculars on each side of the triangle from the 

 opposite angles; a lino drawn through this centre and through o, would 

 be the image of one, perpendicular to the plane of the triangle (97) ; 

 this line would pass through the vertex of the pyramid, or through the 

 angular point in which the other three faces meet, by finding the 

 image of this point, which can be easily done by first determining the 

 intersecting point of the ]>erpendicular, and the intersecting line of any 

 plane in which it lies ; then lines drawn from a, b, and d to this image 

 t would complete the figure. 



115. When a vanishing line is obtained, it is frequently requisite 

 to determine its centre and distance, or its principal radial ; this is 

 done by the construction employed to determine the vanishing line 

 r, P,. Thus, to determine the centre, &c., of vanishing line p, p,, draw 

 a parallel to it through c, making c T" equal c V, the distance of the 

 pi< tore ; also draw c T'" perpendicular to the vanishing line for its 

 auxiliary one, cutting the former in c" its centre. Make c" v'" equal 



the principal radial ; then v'" r,, V'p,, v'"P 4 being drawn, they 

 will be the radials of the three sides, ab,tb,ea, of the face of the 

 Kjlid, and will be found, accordingly, to make angles of 60 with each 

 other (83). The radial v'" p, will also be found equal to v' P,, these 

 line* representing one and the same line, only brought into the plane 

 of the picture by the rotation of two different vanishing planes on their 

 vanishing lines. 



116. The perspective projection of a curve may always be found by 

 means of the images of a sufficient number of points in the original, or 

 )>y the projection of some inscribed or circumscribed polygon ; if the 

 curve be a plane one : . in this case the image of a tangent to the 

 original cunre will be a tangent to the image of that curve. For if the 

 image of the tangent meet that of the curve in more than one point, 

 these points must bo the images of points in the original curve through 

 which the original of the tangent must pass : which U contrary to the 

 Nipnosition. But there are some theorems regarding the perspective 

 projection of a circle, and constructions founded on them, which ought 

 to be well understood by the draughtsman. 



117. The rays from the circumference of a circle, obviously, form a 

 conical surface, the section of which, by the plane of the picture, will 

 be one of the conic sections. If the original circle, or base of the cone 

 of ray*, be parallel to the plane of the picture, the image will bo a 

 circle, the radius of which will be to that of the original in the ratio of 

 the distance of the picture (70) to the distance of the plane of the 

 original circle from the vertex (89). 



118. If an original circle do not touch, or cut, the station line of its 



eqnal to the dUtance of point A in t'.ic original from 1U intersecting point, 

 anil rr equal to the Icnxth of the radial of the line ; then A, v being joined, 

 ' _ v 



['lane, it* imago will be an ellipte wherever tho plane of the ]> 

 may be ; unleo* tho section by the plane of the picture happen to be a 

 subcontrary one, an exception to which we shall recur on a subsequent 

 occasion. If the station line be a tangent to the circle, its image will 

 be a fambula ; and if that line cut the circle, the image will bo the 

 opposite branches of an hyperbola, lying on contrary side* of the 

 vanishing lino of the original plane (80). 



AT will cut v r In n, the image of the point A. For however the linw TA, rv 

 tn)f be drawn, the trluftle* YAO, nrr will be nlnmur ; the antecedents TA, 

 rr bring conunt, the convqurnu TA, ar muftt bo w likewise. When this 

 jTinripIc In applied, the two puallrln may be no takrn that the line VA joining 

 their rxtrrmni': in.iy or M at m..;lv right angle*, and o define thf point 

 with preoUloa. 



119. Let KNLM be an original circle, AB being the station line (81) ; 

 the image of the circle will in this instance, be an ellipse. Draw tho 

 diameter c D to the circle, perpendicular to A B ; and let a be the point 

 in c D through which tho chords of the tangent* from all points in A B 

 pass, according to the well known property of the circle. Let v 

 represent the vertex, the vertical plane being supposed to be 

 round on the station line A B, till it coincide with the plane of the 

 circle ; v v' being the director perpendicular to the station line. Make 

 D E, in D R, equal to the tangent to tho circle drawn from D ; bisect v E 

 by a perpendicular, cutting A B in r ; on F as a centre, with r v or F E 

 for a radius, intersect A B in A and B, and draw lines through these 

 pomts and through o ; K L, M N will ,be the originals of the axis of the 

 elliptic image of the given circle, wherever the plane of the picture may 

 be assumed, and at whatever angle that plane and the vertical one be 

 inclined to the plane of the circle. 



120. If A, B be two points in A B, such that each is in the chord of 

 the tangents from the other point produced, then, from the properties 

 of the circle, A K, n i: will be equal respectively to the tangents A N, 

 B i,, drawn from those points ; and the square on x B is equal to the 

 sum of the squares on A N, B L, or on A E, B E. E therefore lies in the 

 circumference of a circle described on A B as a diameter. Since the 

 angle A v B, made by the directors of A L, B N is a right angle by in- 

 struction ; the image* of AL,BXwill be perpendicular to each other, 

 and parallel, respectively, to those of the tangents AN, AM; B I., n K 

 having the same station points with the chords K L, K N. Again, 

 since A L is harmonically divided in K and o, and B x in M and a, 

 the image of K L will be bisected by that of o, and the image of u x 

 will be also bisected by the image of o (87) : hence those images being 

 diameters to the ellipse, mutually bisecting each other, and parallel 

 reciprocally to the tangents which are the images (86) of A x, A M, u K, 

 B L, the images of K L, M N must be conjugate diameters, and since 

 those diameters are perpendicular to each other, they must be the axes. 



121. If v', the foot of the director v v', coincided with i), or if v v' 

 were in the auxiliary vanishing plane, the perpendicular to v E would 

 be parallel to A B, and r <J, s K would be the originals of the axes, 

 which accordingly would be parallel and perpendicular to the inter- 

 secting line. But in every other position of v v', with reference to tho 

 circle, these axes must be oblique to that intersecting line, while the 

 angles they form with it will vary according to the distance of v' from 

 D, and according to the length of the director v v'. 



122. The points i; and K will not be common to two or more con- 

 centric .circles, the originals of the axes of the elliptic projections "f 

 concentric circles will not be in the same straight lines, nor v. ill they 

 have the same station points, except in the case of v' and D coinciding, 

 when the originals of the axes will bo parallel and perpendicular to A B. 



123. If AB touched or cut the original circle, the originals of the 

 ' axes, ftc., of the parabolic or hyperbolic projections might be found on 



the same principles : but as these curves do not often occur in 

 practical perspective drawing, we shall not dwell on the subject. 



124. The only solids with curved surfaces that need be con- i 

 are, the cylinder, the cone, and the (): 



125. If a line be conceived to pass through the vertex, parallel to 

 the axis of a cylinder, whether right or oblique, two planes p. 

 through this parallel will touch the evlimler in two lines of its surface, 

 also parallel to its axis, which will be the originals of the straight out- 

 line of the perspective projection, or image, of that cylinder. 



126. These two tangential planes will cut tho plane of the base of 

 the solid, or that of any section of it whatsover, in two lines, \\lii, -h 

 will be tangents to the curve of that section. And the parallel 

 axis through the vertex is obviously the radial of that axis, which, l>y 

 its intersection with the plane of the picture, will determine tho 

 vanishing point of that axis; and thin vanishing |>oint, it must be 

 remembered, is the image of the point, in any original plane, ou 



the cylinder in which the two tangents to the curve of the section in 

 that plane meet, which have been shown to be the originals of the 

 outline of the solid. 



127. If therefore tho image of the ba.:o or of any section of the 



