 | 7 The bounds of perspective:
marginal distortions The psychology of perspective and Renaissance art |  |
7 The bounds of perspective:
marginal distortions
Cannot bear very much reality.
T. S. Eliot, from “Burnt Norton,” 1935 (Eliot,
1963, p. 190)
We turn now to a class of pictures that are unacceptable
because they do not conform to the robustness of perspective,
that is, they look distorted to all viewers except
those who look at the picture from the center of projection.
The existence of such pictures, as we shall see, constrains
central projection, forcing artists to compromise in their
methods of representing scenes. The upper-right-hand panel
of Figure 7-1 looks distorted from all vantage points except
the center of projection, just over an inch away from the
page, too close to focus on the lines; the drawing in the
lower-left-hand panel does not look distorted from any
vantage point. The two pictures differ in the distance of
the center of projection from the image plane, which is
equivalent to a difference in visual angle subtended by the
scene: The first subtends 102 degrees, whereas the second
subtends only 19 degrees. It is not known how big the
visual angle can be before such distortions, called marginal
distortions, appear in pictures made using central perspective.
Olmer, in his extensive treatise on perspective, Perspective
Artistique (2 vols.: 1943, 1949), reviewed the
recommendations of artists and writers on perspective and
concluded in favor of a horizontal visual angle of 37 degrees
(and a vertical visual angle of 28 degrees), which he calls
perspective normale. In Figures 7-2 and 7-3, he compares an
Figure 7-1. Two central projections
of a church and cloister. In lower-right-hand
panel (labeled Fig.
169bis) is a plan and elevation of the
scene, showing OE1, the center of
projection used to draw upper-right-hand
panel (labeled Fig. 168), and
OE2, center of projection for lower-left-hand
panel (labeled Fig. 169).
Width of scene in Fig. 168 subtends
a visual angle of 102 degrees from
center of projection; width of scene
in Fig. 169 subtends 19 degrees.
of cubes drawn in what he calls perspective exagerée. In the
latter drawing, he shows that in a central area subtending
37 degrees cubes are not distorted. In an even more dramatic
example (Figure 7-4), he shows that outside the frame
xyx′y′, which encompasses what he calls the normal visual
field (37 degrees by 28 degrees), the cubes are seen as
distorted.
We know that fields exceeding a critical extent cannot
be properly perceived without moving one's eyes. Imagine
a horse standing some distance away presenting his flank
to you. Now image yourself moving toward the horse:
As you move closer to the horse, it looms larger; there
will come a point when you are so close that you will not
Figure 7-2. Variations of pictures of
oblique cubes seen under normal
perspective
your eyes or turn your head. Furthermore, if you are asked
to visualize something, such as an animal, seen at a large
distance, and to imagine yourself moving toward it, there
will come a point when you will imagine yourself so close
to the thing you are visualizing that it seems to “overflow”
your “mental screen.” Estimates of the size of the visual
field that we can encompass in focal attention are difficult
to obtain. Using variants on the mental imagery procedure
just described, Steven Kosslyn (1978) obtained estimates
ranging from 13 to 50 degrees, which bracket Olmer's
estimate of the normal visual field.
A somewhat different procedure, developed by A. Sanders
(1963, Experiment 3, pp. 49–52), required a subject to
look at a fixation point where a column consisting of either
four or five lights would appear, while simultaneously, to
the right of the fixated column of lights, another of column
Figure 7-3. Variations of pictures of
oblique cubes seen under exaggerated
perspective
(Figure 7-5). The angular distance between the two displays
varied from 19 to 94 degrees. Furthermore, there
were two viewing conditions: one in which subjects were
allowed to move their eyes to scan the display, and one in
which they were instructed to keep their eyes on the location
of the left column. The subject's task was to press
one of four keys as quickly as possible after the two columns
of lights were turned on. One key meant that both
columns consisted of four lights, a second key meant that
both had five, and the remaining two keys covered the
remaining two possibilities of unequal numbers in the two
columns. The median reaction times of two subjects are
shown in Figure 7-6. First, look at the reaction times represented
by the filled circles and summarized by the broken
curve (condition I: eye movements forbidden). The larger
the display angle, the longer the reaction time; beyond 34
Figure 7-4. Marginal distortions in
picture of array of cubes seen from
above
times represented by unfilled circles and summarized
by the solid curve (condition II: eye movements required).
