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 toVisual 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.
Box7.1 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. ^{2}La Gournerie (1884, Book VI, Chapter 1), Olmer (1949),
and Adams (1972) discuss such procedures.