324 



THE POPULAR EDUCATOR. 



GEOMETRICAL PERSPECTIVE. XVII. 



PERSPECTIVE OP SHADOWS. 



WE now propose to consider the projection of shadows as they 

 appear under the second conditions mentioned in Lesson XVI. ; 

 viz., when the sun is before, or in front of the picture ; that is, when 

 it is behind the spectator, or when the spectator is between the 

 sun and the object. 



RULE. Draw a line from the station-point, or E, to the hori- 

 zontal line at the same angle with the picture plane at which the 

 horizontal direction of the shadow is said to be inclined ; this 

 will give the VP for the sun's inclination. The length of the 

 shadow is determined according to the sun's elevation (or height 

 in the heavens). Therefore the angle of elevation must be con- 

 structed by drawing a line, at the given angle of elevation, from, 

 the distance point of the vanishing point of the sun's inclination 

 to meet the perpendicular line drawn through the VP of the 

 sun's inclination. This will bo the VP for the sun's elevation, 



a long beam standing on its end, and opposite a point 2 feet 

 from the nearest end of the block. The beam is 1 foot G inches 

 square at the base, 8 feet high, and 1 foot space between the 

 block and the beam. Sun's inclination 38 ; elevation 30, vanisli- 

 ing point of the sun to the left of the eye. Line of sight 5 feet. 

 Distance from the pp 6 feet. 



Trusting our pupils will be able to represent the perspective 

 of the solids, we shall limit our instructions, for that part of the 

 drawing, to merely reminding them of some of the leading par- 

 ticulars in the process of construction, a is 2 feet to the left of 

 the eye, 6 is 3 feet from a, for the purpose of finding the nearest 

 angle of the block within the picture by drawing from b to DE. 

 To find the point in the block to which the beam is opposite, 

 rule a line from the near angle of the block to the BP at c ; make 

 c d equal to 2 feet, and rule from d back again to the base of 

 the block, directed by DVP 1 this is cutting off from the near 

 angle of the block a distance of 2 feet on the line of its base; 

 rule from the point thus found towards the PP, directed by DVP 2 ; 



and will be the point of direction to determine the lengths of 

 the shadows, by drawing to it lines from the angles and pro- 

 jecting parts of the object, to cut tho^e drawn from the object 

 in the direction of the VP for the sun's inclination. When the 

 position of the sun is, as in the present case, before the picture, 

 the line forming the angle of the sun's elevation is drawn down- 

 wards. When the sun is behind the .picture, the line of the angle 

 is drawn upwards ; this latter case will be treated upon in a 

 future lesson. To render the above rule as clear as possible, we 

 have introduced a very simple example (Fig. 79), giving only the 

 vanishing points for the representation of the shadow. Let AB 

 be a pole in a perpendicular position, VPSI is the vanishing point 

 for the sun's inclination at an angle of 35 3 , and VPSE, the 

 vanishing point for the sun's elevation, is at an angle of 30 

 with the horizon ; therefore the shadow of the pole on the 

 ground retires towards its vanishing point on the HL, and its 

 length is determined by a line drawn from the top of the pole 

 towards the vanishing point of the sun's elevation, producing 

 AC, the shadow of AB. Our pupils will perceive' that the prin- 

 ciples of the perspective of shadows closely resemble those which 

 belong to horizontal and inclined planes. 



PROBLEM XLTX. (Fig. 80). A rectangular block of stone 2 

 feet wide, 6 feet long, and 3 feet high, is lying Jiorizontally on its 

 narrowest side; its face is at an angle of 40 with the PP, 3 

 feet within, and 2 feet to the left of the eye. Parallel to it is 



upon the last line a portion of 



1 foot must be cut off to obtain 



the perspective distance between 



the block and the beam, this 



will be between e and /. The 



lines for the production of the 



shadows are dotted, drawn from the projecting angles of tha 



solids to the vanishing point of the sun's elevation (VPSE) to 



cut the lines drawn from the plans or bases of the projecting 



angles towards the vanishing point for the sun's inclination 



(VPSI). The intersection of these lines will limit the extent of 



the shadows, as shown in Fig. 79. 



PROBLEM L. (Fig. 81). A circular board in a perpendicular 

 position, 6 feet diameter, and having a square opening in the centre 

 3 feet icide. The plane of the board is at an angle of 50 with the 

 picture plane. Sun's elevation 30, and inclination 40. Height 

 of the eye, 4 feet 6 inches ; other conditions at pleasure. 



After drawing the HL, and determining the station point, 

 vanishing points, and distance points, the plan of the circle (A) 

 must be made with the additional working lines for the purpose 

 of obtaining the true form of the circle when placed in a retiring 

 and perpendicular positions (see Fig. 31, page 8; Fig. 36, p. 73, 

 and Fig. 40, p. 141 in Vol. III.). It will then appear as a circle 

 in a squaro. If the pupil will turn back to the above figures, he 

 will at once understand why the points in the base of the plan 



