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Perceiving The Window in Order To See The World (page 1)

Perceiving The Window in Order To See The World
"The picture is both a scene and a surface, and
the scene is paradoxically seen behind the
surfase. This duality of information is the
reason the observer is never quite sure how to

answer the question, "What do you see?" For
he can perfectly well answer that he sees a
wall or a piece of paper.
J.J. Gibson, from The Ecological Approasch to Visual Perception (Gibson, 1979, p. 281)

Fig.7.1 Donatello, The Feast of Herod (ca. 1425). Gilded bronze panel, baptismal font, Cathedral of San Giovanni, Siena.


We have seen (in Chapter 5) that pictures drawn in perspective suffer very little distortion when they are not seen from the center of projection. Even though the Renaissance artists did not write about the robustness of perspective, they must have understood that paintings can look undistorted from many vantage points. In fact, soon after the introduction of linear perspective they began to experiment most audaciously with the robustness of perspective. As John White points out, Donatello's relief The Dance of Salome (or The Feast of Herod), shown in Figure 7.1, in the Siena Baptistery, "is less than two feet from the top step leading to the font, and well below eye level even when seen from the baptistery floor itself" (White, 1967, p. 192).1

In this chapter, we will explore the underpinnings of the robustness of perspective, and we will see why the phenomenon does not occur unless the surface of the picture is perceptible. In other words, we will discover that the Alberti window differs from all others in that it functions properly only if it is not completely transparent: We must perceive the window in order to see the world.

Look back at Figure 5.1 and imagine a geometer familiar with Gothic arcades who has been asked to solve the inverse perspective problem given that o as depicted in panel 95 is the most likely center of projection. Our geometer can now do one of two things: accept the suggested center of projection, in which case the solution will be a plan very much like the one shown in panel 95, a plan such as no Gothic architect would envisage in his most apocalyptic nightmares, or assume that the arcade is in keeping with all other Gothic architecture, with respectable right angles and columns endowed with a rectangular cross section, such as is shown in panel 97. The latter assumption implies that the center of projection of the picture does not coincide with the one suggested. Thus the observer is faced with a dilemma: to ignore the rules of architecture, or abandon the suggested center of projection and choose one in keeping with the rules of architecture. This is the geometer's dilemma of perspective, which the visual system too must resolve.

The robustness of perspective shows that the visual system does not assume that the center of projection coincides with the viewer's vantage point. For if it did, every time the viewer moved, the perceived scene would have to change and perspective would not be robust. Indeed, the robustness of perspective suggests that the visual system infers the correct location of the center of projection. For if it did not, the perceived scene would not contain right angles where familiar objects do. We do not know how the visual system does this. I will assume that it uses methods similar to those a geometer might use. Such methods require two hypotheses: (1) the hypothesis of rectangularity, that is, to assume that such and such a pair of lines in the picture represents lines that are perpendicular to each other in the scene, and (2) the hypothesis of parallelism, that is, to assume that such and such a pair of lines in the picture represents lines that are parallel to each other in the scene. Box 7.1 offers a geometric method that relies on the identification of a drawing as a perspectival representation of a rectangular parallelepiped (a box with six rectangular faces).

Box 7.1 How the visual system might infer the center of projection:
If the box shown in Figure 7.2 is assumed to be upright, i.e., its top and bottom faces are assumed to be horizontal, then we must assume a tilted picture plane (as if we were looking at the box from above). Because the picture plane is neither parallel nor orthogonal to any of the box's faces, there are three vanishing points. The two horizontal vanishing points V' and V" are conjugate, as is the vertical vanishing point V'"with each of the other ones. Each pair of conjugate vanishing points defines the diameter of a sphere that passes through the center of projection, which we wish to find. (The diameters of the three spheres form a triangle, V'V"V"', and the intersection of each sphere with the picture plane is a circle; in Figure we show only half of each circle). Because the three spheres pass through the center of projection, the single point they share must be the center of projection we are looking for. Or, to put it in somewhat different terms, the center of projection must be at the point of intersection of the three circles formed by the intersections of the three spheres with each other. But to find this point, we need only determine the point of intersection of two of these circles. First we note that these circles define planes perpendicular to the picture plane. Thus the line of sight (that is, the principal ray) must be the intersection of these two planes. The diameters of two of the sphere intersect circles arc shown in Figure 7.2; the point at which the diameters intersect is the foot of the line of sight (that is, the intersection of the principal ray with the picture plane). To find the center of projection we need only erect a perpendicular to the picture plane from the foot of the line of sight. To find the distance of the center of projection .long this line, we draw one of the sphere-intersect circles, on its diameter we mark the foot of the line of sight, and at that point we erect a perpendicular to the diameter; the perpendicular intersects the circle at the center of projection, at a distance equal to the distance of the center of projection from the picture plane.


Fig.7.2 Perspective drawing of a figure and determination of center of projection.

We have just gone through the steps for finding the center of projection of the most elaborate type of perspectival arrangement, three-point perspective. In general, if one wants to find the center of projection of perspectival pictures, one always needs more than one pair of conjugate vanishing points. For instance, in the case of the construzione legittima (Figure 2.14), often referred to as one-point perspective, .all the sides of box-like objects (interiors of rooms or exteriors of buildings) are either parallel or orthogonal to the picture plane. The vanishing point of the orthogonals is the foot of the line of sight (it plays the role of a pair of conjugate vanishing points). To determine the distance of the center of projection, it is necessary to find another pair of conjugate vanishing points, that is, the so-called distance points (at which the diagonals of a checkerboard pavement converge). Or, consider the somewhat more complicated case of oblique perspective, sometimes called two-point perspective, in which the tops and bottoms of boxes are horizontal (or, more precisely, orthogonal to the picture plane), but the other faces are neither parallel nor orthogonal to the picture plane. To find the center of projection in this case, we must have in the picture at least two boxes whose orientations are different; that is, their sides are not parallel. Then we have two pairs of conjugate vanishing points with which we can find the center of projection.2

1 Chapter 13 of White's book discusses several other frescoes and reliefs that have an inaccessible center of projection. No one has done the inventory of Renaissance art with respect to this phenomenon. We will return to the question of inaccessible centers of projection in the next chapter.

La Gournerie (1884, Book VI, Chapter 1), Olmer (1949), and Adams (1972) discuss such procedures.

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