83. The following problems illustrate short and convenient methods of construction for determining the shadows of lines, surfaces and solids, in the positions in which they commonly occur in architectural drawings. These methods here worked out with regard to the co-ordinate planes apply also to parallel planes.

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FIG36.

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F1G.37

84. They will be found to be of great assistance in casting the shadows in architectural drawings. The latter seldom have the plan and elevation on the same sheet, and these methods have been devised to enable the shadows to be cast on the elevation without using construction lines on the plan or profile projection.

Such distances as are needed and obtained from the plan, can be taken by the dividers and applied to the construction in the elevation.

In casting shadows it will be found convenient to have a triangle, one of whose angles is equal to the true angle which the ray of light makes with the co-ordinate plane. See Fig. 36. "With such a triangle the revolved position of the ray of light can be drawn immediately without going through the operation of revolving the ray parallel to one of the co-ordinate planes.

85. Problem XVII. To construct the shadow on a co=ordi-oate plane of a point.

It will lie on the 45° line passing through the point and representing the projection of the ray of light on that plane. It will be situated on the 45° line at a distance from the given point, equal to the diagonal of a square, the side of which is equal to the distance of the point from the plane.

Given the vertical projection of the point a situated 2 inches from the V plane, to construct its shadow on V. Fig. 37.

From the point av draw the 45° degree line avavs equal in length to the diagonal of a square whose sides measure 2 inches. Then avs is the required shadow.

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FIG-38.

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FIG.39.

86, Problem XVIII. To con= struct the shadow of a line perpendicular to one of the co=ordi nate planes.

(1) It will coincide in direction with the projection of the ray of light upon that plane, without regard to the nature of the surface upon which it falls.

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FIG.40.

FIG'41.

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(2) The length of its projection upon that plane will be equal to the diagonal of a square, of which the given line is one side.

Given the vertical projection of the line ah perpendicular to V, 2 inches long and 1/2 inch from V, to construct its shadow on Y. See Fig. 38. Find the shadow of the point av by Problem XV11.

From the point avs draw the 45° line avs bvsequal to the diagonal of a square 2 inches on each side.

86. Problem XIX. To construct the shadow of a line on a plane to which it is parallel.

(1) It will be parallel to the projection of the given line.

(2) It will be equal in length to the projection of the line.

Given the vertical projection of the line ab, parallel to V, 2 inches in length and 1/2 inch from V, to construct its shadow on.

V. See Fig. 39.

Find the shadow of av by Problem XVII.

Draw avsbvs parallel and equal in length avbv.

87. Problem XX. To construct the shadow of a vertical line on an in= clined plane parallel to the ground line.

It makes an angle with the horizontal equal to the angle which the given plane makes with H

Given the vertical projection of a vertical line abf its lower end resting on a plane parallel to the ground line and making an angle of 30° with H, to construct its shadow on this inclined plane. See Fig. 40. Through the point bv draw the 30° line bvavs. The point avs, the end of the shadow, is determined by the intersection of the 45° line drawn through the end of the line av.

88. Problem XXI. To construct the shadow on a co-ordi-nate plane of a plane which is parallel to it.

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FIG .42,.

(1) It will be of the same form as that of the given surface.

(2) It will be of the same area.

If the plane surface is a circle, the shadow can be found by rinding the shadow of its center, by Problem XVII, and with that as a center describing a circle of the same radius as the given circle.

Given a plane parallel to V, 1/2 inch from V and 1 1/2 inches square, to construct its shadow on Y. See Fig. 41.

Find the shadow of any point av for example, by Problem XVII. On that point of the shadow construct a similar square whose side equals 1 1/2 inches.

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89. Problem XXII. To construct the shadow on V of a circular plane which is parallel to H, or which lies in a profile plane.

Given avovbv, the projection of a circular plane perpendicular to V and H, 2 inches in diameter, its center being 2 1/4 inches from V, to construct the shadow on V. Figs. 42 and 43. The shadow of ov, the center of the circular plane is found by Problem XVII. About ovs as a center, construct the parallelogram ABCD made up of the two right triangles ADB and DBG, the sides adjacent to the right angles being equal in length to the diameter, 2 inches, of the circular plane. Draw the diameters and diagonals of this parallelogram. The diameter TW is equal to the diameter of the given circle and parallel to it.

With ovs as a center and OD and OB as radii, describe the arcs cutting the major diameter of the parallelogram in the points E and F. Through E and F draw lines parallel to the short diameter, cutting the diagonals in the points G, H, M and 1ST. These last four points and the extremities of the diameters R, S, T, and W, are eight points in the ellipse which is the shadow of the given circular plane on V. A similar construction is followed for finding the shadow on V of a circular plane parallel to H. Fig. 43.