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The Bounds of Perspective: Marginal Distortions (page 5)

The Bounds of Perspective: Marginal Distortions

Fig.8.13 Diagram illustrating argument about perspective made by Goodman.

Goodman also developed a second line of attack, which runs as follows:
The source of unending debate over perspective seems to lie in confusion over the pertinent conditions of observation. In Figure [8.13], an observer is on ground level with eye at a; at b, c is the facade of a tower atop a building; at d, e is a picture of the tower facade, drawn in standard perspective and to a scale such that at the indicated distances picture and facade subtend equal angles from a. The normal line of vision to the tower is the line a, f; looking much higher or lower will leave part of the tower facade out of sight or blurred. Likewise, the normal line of vision to the picture is a, g. Now although the picture and facade are parallel, the line a, g is perpendicular to the picture, so that vertical parallels in the picture will be projected to the eye as parallel, while the line a, f is at an angle to the facade so that vertical parallels there will be projected to the eye as converging upward We might try to make picture and facade deliver matching bundles of light rays to the eye by either (1) moving the picture upward to the position h, i, or (2) tilting it to position j, k, or (3) looking at the picture from a but at the tower from m, some stories up. In the first two cases, since the picture must also be nearer the eye to subtend the same angle, the scale will be wrong for lateral (left-right) dimensions. What is more important, none of these three conditions of observation is anywhere near normal. We do not usually hang pictures far above eye level, or tilt them drastically bottom toward us, or elevate Ourselves at will to look squarely at towers. With eye and picture in normal position, the bundle of light rays delivered to the eye by the picture drawn in standard perspective is very different iron the bundle delivered by the facade. (1976, pp. 17-9).

Here Goodman makes several errors. No one after Brunelleschi ever tried to "make picture and facade deliver matching bundles of light rays to the eye" in situ, even though it is very easy, in principle, to do so. What some may want to claim for perspective (and I am one of them, though with much hedging) is that, by using it, one can create a picture that, if viewed from the center of projection, will deliver a bundle of light rays to the eye that matches one bundle of rays delivered by the scene viewed from one vantage point.

For the sake of argument, let us use Goodman's strict notion of matching. Because Goodman does not tell us where the picture plane was when the picture was made, we must guess. It could not have been at d, e, because a cannot be the center of projection that would make d, e, a picture of b, c. If it was at h, i, then the perspective belongs to the same rare class as the base in Uccello's Hawkwood (Figure 8.12) and Mantegna's Saint James Led to Execution, which we will discuss in the next chapter (Figure 9.7). If the artist who created such a picture using central projection also wants the viewer to be able to see it from the center of projection (as Mantegna apparently did), he will place the picture above eye level, notwithstanding Goodman's protestations that such things are not done. If the picture plane was at the height of m,f, then it was the artist who must have elevated himself to paint the tower as it is seen squarely, and the only way to match the bundles of light rays from the facade and the picture exactly is to elevate the viewer to the height of the center of projection. The third possibility is one not hinted at by Goodman, and it is the solution to his problem: Suppose that when the picture was drawn the picture plane was perpendicular to a, f. Then, when the picture was eventually moved to its "normal" position (according to Goodman) at d, e, it would deliver a bundle of light rays matching the one delivered by the facade.

Goodman mistakenly constrained perspective to pictures projected onto vertical picture planes and hung at the height of the eye, but he allowed the height of the center of projection to be chosen at will. Under these constraints, there are indeed pictures that will not deliver a bundle of light rays to match the one delivered by the scene. But those are constraints invented by Goodman on the basis of a misinterpretation of the rules of central projection.

Goodman tried to show that the "choice of official rules of perspective [is] whimsical" (1976, p. 19). This is an extremely pregnant way of putting things. By referring to a choice, Goodman suggests a freedom in the selection of rules of pictorial representation that others have denied. By referring to the choice as whimsical, Goodman suggests that the choice was unwise, to say the least. In the first part of this chapter, I made a point not too far removed from Goodman's, namely, that geometry does not rule supreme in the Land of Perspective. However, we stopped far short of agreeing that the rules of pictorial representation are arbitrary and can be chosen freely. In fact, if in the Land of Perspective geometry plays a role analogous to the role played by Congress in the United States, then perception has the function of the Constitution. Whatever is prescribed by the geometry of central projection is tested against its acceptability to perception. If a law is unconstitutional, it is rejected and must be rewritten to accord with perception.

In consequence, the laws of perspective do not coincide with the geometry of central projection. We have noted two ways in which the practice of perspective deviates from central projection: (1) the restriction of the field of perspective pictures to 37°, and (2) the representation of round bodies (spheres, cylinders, human figures) as if the principal ray of the picture ran through them. This procedure does not preclude foreshortening: It is designed to avoid the rather severe marginal distortions that are perceived when such bodies are not very close to the principal ray.

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