33. The plane of the ground is always represented by its intersection with the picture plane (see VH, Fig. 7). Its only use is to determine the relation between the plane of the horizon and the plane on which the object rests (§ 29). The true distance between these two planes is always shown • 1,y the distance between VII and VII, as drawn on the picture plane.
34. To find the perspective of a point determined by its vertical and horizontal projections.
Fig. 8 is an oblique projection showing the two coordinate planes at right angles to each other. The assumed position of the plane of the ground is indicated by its vertical trace The vertical trace of any plane is the intersection of that plane with the vertical coordinate. The horizontal trace of any plane is the intersection of that plane with the horizontal coordinate.
The assumed position of the station point is indicated by its two projections, SPY and SPH. Since the station point lies in the plane of the horizon (§ 27), it is evident that its true position must coincide with SP", and that (§ 31) its vertical projection must be found in VII, as indicated in the figure. Let the point a represent any point in space. The perspective of point a will be at e, where a visual ray through the point a pierces the pic ture plane (§ 24). We may find ar in the follOwing manner, by using the orthographic projections of the point a'', instead of the point itself. an represents the horizontal projection, and a v repre sents the vertical projection of the point a. . A line drawn horn the vertical projection of the point a to the vertical projection of the station point, will represent the vertical projection of the visual ray, which passes through the point a. In Fig. 8 this vertical projection is represented by the line drawn on the picture plane from e to SP".
A line drawn from the horizontal projection of a to the hori zontal projection of the station point will represent the horizontal projection of the visual ray, which ptisses through the point a. In Fig. 8, this horizontal projection is represented by the line drawn on the plane of the horizon from an to SI'. Thus we have, drawn upon the planes of projection, the vertical and horizontal projections of the point a, and the vertical and horizontal projec tions of the visual ray passing between the point a and tbe station point.
35. We. must now find the intersection of the visual ray with the picture plane. This intersection will be a point in the picture plane. It is evident that its vertical projection must coin cide with the intersection itself, and that its horizontal projection must be in HPP (§ 32). But this intersection must also be on the visual ray through the point a, and consequently the horizon tal projection of this intersection must he on the horizontal pro jection of the visual ray: Therefore, the horizontal projection of this intersection must be the point m", where the line between SPH and aH crosses HPP. The vertical projection of this inter section must be vertically in line with this point, and on the line drawn between SPv and av, and at mv. Since the vertical projection of the intersection coincides with the intersection itself, aP (coincident with mV) must be the perspective of the point a.
36. This is the method of finding the perspective of any point, having given the vertical and horizontal projections of the point and of the station point. The method may be stated briefly as follows :— Draw through the horizontal projection of the point and the horizontal projection of the station point, a line representing the horizontal projection of the visual ray, which passes through the point. Through-the intersection of this line with HPP, draw a vertical line. The perspective of the point will be found where this vertical line crosses the vertical projection of the visual ray, drawn through SPv and ay.
37. It would evidently be inconvenient to work upon two planes at right angles to one another, as shown in Fig. 8. To avoid. this, and to make it possible to work upon a plane sur face, the picture plane (or vertical coordinate) is supposed to be revolved about its intersection with the plane of the horizon, until the two coincide and form one surface. The direction of this revolution is indicated by the arrows s, and s,. After revo lution, the two coordinate planes will coincide, and the vertical and horizontal projections overlap one another, as indicated in Fig. 9.