SHORT METHODS OF CONSTRUCTION.
83. The following problems illustrate short and convenient methods of construction for determining the shadows of lines, stir. faces 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.
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 el6vation.
In casting shadows it will be found convenient to have a tri angle, 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 revolv ing the ray parallel to one of the co-ordinate planes.
85. Problem XVII. To construct the shadow on a co-ordi nate plane of a point.
It will lie on the 45° line passing through the point and rep resenting 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 a' draw the 45° degree line avays equal in length to the diagonal of a square whose sides measure 2 inches. Then ays is the required shadow.
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.
(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 ab perpendicular to V, 2 inches long and 1- inch from V, to construct its shadow on V. See Fig. 38. Find the shadow of the point av by Problem From the point ays draw the line aysbvs equal to the diag. onal 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 pro jection of the line ab, par. allel to V, 2 inches in length and inch from V, to construct its shadow on V. See Fig. 39.
Find the shadow of av by Problem XVII.
Draw aysbv3 parallel and equal in length wk.
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 ab, its lower end resting on a plane parallel to the ground line and making an angle of 30° with H, to construct its on this inclined plane. See Fig. 40. Through the point by draw the 30° line Lvays. The point ay., the end of the shadow, is determined by the intersection • of the 45° line drawn through the end of the line ay.
88. Problem XXI. To construct the shadow on a co-ordi nate plane of a plane which is parallel to it.
(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 finding the shadow of its center, by Problem XVII, and with that as a center describing a circle of the smile radius as the given circle.
Given a plane parallel to V, inch from V and Winches square, to construct its shadow on V. See Fig. 41.