Up to about 30 degrees, reaction times were longer than
those obtained in the absence of eye movements, suggesting
that eye movements were not necessary to see the
right-hand column for smaller visual angles. This then is
an estimate of the size of the field encompassed by the
stationary eye. This estimate of the field normally captured
by a glance is not inconsistent with Olmer's normal visual
field.
The most impressive confirmation of our attempt to link
the extent of Olmer's normal visual field for perspective
drawings with the extent of what we can encompass in a
single glance is provided by an experiment done by Finke
and Kurtzman (1981). Imagine that you are looking at
Figure 7-7 and that you are handed a pointer with a red
Figure 7-5. Four displays and response
keys used by Sanders (1963)
Figure 7-6. Median reaction time
(in seconds) as a function of display
angle and fixation conditions: Condition
I: Eyes immobile, fixating left
column (filled circles, broken line).
Condition II: Eye movements required
(blank circles, solid line).
Panels a and b represent data of two
subjects.
Figure 7-7. Display used by Finke
and Kurtzman (1981) to measure
extent of visual field in imagery and
perception
line while keeping your eyes on the red dot. As you move
the pointer and your gaze away from the center of the
circle, it becomes gradually more difficult to discriminate
the two sets of stripes, until you cannot tell that there are
two distinct sets. The distance from the center at which
this loss occurs is taken as an estimate of the boundary of
the visual field. If the pattern is turned 45 degrees clockwise
and the observer is asked to move the pointer and his eyes
rightward along the horizontal line, the boundary is found
somewhat further from the center of the circle. If the procedure
is repeated six more times, once for each remaining
radial line, a rough estimate of the shape of the visual field
can be obtained.
The size of the visual field estimated by this procedure
varies with changes in the number of bars per inch: The
higher the density of bars in the central pattern, the sooner
the observer will report that the pattern has melted into a
blur. The widest patterns used gave a field of 35 by 28
degrees, gratifyingly close to Olmer's estimate.[1]
This correspondence
suggests that we are comfortable with perspective
drawings only if the scene they encompass does
not subtend a visual angle greater than we would normally
encompass in our visual field.
To find what it is in perspective pictures subtending a
large visual angle that causes us to reject them, let us look
back at Olmer's figures, which subtend large visual angles
(Figures 7-3 and 7-4): Not all the cubes that fall outside
the interior frame that bounds Olmer's normal visual field
(between the two points D/3 in Figure 7-3, and within the
rectangle xx′yy′ in Figure 7-4) look equally distorted. In
Figure 7-4, for instance, compare the cube just below the
line x′y′ to the cube just to the left of x′. The former looks
considerably more distorted than the latter; it violates Perkins's
law for forks, one of the angles of the fork being
less than 90 degrees. Only the cubes that violate Perkins's
laws look distorted; the others do not. Therefore, perhaps
it is not the wide angle of the view per se, but rather local
features of the depictions, that cause these pictures to look
distorted.
We are now in a position to understand the connection
between Perkins's laws and the limited size of our visual
field. We have seen from Olmer's drawings that the perspective
drawings of rectangular objects are likely to violate
Perkins's laws only when they fall outside a field that
subtends 37 degrees by 28 degrees. We have also seen that,
because our visual field subtends about 37 degrees by 28
degrees, we are unlikely to perceive objects in our environment
that fall outside such a field. In other words, the
projections of objects that fall within our field of view all
obey Perkins's laws. Because Perkins's laws are very simple,
we may notice their violation in pictures only because
they constitute a striking deviation from what we are accustomed
to see and not because of the relation of Perkins's
laws to parallel projection, which we observed in Chapter
6.[2]
Up to this point, we have been discussing the marginal
distortion of right angles. Figure 7-8 (the panel labeled Fig.
243) shows that the correct central projection of a sphere
that is not centered on the principal ray is an ellipse. Nevertheless,
if the projectively correct ellipses were substituted
for the circles with which Raphael represented the spheres
in his School of Athens[3]
(Figure 7-9 and the detail in Figure
7-10), they would not look like spheres (unless the fresco
were viewed through a peephole at the center of projection).
