It will be noticed that the coincidence of the two planes in no way interferes with the method given in § 36, of finding the perspective (ar) of 'the point a, from the vertical and horizontal projections of the point. Thus, the horizontal projection of the visual ray through the point willbe seen, drawn from SPH to•a", and intersecting HPP in the point mn. The vertical projection of the visual ray through the point will be seen pasSing from SPv to ay. And e is found upon the vertical projection of the visual ray, directly under rn.H.
It will readily be understood that in a complicated problem, the overlapping of the vertical and horizontal projections might result in some confusion. It is, therefore, usually customary, after having revolved the two coordinate planes into the pOsition shown in Fig. 9, to slide them apart in a direction perpendicular to their line of intersection, until the two planes occupy a position simila'r to that shown in Fig. 9a.
38. It will be remembered from the, course on projections which the student is supposed to have taken, that horizontal pro jections must always be compared with horizontal, and never with vertical projections, and that in the same way, vertical projections must always be compared with vertical, and never with horizontal projections. It is evident that in sliding the planes apart, the relations between the projections on the vertical plane will not be disturbed, nor .will the relations between the projections on the horizontal plane, and consequently it will make no difference how far apart the two coordinate planes are drawn, provided that horizontal and vertical projections of the same points are always kept in line. Thus, in Fig. 9a, it will be seen that in drawing the planes apart, av has been kept in line with an, my with mil, SPv with SP", etc.
39. It will be observed that in sliding the planes apart, their line of intersection has been separated into its two projections (§§ 31 and 32). HPP, being on the plane of the horizon or hori zontal coordinate, is the horizontal projection of the intersection of the two planes, while VH, being on the picture plane or vertical coordinate, is the vertical projection of the intersection of the two planes. In the original position of the planes (Fig. 9) these two projections were coincident. The distance between IIPP and VII always represents the distance through which the planes have been slid. This distance is immaterial, and will have no
effect on the perspective drawing. VII, represents the vertical trace of the plane of the ground.
In this figure, as in the case of Fig. 9, the student should follow through the construction of the perspective of the point a, applying the method of § 36.
40. Fig. 10 shows the position of the two coordinate planes and of the plane of the ground, as they are usually represented on the drawing board in making a perspective drawing. It is essentially the same thing as Fig. 9a, except that the latter was shown in oblique projection in order that its development from the original position of the planes (Fig. 8) might be followed more readily. The two coordinate planes are supposed to lie in the plane of the paper.
hpp represents the horizontal projection of the picture plane, and VH represents the vertical projection of the plane of the horizon.
As horizontal projections are never compared with vertical projections (§ 38), HPP may be drawn as far from, or as near, VII as desired, without in any way affecting the resulting per spective drawing. HPP and VII were coincident in Fig. 9, and the distance between them in Fig. 10 simply shows the distance that the two planes have been slid apart, as illustrated in Fig. 9a. As already stated, this distance is immaterial, and may be made whatever.is most convenient, according to the nature of the problem.
If HPP should be placed nearer the top of the sheet, a" and SPH, both being horizontal projec tions, would follow it, the relation between these horizontal projec tions always being preserved.
On the other hand, SP", av, a", VII, and all being pro jections on the vertical plane, must. preserve their relation with one another, and will in no way he affected if the group of projections on the horizontal coordinate is moved nearer or farther away. It must be borne in mind, however, that, in all cases, the vertical' and horizontal projections of corresponding points must be kept vertically in line. Thus, aw must always be vertically in line with ay. The vertical distance between these two projections does not matter, provided the distance from all to HPP,'or the distance from av to VH, is not changed. This point 'cannot be too strongly emphasized.