Thus, mab is to the left of SPV. The point bp is more distant than the point ap. Therefore, b1, which shows the true measure-ment for the point bp, must be laid off to the right of ap.

104. It sometimes happens that a line extends in front of the picture plane, as has already been seen in the lines of the nearest corner of the porch in Fig. 22. It may be desired to extend the line apcp in Fig. 27, in front of the picture plane, to the point yp, as indicated in the perspective projection. In this case, the point ap being more distant than the point yp, and ;mad being to the right of SPV, the true measurement of apyp must be laid off on VH l, in such a manner that the measurement for ap will be to the left of the measurement for yp. In other words, y1 must be on VH1, to the right of ap, the distance apyl showing the true length of apyp.

Note. - The true length of any line which extends in front of the picture plane will be shorter than the perspective of the line.

105. Having determined the perspective of any line, as dpcp, its true length may be determined by drawing measure lines through dp and cp. The distance intercepted on VH, by these measure lines will show the true length of the line. Thus, dpcp vanishes at vab. Its measure point must-therefore be mab. Two lines drawn from mah, and passing through cp and dp respectively, will intersect VH1 in the points c1 and d2. The distance between c1 and d2 is the true length of cpdp. This distance will be found equal to apb1which is the true measure for the opposite and equal side (apbp) of the rectangle.

In a similar manner, the true length of bpcp may be found by drawing measure lines from mad through bp and cp respectively. b2c2, will show the true length of cpbp, and should be equal to apd1 which is the true length of the opposite and equal side (VpdP) of the rectangle.

106. The perspective (wp) of a point on one of the rear edges of the card may be determined in either of the following ways: -

1st. From b2„ which is the intersection with VHt of the measure line through bp, lay off on VH,, to the left (§103), the distance bw2 equal to the bhwh taken from the given plan. A measure line through w2,, vanishing at mad, will intersect cpbp at the point wp

2d. In the given plan draw a line through wh, parallel to ahbh, intersecting ahdh in the point w4. On VH1 make apw1 equal to ahw4, as given in the plan. A measure line through w1, vanishing ;at mad, will determine w3 on apdp. From w3, a line parallel to apbp (vanishing at vab) will determine, by its intersection with bpcp, the position of wp.

107. In making a perspective by the method of perspective plan, it is generally customary to assume VH and HPP coincident. That is to say, the coordinate planes are supposed to be in the position shown in Fig. 9, instead of being drawn apart as indicated in Fig. 9a. This arrangement simplifies the construction somewhat.

This is illustrated in Fig. 28, which shows a complete problem in the method of perspective plan. Compare this figure with Fig. 27, supposing that, in Fig. 27, HPP with all its related horizontal projections could be moved downward, until it just coincides with VH. The point n would coincide with vab, h with vad, and the arrangement would be similar to that shown in Fig. 28. All the principles involved in the construction of the measures, points, etc., would, remain unchanged.

108. The vanishing points in Fig. 28 have first been assumed, as indicated at vab and vad. As the plan of the object is rectangular, SPH may be assumed at any point on a semicircle constructed with vabvad as diameter. By assuming SPH in this manner, lines drawn from it to vab and vad respectively must be at right angles to one another, since any angle that is just contained in a semicircle must be a right angle. These lines show by the angles they make with HPP, the angles that the vertical walls of the object in perspective projection will make with the picture plane (§ 102).

109. mad and mab have been found, as explained in §97, in accordance with the rule given in § 98.

VH2 should next be assumed at some distance below VH, to represent the vertical trace of the horizontal plane on which the perspective plan is to be made (§ 91). The position of ap (on VH.,) may now be assumed, and the perspective plan of the object constructed from the given plan, exactly as was done in the case of the rectangular card in Fig. 27.

Fig.27.

Fig.28.

110. Having constructed the complete perspective plan, every point in the perspective projection of the object will be found vertically above the corresponding point in the perspective plan.

VH1 is the vertical trace of the plane on which the perspective projection is supposed to rest. ap is found on VH1 vertically over ap in the perspective plan. a1pe1P is a vertical line of measures for the object, and shows the true height given by the elevation.

To find the height of the apex (k1P) of the roof, imagine a horizontal line parallel to the line ab to pass through the apex, and to be extended till it intersects the picture plane. A line drawn through kp, vanishing at vab, will represent the perspective plan of this line, and will intersect VH2 in the point m, which is the perspective plan of the point where the horizontal line through be apex intersects the picture plane. The vertical distance n1m1, laid off from VH1, will show the true height of the point k above the ground, k1p will be found vertically above kp, and on the line through n1 vanishing at vab. The student should find no difficulty in following the construction for the remainder of the figure.

111. Fig. 29 illustrates another example of a similar nature to that in Fig. 28. The student should follow carefully through the construction of each point and line in the perspective plan and in the perspective projection. The problem offers no especial difficulty.

Plate VI. should now be solved.