This misperception of the correct projection of a
sphere is a marginal distortion very much like the misperception
of projectively correct representations of the
vertices of cubes when they are outside the area of normal
perspective (because they are likely to violate Perkins's
laws). There is, however, one major difference: A cube
can be anywhere within the area of normal perspective and
still look like a cube; a sphere that is not on the principal
ray will look distorted. The visual system, so tolerant of
Figure 7-8. Panel [Fig.] 242: Central
projection of sphere centered on
principal ray is a circle. Panel 243:
Central projection of sphere not on
principal ray is an ellipse and does
not look like a sphere. Panels 246246ter:
Three projections of Raphael's
Aristotle (see Figure 7-9) as
they should be drawn at different
displacements to the right of the
principal ray.
completely intolerant of variations in the representations
of spheres. As discussed earlier in this chapter, the link
between perspective exagérée and Perkins's laws is that the
latter are a convenient rule of thumb that separate pictures
that could represent objects in our normal field of view
from those that could not. Vertices of cubes that are on
the principal ray vary in their appearance depending on the
distance of the center of projection from the picture plane;
the projection of a sphere whose center is on the principal
ray is always a circle. Furthermore, there is no convenient,
easy to perceive, rule of thumb (analogous to Perkins's
laws) to separate the unlikely projections of spheres from
Figure 7-9. Raphael, The School
of Athens (1510-1). Fresco.
Stanza della Segnatura, Vatican,
Rome.
a sphere that falls just within the area of normal perspective
and one that falls just outside is a purely quantitative difference
in the ratio of the long dimension of an ellipse
(major axis) to its shorter dimension (minor axis). As a
result, only circles are considered acceptable projections of spheres.
And because artists have always accepted the primacy of
perception over geometry, whenever they represented
spheres in their paintings (which was not often), they always
represented them as circles. In other words, it is as
if whenever a sphere had to be represented, an ad hoc center
of projection and a new principal ray (which passed through
the center of the sphere) was created.
Just as the correct central projection of a sphere becomes
a more elongated ellipse the further the center of the sphere
Figure 7-10. Detail of Figure 7-9
showing Ptolemy, Euclid, and
others.
projection of a cylindrical column becomes under these
circumstances[4] (see Figure 7-11). This marginal distortion
is mostly academic, because I have not found any Renaissance
paintings of colonnades that could have been subject
to this sort of distortion and were corrected to accord with
perception. Nevertheless, Leonardo was aware of the problem,
and he correctly pointed out that, although the progressive
thickening of the pictures of columns the further
they are from the principal ray (and the concomitant narrowing
Figure 7-11. Plan of four cylindrical
columns, C, C1, C2, C3, projected
onto picture plane AB using O as
center of projection. Although frontal
chords of circular cross sections of the
columns (mn, m1n1, m2n2) project as
constants (MN, M1N1, M2N2), diameters
of columns project wider images
the further away they are from
principal ray.
projection, this “good” method (as he puts it) is “satisfactory”
only if the picture is viewed through a peephole
located at the center of projection. He concludes that when
the picture “is to be seen by several persons” the only
perceptually acceptable solution (which is “the lesser fault,”
i.e., not as good as using a peephole) is analogous to what
Raphael did with the sphere: to ignore the rules of geometry
and to represent the columns in the colonnade “in
their proper size,” that is, with equally wide projections
(Leonardo da Vinci, 1970, §544, pp. 326-7).
If spheres and cylinders are treated in a special way by
the practice of perspective, it should not come as a surprise
that the same is true of human bodies. If we think of the
human body as a flattened sphere on top of a flattened
cylinder, we can appreciate the distortions its picture
undergoes as it is displaced away from the principal axis
of the projection. In Figure 7-8, the panels labeled Fig. 246,
246bis, 246ter, Olmer shows three versions of the figure of
Aristotle from Raphael's School of Athens, successively displaced
to the right from the principal ray. Needless to say,
artists never complied with this implication of geometry.
Let us examine the famous fresco by Paolo Uccello Sir
Figure 7-12. Paolo Uccello, Sir
John Hawkwood (1436). Fresco,
transferred to canvas. Cathedral of
Santa Maria delle Fiore, Florence.
of the geometric rules of central projection (Figure 7-12).
Here is Hartt's description of the work:
[Uccello's] earliest dated painting is the colossal fresco in the
Cathedral of Florence, painted in 1436 on commission from the
officials of the Opera del Duomo, an equestrian monument to
the English condottiere Sir John Hawkwood, known to the Italians
as Giovanni Acuto, to whom a monument in marble had
been promised just before his death in 1394. ...
