In a similar manner the vertical edges of the steps, where they intersect the plane M, might have been found by laying off from u, on Or, the' divisions we and vw taken from the plan. These divisions could have been carried along the floor by hori zontal lines parallel to the sides of the room (vanishing at vas), to the plane M, and then projected vertically_ upward on the plane M, as indicated in the figure.
Solve Plate V.
91. In the foregoing problems the perspective projection has been found from a diagram of the object. Another way of con structing a perspective projection is by the method of Perspective Plan. In this method no diagram is used, but a perspective plan of the object is first made, and from this perspective plan the per spective projection of the object is determined. The perspective plan is usually supposed to lie in an auxiliary horizontal plane below the plane of the ground. The principles upon which its construction is based will now be explained.
92. In Fig. 26, suppose the rectangle aHb"eli du to represent the horizontal projection of a rectangular card resting upon a horizontal plane. The diagram of the card is shown at the upper part of the figure. It will be used only to explain tke construc tion of the perspective. plan of the card.
First consider the line ad, which forms one side of the card. On HPP lay off from a, to the left, a distance (ae) equal to the length of the line ad. Connect the points e and d. ead is by construction an isosceles triangle lying in the plane of the card, with one of its equal sides (ae) in the picture plane. Now, if this triangle be put into perspective, the side ad, being behind the pic - ture-plane, will appear shorter than it really is; while the side ae, which lies in the picture plane, will show in its true length.
Let VII, be the vertical trace of the plane on which the card and triangle are supposed to rest. The position of the station point is shown by its two projections SPH and SPv. The vanish ing point for the line ad will be found at .vad in the usual man ner. In a similar way, the vanishing point for the line ed, which forms the base of the isosceles triangle, will be. found at ved, as indicated. aP will be found on VII, vertically under the point a, which forms the apex of the isosceles triangle ead. The line aPdP will vanish at tied. The point er will be found vertically below the point e. eP olP will vanish at ved, and determine by its
intersection with aPdP the length of that line. ePaPdP is the perspective of the isosceles triangle ead.
If the line ad in the diagram is divided in any manner by the points t, s, and r, the perspectives of these points may be found on the line aPdP in the following way. If lines are drawn through the points t, s, and r in the diagram parallel to the base de of the isosceles triangle (ead), these lines will divide the line ae in a manner exactly similar to that in which the line ad is di vided. Thus, ate will equal at, wv will equal ts, etc. Now, in the perspective projection of - the isosceles triangle, aPeP lies in the picture plane. It will show in its true length, and all divisions on it will show in their true size. Thus, on aPeP lay off arwP, evP, and vPur equal to the corresponding distances at, ts, and sr, given in the diagram. Lines drawn through the points qv'', vP, and uP, vanishing at ved, will be the perspective of the lines tet, vs, and ter in the isosceles triangle, and will determine the positions of tP, sP, and rr, by their intersections with aP dP .
93. It will be seen hat after having found vad and ved, the perspective of the isosceles triangle can be found without any reference to the diagram. Assuming the position of a" at any desired point on VH,, the divisions a", wP , vP, uP, er may be laid off from e directly, making them equal to the corresponding divisions a", tH, rH, ell', given in the plan card. A line through aP vanishing at vad will represent the perspective side of the isosceles triangle. The length of this side will be determined by a line drawn through eP, vanishing at ved. The positions of tP, sP, and rP may be determined by lines drawn through vP, and uP, van ishing at ed.
94. It will be seen that the lines drawn to ved serve to measure the perspective distances ce tP, tP sP, sP rP, and rP dP, on the line aPdP, from the true lengths of these distances as laid off on the line aPeP. Hence the lines vanishing at ved are called Measure Lines for the line aPdP, and the vanishing point ved is called a Measure Point for aPdP.
95. Every line in perspective has a measure point, which may be found by constructing an isosceles triangle on the line in a manner similar to that just explained.