19. Problem I. To find the shadow of a given point on a given plane.

Fig. 5 shows the plan and elevation of a given point a. It is required to find its shadow on a given plane, in this case the V plane. The shadow of the point a on V will be the point at which the ray of light passing through a intersects V.

Through the H projection ah of the given point, draw Rh until it intersects the lower ground line. This means that the ray of light through a has pierced V at some point. The exact point will be on the perpendicular to the ground line, where Rv drawn through av intersects the perpendicular. The point axs is, therefore, the shadow of a on the V plane.

Rv is also the V projection of the umbra of the point a and it will be seen that the shadow of a on V is the intersection of its umbra with Y.

20. Fig-. C shows the construction for finding the shadow of a point a on H.

21. Fig. 7 shows the construction for finding; the shadow of a point a, which is at an equal distance from both V and H. Its shadow, therefore, falls on the line of intersection of V and H.

22. Fig. 8 shows the construction for finding the imaginary shadow of the point a, situated as in Fig. 5, that is, nearer the V plane than the II. The actual shadow would in this case fall on V, but it is sometimes necessary to find its imaginary shadow on H. The method of determining this is similar to that explained in connection with Fig. 5.

FIG.5.

Draw Rv until it meets the ground line of H.

Erect a perpendicular at this point of intersection.

Draw Rh.

The intersection ahs, of the latter and the perpendicular, is the required imaginary shadow on H of the point a.

FIG-6.

23. The actual shadow of a given point, with reference to the two co-ordinate planes, will fall on the nearer co-ordinate plane.

24. Fig. 9 shows the construction for finding the shadow of a given point a on the V plane when the vertical and profile projections of the point are given.

25. In general, the finding of the shadow of a given point on a given plane is the same as the finding of the point of intersection of its umbra with that plane. To obtain this, one projection of the given plane must be a line and that is used as the ground line. It is necessary to have a ground line to which is drawn the projection of the ray of light, in order that we may know that the ray of light has pierced the given plane.