The pedestal rests on a base that is supported by three consoles
... The simulated architecture is projected in perspective from
eye level of a person standing in the side-aisle.[5] But the horse
and rider are seen from a second point of view, at about the
middle of the horse's legs. One is tempted to speculate as to why
Uccello changed the perspective system. If he had projected the
horse and rider from below, in conformity with the pedestal,
the observer would have looked up to the horse's belly, and have
seen little of the rider but his projecting feet and knees and the
underside of his face. But might not Uccello, a lifelong practical
joker, have done exactly that? Perhaps at first he did. The officials
of the Opera objected to his painting of the horse and rider and
compelled him to destroy that section of the fresco and do it
over again. The explanation of this oft-noted circumstance[6] may
well have been Uccello's view of the great man from below.
(1969, pp. 212-13)
John White writes in a similar vein:
The advantages of [using several viewpoints in a single composition]
— sometimes even the necessity for it, are shown most
obviously in Uccello's Hawkwood ... A fairly high degree of
realism was desirable in frescoes which were substitutes for more
expensive marble monuments, and this element of illusion is
supplied by the steep foreshortening of the architectural [base].
and a general's feet can be at best a dubious tribute to his memory.
The realism of the low-set viewpoint is therefore restricted
to the architecture. In Uccello's fresco there is no foreshortening
of the horse or rider ... (1967, p. 197)
Peter and Linda Murray attribute the effect to Uccello's
incompetence:
During the 1430S [Uccello] became fascinated by the new ideas
in perspective and foreshortening, although he never really mastered
the full implications of the system, which became for him,
eventually, no more than another form of elaborate pattern-making.
Even when the impact of the new ideas was fresh, his
treatment of them was quite arbitrary, as can be seen in the
[fresco of] Sir John Hawkwood ... This has two separate viewpoints,
one for the base and another for the horseman ...; a
similarly irrational approach was also used in his Four Heads of
Prophets of 1443 in the roundels in the corners of the clock of
Florence Cathedral. (1963, pp. 113-14)
In view of our analysis of marginal distortions, I believe
that Hartt and White are only partially correct in their
analysis of why Uccello chose two inconsistent centers of
projection, and, a fortiori, I believe that Murray and Murray
err in their attempt to debunk Uccello's mastery of
perspective. Hartt and White are mistaken in thinking that,
as Hartt puts it, if Uccello “had projected the horse and
rider from below, in conformity with the pedestal, the
observer would have looked up to the horse's belly, and
have seen little of the rider but his projecting feet and knees
and the underside of his face.” Hartt's and White's analyses
are based on a failure to appreciate the importance of the
distinction between the central projection of a scene (in
our case, the monument) from a low vantage point onto
a vertical picture plane, and its projection onto a tilted picture
plane. As long as the picture plane is, on the whole, parallel
to the important surfaces of the objects represented, such
as the side of the horse, none of the features of these important
surfaces is lost by moving the center of projection. To better
understand this point, let us ask the question in a slightly
different way: How would the appearance of the horse and
rider have changed had they been depicted in a manner
low vantage point onto a vertical picture plane? It is true
that more of the horse's underbelly would be visible in the
picture, and that the soles of the figure's boots would be
seen, but that is true of any equestrian monument erected
on a tall pedestal. But Hartt and White are wrong to think
that the horse's underbelly and the figure's soles would be
visible to the exclusion of the side of the horse and the side
of the rider. That would happen only if the picture plane
were tilted, which would not be consistent with Uccello's
representation of the base of the statue. I do not think that
a representation of the horse and rider that would be consistent
with the representation of the base would have been
“a dubious tribute” to the general's memory and therefore
do not believe that the officials of the Opera del Duomo
who viewed the first version of Uccello's fresco were angered
by having been the butt of a practical joke (an unlikely
action on the part of an aspiring young artist,
dependent on further commissions). What is at stake here
is marginal distortion: I believe that Uccello's first attempt
was a correct central projection of the pedestal, the horse,
and the rider, which suffered from extreme marginal distortion;
that his second attempt was a partial compromise,
which was still afflicted with too much distortion; and that
his third attempt — which is the masterpiece we know so
well — was perceptually acceptable. Leonardo elevated
Uccello's procedure to the level of principle:
In drawing from the round the draughtsman should so place
himself that the eye of the figure he is drawing is on a level with
his own. This should be done with any head he may have to
represent from nature because, without exception, the figures
or persons you meet in the streets have their eyes on the same
level as your own; and if you place them higher or lower you
will see that your drawing will not bear resemblance. (Leonardo
da Vinci, 1970, §541, p. 325)
In conclusion, we have seen that nonrectangular bodies
that are not on the principal axis of a central projection
cause problems for the would-be orthodox user of this sort
of projection. In general, such bodies — including humans
and animals — are not drawn in accordance with the geometry
from a center of projection on a line perpendicular to the
picture plane intersecting the picture at a point inside the
contour of the body. Only the size of the nonrectangular
objects and their position in the two-dimensional space of
the picture are subject to the rules of central projection.
