66. The perspective of the house in the last problem was completely drawn, using only the vanishing points for the two principal systems of horizontal lines. By this method it is possible to find the perspective projection of any object. But it is often advisable, for the sake of greater accuracy, to determine the vanishing points for systems of oblique lines in the object, in addition to the vanishing points for the horizontal systems.

Take, for example, the lines gvyv and xpzp in Fig. 22. The perspective projections of those two lines were obtained by first finding the points gp,yp,xp, and zp, and then connecting gp with yp, and xv with zP. As the distances between g and y, and x and z, are very short, a slight inaccuracy in determining the positions of their perspectives might result in a very appreciable inaccuracy in the directions of the two lines gp yp and .r'V. These two lines belong to the same system, and should approach one another as they recede. Unless the points which determine them are found with great care, the two lines may approach too rapidly, or even diverge, as they recede from the eye. In the latter case, the drawing would be absolutely wrong in principle, and the result would be very disagreeable to the trained eye. If, however, the perspective of the vanishing point of the system to which these two lines belong, can be found, and the two lines be drawn to meet at this vanishing point, the result will necessarily be accurate.

The line through rp, which forms the intersection between the roof of the house and the left hand face of the chimney, is a still more difficult one to determine accurately. Its length is so short that it is almost impossible to establish its exact direction from the perspective projections of its extremities. If the perspective of its vanishing point can be found, however, its direction at once becomes definitely determined.

67. It is not a difficult matter to find the perspective of the vanishing point for each system of lines in an object. The method is illustrated in Fig. 23. The general method for finding the perspective of the vanishing point for any system of lines has already been stated in § 24 g, and illustrated in Figs. 16, 17, and 18, §§ 50, 51, and 52. It remains only to adapt the general method to a particular problem, such as that shown in Fig. 23.

The plan and elevation of a house are given at the left of the figure. The diagram has been drawn at the top of the sheet, turned at the desired angle. The assumed position of the station point is indicated by its two projections, SPV and SPH. VH necessarily passes through SPV.

68. In order to find the perspective of the vanishing point of any system of lines, the vertical and horizontal projections of some element of the system must be known (see method of Problem III.). The diagram gives the horizontal projection of every line in the object which is to appear in the perspective projection. The diagram, however, has been turned through a certain horizontal angle in order to show the desired perspective view, and there is no revolved elevation to agree with the revolved position of the diagram. A revolved elevation could, of course, be constructed by revolving the given plan of the object until all its lines were parallel to the corresponding lines in the diagram, and then finding the revolved elevation of the object corresponding to the revolved position of the plan.

Note. - The method of constructing a revolved elevation has been explained in detail in the Instruction Paper on Mechanical Drawing, Part III., Page 12.

Having constructed the revolved plan and elevation of the object to agree with the position of the diagram, we should then have the vertical and horizontal projections of a line parallel to each line that is to appear in the perspective drawing, and the method of Problem III. could be applied directly.

This is-exactly the process that will be followed in finding the vanishing points for the oblique lines in the object, except that instead of making a complete revolved plan and elevation of the house, each system of lines will be considered by itself, and the revolved plan and elevation of each line will be found as it is needed, without regard to the remaining lines in the object.

69. All the lines in the house belong to one of eleven different systems that may be described as follows : -

A vertical system, to which all the vertical lines in the house belong. The perspective of the vanishing point of this system cannot be found within finite limits (§ 51).

Two horizontal systems parallel respectively to ob and ad (see diagram). The perspectives of the vanishing points of these systems will be found in VH (§ 24 c, note).

Five systems of lines vanishing upward, parallel respectively to af, bg, nm, on, and hk (see diagram). The perspectives of the vanishing points of these systems will be found to lie above VII (§ 51, note).

Three systems of lines vanishing downward, parallel respectively to fd, gc, and kl (see diagram). The perspectives of the vanishing points of these systems will be found below VH (§ 51, note).

NOTE. - To determine whether a line vanishes upward or downward, proceed as follows :-

Examine the direction of the line as shown in the diagram. Determine which end of the line is the farther behind the picture plane. If the more distant end of the line is above the nearer end, the line vanishes upward, and the perspective of its vanishing point will be found above VH.

If, on the other hand, the more distant end of the line is lower than the nearer end, the line vanishes downward and the perspective of its vanishing point will be found below VH.

For illustration, consider the line bg. The diagram shows the point g to be farther behind the picture plane than the point b, while the given elevation shows the point g to be higher than the point b. Therefore the line rises as it recedes, or, in other words, it vanishes upward.