Its perspective projection will evidently be smaller than if the vertical edge ae were in the picture plane, as was the case in Figs. 19 and 20, and the perspective of ae will evidently be shorter than the true length of ae. There is, therefore, no line in the object that can be used for a line of measures. It becomes necessary to extend one of the vertical faces of the block until it intersects the picture plane, and shows by the intersection its true vertical height. Thus, the plane abfe has been extended, as indicated in the diagram, until it intersects the picture plane in the line mn. This intersection is an auxiliary line of measures for the plane abfe, and mpnp shows the true vertical height of this plane.

Either of the other vertical faces of the block, as well as the face abfe, might have been extended until it intersected the picture plane, and formed by this intersection a line of measures for the block.

The vanishing points for the various systems of lines have been found as in the previous cases.

From mp and np, the horizontal edges of the face abfe vanish to vab. ap will be found on the upper edgt of this face, vertically below the intersection of HPP with the horizontal projection of the visual ray through the point a in the diagram. A vertical line through ap will represent the perspective of the nearest vertical edge of the block, and will establish the position of ep.

In a similar manner, bp will be found vertically below the intersection of HPP with the horizontal projection of the visual ray through the point b in the diagram. A vertical line through bp will establish fp, and complete the perspective of the face abfe. Having found the perspective of this face, the remainder of the block may be determined as in the previous problems.

Note. - Instead of being some distance behind the picture plane, the block might have been wholly or partly in front of the picture plane. In any case, find the intersection with the picture plane of some vertical face of the block (produced, if necessary). This intersection will show the true vertical height of the block.

At this point the student should solve Plate II.

60. PROBLEM V. Fig. 22. To find the perspective of a house, the projections of which are given.

The plan, front, and side elevations of the house are shown in the figure. The side elevation corresponds to the projection on the profile plan, used in the study of projections. This problem is a further illustration of the method of revolved plan and of the use of horizontal vanishing points and auxiliary lines of measures. It is very similar to the three previous problems on the rectangular blocks.

The first step in the construction of the perspective projection is to make a diagram (§ 53) which shall show the horizontal projections of all the features that are to appear in the drawing. The diagram should be placed at the top of the sheet, and turned so that the sides of the house make the desired angles with the picture plane. In Fig. 22 the diagram is shown with the long side making an angle of 30° with the picture plane. The roof lines, the chimney, and the positions of all windows, doors, etc., that are to be visible in the perspective projection, will be seen marked on the diagram.

The nearest vertical edge of the house is to lie in the picture plane. This is indicated by drawing HPP through the corner of the diagram which represents this nearest edge.

VH may be chosen at any convenient distance below HPP.

The position of the station point is shown in the figure by its two projections SPV and SPH. SPV must always be in VH. The distance between SPH and HPP shows the distance of the observer's eye in front of the picture plane (§ 43).

vab and vad may be found as in the preceding problems.

The position of the plane on which the object is to rest should next be established by drawing VHl, the distance between VH and VHt showing the height of the observer's eye above the ground (§ 44).

In addition to the plane of the ground represented by VH1, a second ground plane, represented by VH2, has been chosen some distance below VH1. In the figure, two perspective projections have been found, one resting on each of these two ground planes. The perspective which rests upon the plane represented by VHI, shows the house as though seen by a man standing with his eyes nearly on a level with the tops of the windows (§ 29). The view which rests on the plane represented by VH2 shows a bird's-eye view of the house, in which the eye of the observer (always in VII) is at a distance above the plane on which the view rests, equal to about two and one-half times the height of the ridge of the house above the ground.

These two perspective projections illustrate the effect of changing the distance between VH and the vertical trace (§ 34, note) of the plane on which the perspective projection is supposed to rest. The construction of both views is exactly the same. The following explanation applies to both equally well, and the student may consider either in studying the problem.

61. We will first neglect the roof of the house, and of the porch. The remaining portion of the house will be seen to consist of two rectangular blocks, one representing the main body of the house, and the other representing the porch.

The block representing the main part of the house occupies a position exactly similar to that of the block shown in Fig. 19. First consider this block irrespective of the remainder of the house. A vertical line dropped from the corner of the diagram that lies in HPP will be a measure line for the block, and will establish, by its intersection with VH1 (or VH)the position of the point ep, in exactly the same way that the point ep in Fig. 19 was established. epap shows the true height of the part of the house under consideration, and should be made equal to the corresponding height avev, as shown by the elevations. The rectangular block representing the main part of the house may now be drawn exactly as was the block in Fig. 19, Problem IV.