We have argued in this chapter that this convention of
painting reflects the perspectivists' acceptance of the primacy
of perception and that central projection is applied
principally to architectural settings of scenes. So perspective,
as it was practiced by artists, was far from being an
inflexible system. Because it was subordinated to perception
and because different kinds of objects were made to
obey the laws of central projection to different extents, a
unifying concept such as Alberti's window cannot do justice
to the subtleties and complexities of Renaissance
perspective.
Some artists and scholars, who did not recognize the
richness and elaborateness of perspective, have thought of
it as an awesome monster unleashed on the art of the Renaissance,
a geometric system so truculent that it confined
the imagination of artists to an inescapable four-square
grid. Here, for instance, is how Carlo Carrà wrote in his
1913 manifesto of Futurism, The Painting of Sounds, Noises,
and Odors:
The old running perspective and trompe l'oeil, a game worthy
at most of an academic mind such as Leonardo's, or of a designer
of sets for realist melodramas.[7]
The Gestalt psychologist Rudolph Arnheim expresses a
similar disdain for perspective in his classic Art and Visual
Perception:
[Perspective] must distort sizes, shapes, and spatial distances and
angles in order to convey depth, thus doing considerable violence
not only to the character of the two-dimensional medium but
also to the objects in the picture. We understand why the film
critic André Bazin has called perspective “the original sin of
Western painting.” In manipulating objects to foster the illusion
of depth, picture-making relinquishes its innocence. ...
The discovery of central perspective bespoke a dangerous development
in Western thought. It marked a scientifically oriented
preference for mechanical reproduction and geometrical constructs
in place of creative imagery. William Ivins [1973, p. 9]
has pointed out that, by no mere coincidence, central perspective
was discovered only a few years after the first woodcuts had
been printed in Europe. The woodcut established for the European
mind the almost completely new principle of mechanical
reproduction. It is to the credit of Western artists and their public
that despite the lure of mechanical reproduction, imagery has
survived as a creation of the human spirit. ... Nevertheless, the
lure of mechanical faithfulness has ever since the Renaissance
tempted European art, especially in the mediocre standard output
for mass consumption. The old notion of “illusion” as an artistic
ideal became a menace to popular taste with the beginnings of
the industrial revolution. (1974, pp. 258, 284-5)
Perhaps it is this mimetophobia, the morbid fear of slavish
imitation, that impelled scholars like Herbert Read, Nelson
Goodman, and Rudolph Arnheim, to name a few, to look
for flaws in central projection as a method for the representation
of space. Let us consider the most sustained critique,
by Nelson Goodman in his important book Languages
of Art. One line of Goodman's attack concentrates on what
has been called the projective surrogate[8]
conception of perspective,
namely
that pictorial perspective obeys laws of geometrical optics, and
that a picture drawn according to the standard pictorial rules
will, under the very abnormal conditions outlined above [viewed
with one eye only, through a peephole] deliver a bundle of light
rays matching that delivered by the scene portrayed. Only this
assumption gives any plausibility at all to the argument from
perspective; but the assumption is plainly false. By the pictorial
rules, railroad tracks running outward from the eye are drawn
converging, but telephone poles (or the edges of a facade) running
upward from the eye are drawn parallel. By the 'laws of
geometry' the poles should also be drawn converging. (1976,
pp. 15-16)
Although criticism along these lines is fairly widespread,[9]
it rests on a misunderstanding of the basis of perspective.
Goodman erroneously assumes that when one talks about
to which the further an object is from the viewer the smaller
the visual angle it subtends, which is correct, but is not
the basis of perspective. According to the geometric rules
of central projection, the projection of any two lines that
are parallel to the picture plane, such as two telephone
poles, or the edges of an appropriately oriented facade, will
be two parallel lines.
