392 



THE POPULAR EDUCATOR. 



Pig. 83. 



GEOMETRICAL PERSPECTIVE. XVIII. 



PEESPECTIVE OF SHADOWS (continued). 



WE will now give some examples of the third position of the 

 sun, as given in Lesson XVI., page 260, that is, when the sun is 

 behind the picture, or in other words, when the object upon 

 which the light falls is between the sun and the spectator. 



We have said before that, when the sun is behind the picture, 

 the vanishing line for the sun's elevation is inclined upwards, 

 determining the VP for the sun's elevation over the VP for the 

 sun's inclination. To assist in explaining this, we will again 

 make use of a line to represent 

 a pole. 



Let a 1) (Fig. 83) represent a 

 pole. VPSI is the vanishing 

 point of the sun's inclination 

 at an angle of 50 ; DVPSI is 

 its distance point, from which is 

 drawn the vanishing line at an 

 angle of 40 with the horizon 

 to VPSE, the vanishing point of 

 the sun's elevation. Draw a 

 line through a directed from 

 VPSE until it cuts another line 

 drawn through b directed from 



VPSEV 



a o to be a c. The other corresponding ray of elevation through 

 d, meeting the ray of inclination at c, determines the shadow of 

 b d to be c e, which has the same vanishing point as b d, 

 viz., VP 1 . The shadow of the perpendicular edge in n is mo; 

 join e o, which will be the shadow of d n ; then upon the same 

 principle as in the case of the shadow of b d will the shadow 

 of d n, viz., e o, vanish at VP 2 . To determine the edge of the 

 shadow as it crosses the slab, draw a perpendicular line from 

 j the point in which the shadow line a c cuts the base line of 

 the slab, either from h or i, to meet the upper edge of the slab 

 ! in s or v ; if- preferred, both lines may be drawn, then by join- 

 ing s and v the edge of the 

 shadow of the block crossing 

 the slab will be determined ; if 

 one of the lines only is drawn, 

 then the edge of the shadow 

 represented by s v must be di- 

 rected towards VPSI. 



The rays of the sun's incli- 

 nation and those of the eleva- 

 tion must always be drawn in 

 order to obtain the termination 

 or extent of the shadow, let 

 the position of the object be 

 as it may, as in Fig. 85, where 



VPSI 



VPSI ; these lines intersecting at c will give the extent and 

 position of the shadow c b. 



PROBLEM LII. (Fig. 84). A Slab and a Block. The slab 

 is placed on its edge at an angle of 30 with the PP. The 

 block is in an upright position, and parallel with the slab at 

 some distance beyond it; sun's elevation 50, and inclination 

 40. 



This problem being given as an exercise for our pupils, we 

 will leave them to decide for themselves upon the proportions 

 of the blocks, and upon the remaining perspective conditions ; 

 they will understand that the process for casting the shadow 

 will be the same as shown by Fig. 83. At the same time we must 

 draw attention to some parts of the shadows where the con- 

 struction may not be clearly understood. The ray of the sun's 

 Inclination, through a of the block, meets the one through b 

 rom the sun's elevation, making the extent of the shadow of 



the chimney which is behind the building is brought down 

 to the ground at a, although the line b a coincides vrith 

 the further side of the building, and thus, being on the same 

 plane, is brought down to Uhe ground-line of the wall, yet 

 the edge of the shadow of the chimney is found by drawing 

 a line through a from the vanishing point of the sun's incli- 

 nation to meet the ray from the one of the sun's elevation ; 

 the same as though the chimney had projected from the wall, 

 when the ray of inclination must be produced from the base of 

 the chimney to obtain the edge of the shadow at c ; because the 

 plane of the wall does not vanish at the same vanishing point 

 as the sun's inclination, the retiring edge of the shadow at c 

 will vanish at the same VP for the corresponding edge of the 

 chimney. The pupil will notice that the building being placed 

 j:t an angle of 45 with the picture plane, the distance points of 

 the station point are its vanishing points. If the sun's inclL- 



