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42. The methods for finding the shadows of solids vary with the nature of the given solid. The shadows of solids whiCh are bounded by plane surfaces, none of which are parallel, or perpendicular, to the co-ordinate planes, can in general, be found only by finding the shadows of all the boundhig planes. These will form an enclosed polygon, the sides of which are the shadows of the shade lines of the object, and the shade lines of the solid are deter mined in this way. The following is an illustration of this class of solids.

43. Problem IV. To find the shade and shadow of a polyhedron, none of whose faces are parallel or perpendicular to the co-.or= dinate planes.

Fig 20 shows a poly hedron in such a position and of such a shape that none of its faces are per pendicular or parallel to the co-ordinate planes. It is impossible, therefore, to apply to this figure the projections of the rays of light and determine what faces are in light and what hi shade. Consequently we cannot determine the shade line whose shadows would form the shadow of the object.

Under these circumstances we must east the shadows cf all the boundary edges of the object. Some of these lines of shadow will form a polygon, the others will fall inside this polygon. The edges of the object whose shadows form the bounding lines of the polygon of shadow are the shade lines of the given object. Know ing the shade lines, the light and shaded portions of the object can now be determined, since these are separated by the shade lines.

In a problem of this kind care should be taken, to letter or number the edges of the given, object.

44. The edges of the polyhedron shown in Fig. 20 are ab, be, cd, da, ae and bd.

Cast the shadows of each of these straight lines by the method shown in Problem II.

We thus obtain a polygon bounded by the lines bvsevs, cvaays, aysbvs, and this polygon is the shadow of the given solid.

The lines which cast these lines of shadow, ekes, and (MbTs are therefore the shade lines of the object, and, therefore, the face abc is in light and the faces abd, bed and aed are in shade.

The shadows of the edges bd, dc, and ad falling within the polygon, indi cate that they are not shade lines of the given object, and, therefore, they separate two faces in shade or two faces in light. In this example bd

and ed separate two dark faces.

In architectural drawings the object usually has a sufficient number of its planes perpendicular or parallel to the co-ordinate planes, to permit its shadow being found by a simpler and more direct method than the one just explained.

45. Problem V. To find the shade and shadow of a prism on the co-ordinate planes, the faces of the prism being perpen dicular or parallel to the V and II planes.

In Fig. 21 such a prism is shown in plan and elevation. The elevation shows it to be resting on .H, and the plan shows it to be situated in front of V, its sides making angles with V. Since its top and bottom faces are parallel to II and its side faces per pendicular to that plane, we can apply the projections of the rays of light to the plan and determine at once which of the side faces are in light and shade. The projections of the rays R' and 11° show that the faces abyf and adif receive the light directly, and that the two other side faced do not receive the rays of light and are, therefore, in shade. The edges- and di are two of the shade lines. 13,' and are the projections of the rays which are tangent to the prism along these shade lines.

Applying the projection PO in the elevation makes it evident that the top face of the prism is in light and the bottom face is in shade since the prism rests on H. This determines the light and shade of all the faces of the prism, and the other shade lines would therefore be be and ed.

Casting the shadow of each of these shade lines, we obtain the required shadow on V and IL It is evident that the shadows of the edges by and di on II will be 45° lines since these edges.are perpendicular to H (§ 31) Also, their shadows on V will be parallel to the lines themselves since these shade lines are parallel to (§ 30) 40. In general, to find the shadow of an object whose planes are parallel or perpendicular to H or V: (1) Apply to the object the projections of the ray of light to determine the lighted and shaded faces.

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