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ELEMENTARY PROBLEMS.

To find the perspective of a point. The point to be situated behind the picture plane, and A" above the plane of the horizon. The observer's eye to be A" in front of the picture plane.

First assume IIPP and VII (§ 40). These lines may be drawn anywhere on the paper, IIPP 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 verti cal and horizontal projections. av located r above VH will rep resent the vertical projection of the point. Its horizontal projection must be vertically in line with av ; and since the point is to be I" behind the picture plane, its horizontal projection must be f" 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 (SPY) may be assumed anywhere in VH. Its horizontal projection (SPH) must be verti cally in line with SPY and i" 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 aff will be the horizontal projection of this visual ray. Its verti cal projection will be the line R.v drawn through SPv and ay. The perspective aP of the point will be found on Rv vertically in line with the intersection of RH and IIPP (§ 45, note). Com pare with the construction shown in Fig. 10 and Fig 8.

48. Figs. 12, 13, and 14 illustrate this same problem.

In Fig. 12, the point a, as shown by its vertical and horizon tal projections, is situated 1" below the plane of the horizon and i" behind the picture plane. als is the perspective of the point.

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

In Fig. 14, the point a is i" below the plane of the horizon and *" in front of the picture plane. aP is its perspective.

To find the perspective of a line, the line being determined by its vertical and horizontal projections.

Let HPP and Vfl be given as indicated in the figure. Let

A" represent the horizontal projection of the line, its two ex tremities being represented by au and b", respectively. Similarly, let Av be the vertical projec tion of the line, av and by being the vertical projections of its extremities. Let, the position of the observer's eye be as indicated by SP v and SPH, The perspective of the point a has been found by Problem I. at a". The per spective of the point b has been found by Problem I. at bP. The line (AP), joining a" and b", will be the perspective of the, given line. (See note under § 23.) 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 (A° and An), as indicated in the figure. SP° and SP" 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 SP" draw A," parallel to AH, and through SP° draw A,° parallel to A°. A11' and represent the two projections.of a line passing through the observer's eye and parallel to A"A°.

This line pierces the picture plane at v^, giving the perspective of the required vanish ing point (§ 45, note). The perspectives of all lines parallel to A°A" will meet at e.

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, A; drawn through SP° coincides with VII, and the vanishing point for the system of the lines must be found on VII at e, as indicated (§ 24 c).

Systems of lines which vanish upward will have their vanishing points above VII. Systems of lines which vanish downward will have their vanishing points below VEI (§ 16).

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