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 (§ 54).
Two horizontal systems parallel respectively to ab 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, nt, on, and hk (see diagram). The perspectives of the vanishing points of these systems will • to lie above VH (§ 51, note).
-Three systems of lines vanishing downward, parallel respec tively to fd, gc, and kl (see diagram). The perspectives of the vanishing points of these systems will be found be low VH (§ 51, note).
To determine whether a line vanishes upward or downward, proceed as follows :— Examine the direction of the line as shown in the dia gram. 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 dowtheard, and the perspective of its vanishing point will be found below VH.
For illustration, consider the line bg. Tim 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.
In the case of the line fd, the diagram shows the point d to be farther behind the picture plane than the point f, while the elevation shows the point d to be lower than the point f. There fore the line must vanish downward, and its vanishing point be found be'ow If `.ne horizontal projection of a line, as shown by the dia, gram, is parallel to IIPP, the line itself is parallel to the picture plane, and the perspective of its vanishing point Cannot be found within finite limits (§ 54, note). The perspective projections of such a
system of lines will show the true angle which, the elements of the system make with the horizontal- coordinate.
70. The construction for the vanishing points in Fig. 23 is shown by dot and dash lines.
The vanishing points for the two systems of horizontal lines have been found at vab and vad respectively, as in the preceding problems.
Next consider the line af. The first step is to construct a revolved plan and elevation of this line to agree with the position of the diagram. Revolve the horizontal projection (aufH) of the line in the given plan about the point until it is parallel to the line .af in the diagram. During this revolution, the point f" remains stationary, while the point a" describes a horizontal are, until aufH has revolved into the position shown by the red line a,H fH, which is parallel to the line af in the diagram. The vertical projection avf° must, of course, revolve with the horizonthl projec tion. The point fv remains stationary, while the horizontal arc described by the point a shows in vertical projection as a horizon tal line. At every point of the revolution the vertical projection of the point a most be vertically in line with its horizontal pro. jection. When a" has reached the position a,",,e will be vertically above a,' at the point a,°, and ct,yv will be the revolved elevation of the line.
We now have the vertical and horizontal projections (aNv and aff R) of an element of the system to which the roof line, rep resented in the diagram by af, belongs. The vanishing point of this system may be determined as in Problem III. Draw through SPH a line parallel to (Of" (or af in the diagram), representing the horizontal projection of the visual element of the system. Draw through SPv a line parallel to representing the verti cal projection of the visual element of the system. The visual element, represented by the two projections just drawn, pierces the picture plane at 'Of (§ 45, note), giving the perspective of the vanishing point for the roof line, represented by af in the diagram.