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
[7-13], 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
Figure 7-13. Diagram illustrating
argument about perspective made by
Goodman
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
from the bundle delivered by the facade. (1976, pp. 17–19)
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 7-12) and Mantegna's Saint James Led to Execution,
which we will discuss in the next chapter (Figure 8-7). 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
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
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.
Finke and Kurtzman went further. They also trained observers to imagine
the grating and then asked them to move their eyes away from the
position of the imagined pattern until it was too blurred to be seen by
the mind's eye. The results were extremely close to the results obtained
for perceived patterns.
Hagen and Elliott (1976) have made unwarranted claims in favor of the
hypothesis that parallel projection is more natural than central projection
(and predates it by about two millennia). They showed subjects pictures
of 7 different objects (2 cubes and 5 regular pentagonal right prisms)
using 6 different degrees of “perspective convergence from conical (traditional
linear perspective) to axonometric (parallel) projection” (p. 481).
They claim that “for a given object of fixed dimension observed from
a fixed station point, a family of perspective views may be generated
... ” (p. 481). Among the problems that invalidate this experiment and
the authors' interpretation of it, I will mention four: (1) Changes in
convergence are equivalent to changes in the location of the center of
projection (the station point). It is meaningless to speak of a change in
convergence without a concomitant change in the center of projection.
(2) In their experiment, not all the pictures that were meant to depict
different projections of one object showed the same number of the
object's faces, and many of these pictures were degenerate to the extent
that they precluded the recognition of the object (for example, one
picture of a cube was a rectangle divided into two rectangles by a vertical
line). At least 10 of the 42 pictures suffered from such extreme degeneracy,
and 4 of the 7 objects depicted had at least 1 degenerate picture.
Because the purported differences in “degree of perspective convergence”
were inextricably confounded with large variations in the amount
of visual information these pictures conveyed, it is impossible to interpret
subjects' preferences for some of the representations. (3) Of the
3 objects whose 6 pictures did not include cases of extreme degeneracy,
1 (the most convergent central projection of a cube, labeled A in their
Figure 1) was a borderline violation of Perkins's law; hence it was fated
to be rejected by subjects, but not because of their putative preference
for parallel projection. (4) Among the 3 objects whose pictures did not
include cases of degeneracy, only 1 yielded data unequivocally in support
of Hagen and Elliott's conclusion that parallel-perspective drawings
were the most natural or realistic drawings.
An experiment credited by Pirenne (1970, p. 122) to La Gournerie (1859,
p. 170). The second edition of La Gournerie's treatise (1884) does not
mention the experiment.
This kind of marginal distortion was first discussed by Uccello and
analyzed extensively by Leonardo. For a review, see White (1967, Chapter
XIV). A more detailed analysis was published in 1774 by Thomas
Malton; see Plate 144 (Fig. 34) in Descargues (1977). La Gournerie (1884)
also discusses it in detail.
Actually, the present viewing level is near the floor. Hartt is describing
the original state of affairs as if it were current.
Described by Pope-Hennessy (1969, p. 7) as follows: “On 30 May [1436]
Uccello was ordered to replace [Agnolo] Gaddi's fresco [of Hawkwood,
commissioned in 1395] with a new fresco in terra verde [meaning green
earth, a natural earth color], on 28 June he was instructed to efface the
horse and rider he had executed on the wall 'because it was not painted
as it should be,' on 6 July he was told to make a new attempt, and by
the end of August the fresco was complete. The erasure of the first
version was due probably to some technical defect in the preparation
of the ground, and not, as is often implied, to dissatisfaction with
Uccello's cartoon [full-size drawing used for transfer to a wall on which
a fresco is to be painted].” I find Pope-Hennessy's attribution of the
erasure of the first version to a “technical defect in the preparation of
the ground” implausible: Why would there be such a defect in the
preparation of the ground of the horse and rider and not a similar defect
in the ground of the base? Furthermore, what prompted the erasure of
the second version? I think, as I explain later in this chapter, that Uccello
had discovered that strict adherence to the laws of perspective made for
unacceptable paintings and that he had to compromise twice before the
result was acceptable to viewers. I also think that the tolerance of his
employers was due to the avant-garde nature of Uccello's application
of perspective.
| 7 The bounds of perspective:
marginal distortions The psychology of perspective and Renaissance art | |