OEOMKTIMr A I. PERSPECTIVE. 



it directly oppoaite the eyo, the vr for it* elevation will 



be over it, that is, over the point of sight, I-H (Fig. 86). Then 



VPSB ia found by drawing the angle of inclination from the 



diatanoe point of the eye or station point, and the ryi of 



Mition are ruled to the PS. 



Wo odriae our pupils to draw the OTOM 

 ami I. look of Fig. 86 at an angle with the 

 : -i. -turn piano, retaining the aamo < 

 .iii.l inclination of tho HUH ; it will bo an 

 for drawing the edges of the aha- 

 dow of the retiring 

 sides, aa previously 

 explained in Problem 

 LII. (Fig. 84). 



It the position of 

 both be the same as 

 those of Fig. 84, it 

 will be aeen how tho 

 vanishing edges of 

 the shadows retire to 

 tho vanishing points 

 of tho solids ; all this 



Fig. 86. 



regard to the ahadow that fall* OB the second, if we draw perpsn. 

 dioolar linea from the points whet* the linea from the vr of the 

 *an' inclination intersect the edge* of the second solid, to its sor- 

 face, the extent of the ahadow falling upon it will be decided. 

 Dt . It may oocor that the object casting the 



polo in Fig. 83 is inclined, say at aa angle 

 of 40, the rays from the vp of the ron's 

 elevation most be drawn M usual; bat in* 

 stead of directing the lines that are drawn 

 through the base of 



from the TP of the 

 snn's inclination to 

 intersect those from 

 the elevation, we must 

 first project the upper 

 end of the pole on 

 the ground (see Fig. 

 87, Vol. III., page 72, 

 where / is the project 

 tion of m), and draw 



VPM 



can bo proved by the \ 

 rays being drawn \ 

 from the sun's eleva- 

 tion to meet the lines 

 through the angles of 

 the bases of the solids 

 from the sun's inclina- 

 tion ; the result would 

 be the same for pro- 

 ducing tho extent of the shadow as if we drew the retiring edges 

 to the vanishing points. If the shadow projected by a solid 

 crosses a second solid, and partly loses its shadow in that of 

 the second, the rays drawn from the sun's elevation through 

 the angles of the first solid will always determine the extent of 

 the shadow that falls, in the first place, upon the second solid, 

 and determine that part of tho shadow upon the ground which 

 is visible, and if necessary also that which ia lost ; and with 



the line from the vr 

 of the arm's inclina- 

 tion through the pro- 

 jected point (/) on the 

 ground, to meet the 

 ray of elevation drawn 

 through the upper 

 end of the pole; then 

 join the intersection 



of these two lines with tho base of the pole, which will be the 

 shadow. Let the pyramid (Fig. 35, Vol. III., page 72) be recon- 

 structed, the same rule applies in this case aa in that of the pole; 

 for if after finding the vanishing points for the snn'a inclination 

 and elevation, we draw a line from the YPSI through the centre 

 of the base (the plan of the vertex) to intersect a line drawn from 

 VPSK through the vertex, and join the intersection with the 

 angles at the base, the form of the shadow will bo given. 



