47. PROBLEM I. Fig. 11. To find the perspective of a point. The point to be situated 1/4" behind the picture plane, and 1/8" above the plane of the horizon. The observer's eye to be 1/2" in front of the picture plane.

First assume HPP and VH (§ 40). These lines may be drawn anywhere on the paper, HPP usually being placed some distance above VH, in order to avoid confusion between horizontal and vertical projections. The position of the point with respect to the coordinate planes must now be established by means of its vertical and horizontal projections. aV located 1/8" above VH will represent the vertical projection of the point. Its horizontal projection must be vertically in line with aV; and since the point is to be 1/4" behind the picture plane, its horizontal projection must be 1/4" behind the horizontal projection of the picture plane, i.e., i" behind HPP. Next establish the position of the observer's eye, or station point. Its vertical projection (SPV) may be assumed anywhere in VH. Its horizontal projection (SPH) must be vertically in line with SPV and 1/2" in front of HPP. The perspective of the point a will be where the visual ray through the point pierces the picture plane. A line RH drawn through SPH and aH will be the horizontal projection of this visual ray. Its vertical projection will be the line RV drawn through SPV and aV. The perspective aP of the point will be found on RV vertically in line with the intersection of RH and HPP (§ 45, note). Compare with the construction shown in Fig. 10 and Fig 8.

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48. Figs. 12, 13, and 11 illustrate this same problem.

In Fig. 12, the point a, as shown by its vertical and horizontal projections, is situated 1/4" below the plane of the horizon and 1/4" behind the picture plane. ap is the perspective of the point.

Fig. 13.

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Fig. 14.

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In Fig. 13, the point a is 1/4" above the plane of the horizon and 1/8" in front of the picture plane. ap is its perspective.

In Fig. 14, the point a is 1/8" below the plane of the horizon and 1/4" in front of the picture plane. ap is its perspective.

49. PROBLEM II. Fig. 15. To find the perspective of a line, the line being determined by its vertical and horizontal projections.

Let HPP and VH be given as indicated in the figure. Let AH represent the horizontal projection of the line, its two extremities being represented by aH and bh , respectively. Similarly, let Av be the vertical projec tion of the line, av and bv being the vertical projections of its extremities. Let the position of the observer's eye be as indicated by SPV and SPH.

The perspective of the point a has been found by Problem I. at ap. The perspective of the point b has been found by Problem I. at bP. The line (Ap), joining ap and bP, will be the perspective of the given line. (See note under § 23.)

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50. PROBLEM III. Fig. 16. Having given the vertical and horizontal projection of any line, to find the perspective of its vanishing point.

Let the line be given by its vertical and horizontal projections (Av and AH), as indicated in the figure. SPV and SPh represent the position of the observer's eye. To find the perspective of the vanishing point of any line, draw through the observer's eye an element of the system to which the line belongs, and find where this element pierces the picture plane (§ 24 g). Through SPH draw AiH parallel to AH, and through SPV draw A1v parallel to Av. A1H and A1v represent the two projections of a line passing through the observer's eye and parallel to AHAV. This line pierces the picture plane at vA, giving the perspective of the required vanishing point (§ 45, note). The perspectives of all lines parallel to AVAH will meet at vA

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Figs. 17 and 18 illustrate this same problem.

51. In Fig. 17, the line, as shown by its two projections, is a horizontal one; hence, A1v drawn through SPV coincides with VH, and the vanishing point for the system of the lines must be found on VH at vA, as indicated (§ 24 c).

Note.-Systems of lines which vanish upward will have their vanishing points above VH. Systems of lines which vanish downward will have their vanishing points below VH (§ 16).

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52. In Fig. 18, the given line is perpendicular to the picture plane; hence, A1v must be a point coincident with SPV; and as vA will always be found on A1v, the vanishing point of the line must coincide with SPV.

Note. - In a perspective drawing, the vanishing point for a system of lines perpendicular to the picture plane will always coincide with the vertical projection of the observer's eye.