THE POPULAR EDUCATOR. 



GEOMETRICAL PERSPECTIVE. XIX. 



SHADOWS CAST UPON INCLINED PLANES. 



PLANES or surfaces upon which shadows are cast may be in any 

 position. We have in the previous lessons considered those 

 planes only which are horizontal or vertical, and we now intro- 

 duce those that are inclined. One or two important and leading 

 principles will first engage our attention. 



Tho indefinite projection of the shadow of a given line 

 coincides with a plane passing through the source of light (the 

 sun) and the given line ; this we call the plane of shade. 

 Suppose in Fig. 87, s to be the sun, a I an object, say a post, 

 casting a shadow, the ray from s through a to c will determine 

 the length of the shadow b c (see Lesson XVI.) ; then the space 

 inclosed by a b c is deprived of light by the object a b, therefore 

 the triangle a b c is the plane of shade. When the plane of 

 shads is intersected by any surface, the form and extent of the 



culty presents itself, the meaning of the trace of the plane of 

 shade, and how it represents the plane. Planes in space in 

 projection are represented by their traces only. Thus, in Figs. 

 88, 89, 90, the traces h n and h c are the vertical and horizontal 

 traces of the plane abed; and according to the positions of 

 these traces, we understand the positions of the planes. In 

 Fig. 88 the plane is at an angle with the ground, and perpen- 

 dicular to the vertical plane ; in Fig. 89 it is at an angle with 

 both planes of projection ; in Fig. 90 it is perpendicular to both 

 planes of projection. In Linear Perspective the line PP, the 

 picture plane, is the horizontal trace of an indefinite perpendi- 

 cular plane ; the line HL, horizontal line, is the vertical trace on 

 the picture plane of a plane passing through the eye and 

 parallel with the ground. To determine the plane of shade we 

 must necessarily project its trace, by drawing a, straight line 

 through the vanishing point of the line projecting the shadmv 

 and the vanishing point of the sun's rays ; because both these 



92. 



shadow upon that surface are determined according to the 

 inclination of the surface with the plane of shade. Thus, in 

 Fig. 93 the trace of the plane of the shade of the pole is A B. 

 The pole and its shadow are both lying in this plane ; the 

 zigzag form the shadow takes arises from the surfaces (the 

 walls and roofs), which cut this plane, being irregular, or in 

 other words, forming various angles with the plane of shade. 

 To illustrate this change in the direction of the course of the 

 shadow that is, to show why the shadow of the pole is so 

 angular let the pupil hold a pencil in an inclined position under 

 a lamp, and allow the shadow to fall upon a slip of cardboard, 

 placing the board first in an horizontal position, then in a per- 

 pendicular one, then at an angle with the table, afterwards 

 turn it, so that it shall be parallel with the pencil, he will at 

 once see that according to the position of the cardboard, as it 

 intersects the plane of shade, so will the inclination, position, 

 and length of the shadow be affected, and he will also see the 

 reason for the varied form of the shadow of the pole in Fig. 93. 

 It will now be evident that in order to project the shadows of 

 objects upon inclined planes we must determine the plane of 

 shade, which is accomplished by drawing its trace ; here c diffi- 



vanishing points are in the plane of shade. Then the vanishing' 

 points for the shadow of a line, projected upon various inclined 

 planes, will be found upon the trace of shade at the intersections 

 of the traces of the inclined planes upon which the shadow falls, 

 We shall refer to this again in a problem to illustrate it. 



PROBLEM LIII. (Fig. 91) is a square block of masonry be- 

 yond which a, beam projects. The sun is in front of the picture. 



It will be observed that a line is drawn from the vanishing 

 point, VP 2 , to which the beam, the object that causes the shadow, 

 retires, through the vanishing point of the sun's elevation, 

 VPSE, to a perpendicular line drawn from the vanishing point of 

 the object upon ivhich the shadow is cast (the block), meeting 

 at VP 3 . This is the trace of the plane of shade. Consequently 

 the shadow from the beam on the block is drawn in the direc- 

 tion of VP 3 ,* whilst the rays which determine its length are 



* Because the plane or surface of the block upon which the shadow 

 falls vanishes in a perpendicular line through VP 1 . Therefore, any 

 line lying upon that plane will have its vauibhing point somewhere ia 

 that perpendicular line, according to the angle of inclination ; those 

 which are horizontal, like the upper and lower edges of the face, 

 vanish on the HL at VP 1 j VP 3 is in the plane of shade. 



