423 



PERSPECTIVE. 



PERSPECTIVE. 



430 



projecting plane (9) of the given ray. Thia perpendicular plane must, 

 therefore, be the elevation-projecting plane of the given ray. 



154. By the same construction applied to the other projection, the 



elliptic elevation adbe, of the circle separating the light and shade on 

 the sphere, and the elliptic shadow of the sphere on the vertical 

 co-ordauate plane, may be obtained. 



155. It is clear that in this example the two elliptic outlines of the 

 shadows of the sphere on the co-ordinate planes, must cut Y z in two 

 common points ; because the segments of the ellipse on either side of 

 T z of each outline is the projection on the one co-ordinate plane of 

 that portion of the cylinder of rays which form? on the other co-ordi- 

 nate plane the portion of the outline of the shadow on the same side 

 of T z. ( L, perpendicular to Y z, is the trace of the elevation pro- 

 jecting plane of c s, c ; o, o, is the point in which this same plane cuts 

 the trace of the given plane, consequently L o is the plan of the inter- 

 section of those two planes, and T', in which this line is cut by the plan 

 of the ray c s is the intersection of that ray, and the given plane ; the 

 elevation ( of the same intersection may be obtained by applying the 

 same constructions to the other traces and projections. 



156. The two pair of parallel planes, which are respectively perpen- 

 dicular to the co-ordinate planes, and therefore to each other, and 

 which are parallel to the given ray, touch the sphere in the points A, a ; 

 B, 6 ; D, d ; r.,e. These four planes will be cut by the plane L M n in 

 a parallelogram, the sides of the projections of which must be parallel 

 to those of the ray c 8, c, i, and to the lines L o, to n. Draw t I/ per- 

 pendicular to o t, and make ( L' equal to t L ; join o I/, which will 

 represent the intersection of the projecting plane with L M ; draw 

 lines through if, e , parallel to c' *, and from the points in which tbeae 

 parallels cut o I/ draw parallels to L' ( to cut o ( ; again lines drawn 

 through these last intersections parallel to ir n will be the two sides of 

 the elevation of the rectangle above mentioned ; the parallel tangents 

 at a and 6 will complete the figure ; and ol,>cn, will cut the opposite 

 tides in the points in which the elliptic outliue of the shadow of the 

 sphere will touch those sides, or the points which represent the shadows 

 of d, e, a, and 6. 



157. The plan of this parallelogram may be determined in the same 

 manner, or by the other constructions explained for determining the 



ions on the other co-ordinate plane from those already deter- 

 mined on the first, and which are sufficiently indicated in the figure 

 to render further description of them unnecessary. 



158. If L represent a luminous body, and P a point, then, by 



imagining a plane to pass through them, the intersection of that plane 

 with the plane on which the shadow is cast will cut the ray L p in Q, 

 the shadow of the point. To determine this intersection, we have only 

 to draw two parallel lines through L and t, in any direction, and deter- 



mine the points I and p, or t, p', in which these parallels meet the 

 plane of the shadow : then Ip, L P being drawn, they will cut each 

 other in Q, the shadow of the point. This is the principle employed 

 in the following construction. 



159. Let abcdefg be the perspective projection of a cube, c being 

 the centre of the picture, cv the distance of the picture, p, x the 

 vanishing line of the face abed, and Y z its intersecting line ; while 

 Y 7 z' is that of the face efg, parallel to the former. Let Y z and w z ba 

 given as the vanishing and intersecting lines of a plane, on which the 

 shadow of the cube, as cast by the luminous * body given in position, 

 Is to be determined. 



160. X z, X w, being drawn, will represent the lines in which the 

 plane of the shadow intersects those of the parallel faces of the solid 

 (94). If we suppose planes parallel to that of the picture to pass 

 through the various points of the cube, as a, these will intersect the 

 two original planes in lines, saaa, a a', parallel to Y z, z w ; and a line, 

 a a', through the point of the cube, parallel to the auxiliary vanishing 

 line, will meet a a' in the point a',* which will be the oblique pro- 

 jection of the point a on the plane of the shadow. Therefore, by 

 drawing lines through the points a, b, c, d, parallel to Y z, to cut X z in 

 o, 3, . . !, then lines parallel to w z, through a, $, .. .$, will cut lines 

 parallel to w c', drawn through a, b, c, d, in the oblique projections of 

 those pointa on the plane of the shadow, and by referring e,f,g...io 

 w x, in the same way, we obtain the oblique projections of the other 

 angles of the cube. 



161. Since the sides of the cube at, cd, ef, &c., are parallels, their 

 oblique projections will be parallels (59), consequently the images of 

 these parallels a'V, c' d', <i '/', &c., will have a common vanishing point 

 p' in the vanishing line of the plane in which the oblique projections 

 lie ; for the same reason, a'd', 6V,/.'/', c., will have a common vanish- 

 ing point p*, in Y x. Now it is obvious that the vanishing points p',, 

 p'j, are, by an extension of the principle, the oblique projections on the 

 plane of the shadow of the vanishing points p, p, of the original sides 

 of the cube ; consequently the former may be determined from the 

 last-named vanishing points by simply drawing lines through them 

 parallel to w c to cut Y x in p',, p",. 



162. If had been given as the image of the point in which a line 

 through the luminary perpendicular to the plane y z met that plane, 

 the image * of the luminary would be determined by making i *, drawn 

 to the auxiliary vanishing point Q, the image of the given perpendicular 

 height of the luminary above the original plane. A line through * 



| parallel to w d will meet cfk produced in /, the oblique projection of 

 j the luminary on the plane YZ.* Its oblique projection I' on the plane 

 of the shadow may be either determined as those of a, b, c, d, &c , were, 

 or by drawing a line, as at, at pleasure, to cut the vanishing line x p 

 in some vanishing point ; this vanishing point may be transferred to 

 x Y by a parallel to c w ; then a line drawn through a ', the oblique 



That is to say, (t i the peripectict image of the oblique projection of the 

 original point of which a is the perspective image* 



