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This Curving World: Hyperbolic Linear Perspective Author(s): Robert Hansen Source: The Journal of Aesthetics and Art Criticism, Vol. 32, No. 2 (Winter, 1973), pp. 147161 Published by: Wiley on behalf of The American Society for Aesthetics Stable URL: http://www.jstor.org/stable/429032 . Accessed: 11/12/2014 11:22 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. .
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ROBERT HANSEN
This Curving World: Hyperbolic Linear Perspective

.. sag ich, dass alle, auch die geradesten Linien, so nit directe contra pupillam stracks vor dem Aug stehen, oder durch sein Ax gehn, nothwendig umb etwas gebogen erscheinen. Das glaubt gleichwol kein Mahler, darumb mahlen sie die gerade Seitten eines Gebaws mit geraden Linien, wiewol es nach der wahren Perspectiffkunst eigentlich zu reden nit recht ist.... Das Niisslein beisset auf Ihr Kiinstlerl
Wilhelm Schickhardt, 1624
... the more the eye approaches the subject [a sphere] the more one believes to see and the less one does see.
... our author's perspective images are straight reproductions of the natural visual process, which is determined by angles and distances and thus results in an image projected on the interior of a sphere rather than on a plane surface.
Codex Huygens (c. 1580)
In point of fact (modern central perspective) is a mathematically exact abstraction substituted for a physiological image which is wholly deceptive. We see not with one fixed eye, but with two constantly moving eyes; the image which we receive on the retina is a spheroid world projected on a concave plane. Thus, in a perspective drawing, straight lines are presented as straight, but in our visual image they are actually curved. Through the use of the printed page and mathematical perspective we are now accustomed to discount mentally this image; but for the ancients this curving world was an accepted phenomenon.
A. G. M. Little

I

testing my observations with the help of a

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ROBERT HANSEN is a painter and professor of art at significant refinement of previous diagrams

Occidental College.

of curvilinear perspective and a correction

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148

ROBERT HANSEN

of generally accepted perspective "laws." It may also suggest a new application of the views of Leonardo da Vinci as interpreted by art historians, but there is no intention here to question those interpretations. I do hope to challenge certain traditional ideas about vision, particularly the assumption that classical linear perspective represents the way the world appears.
We see curves wherever we look at
straight lines. Or is it more accurate to say that we are looking at curves wherever we believe we see straight lines? We tend to deny the empirical validity of these curves and feel that we must explain them away and correct them. It is my intention first to discuss the general nature of our perception of straight lines, briefly review the history of linear perspective in recent centuries, and then to present a system of five-point hyperbolic perspective that will come closer to representing "this curving world" than

do our inherited rules; closer, I believe,
than any system heretofore proposed. It is true that in many situations, when I
am at a "comfortable distance" from the
focus of my attention, this curvature is negligible. But if I am quite close to a large cube or rectangular solid, or, as is the case with most of us the greater part of every day, enclosed within the walls of a box, the lines of that box swell and bend on all
sides-especially if I enlarge my focus and give attention to the whole of my field of vision. The curves are most subtle directly
in front of my eyes, but they are emphatic
at the periphery of vision. A rectangle's curvature becomes even
more apparent when I move, or when the rectangle moves past me. Indeed, a street or a corridor veritably ripples as I walk along it, its apparently largest section accompanying me precisely as I move. Standing on a railway platform watching a passing train

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149

presents the simplest experience of the by attending to peripheral areas of the field

swelling of the nearest section; but every of vision.

vehicle, pedestrian, or animal moving past There are exceptions, lines that do not

us on a straight street exhibits the same appear to curve. Some are "orthogonals," to

curved passage.

be discussed later, the lines moving toward

In your own room, you can simulate the the horizon directly in front of me. There

moving rectangle by moving your eyes or are also the lines that pass through the

your head. It is most readily done by center of vision, intersecting that line pro-

approaching to within a couple of feet of a jected straight forward from the point be-

long wall, preferably one containing a num- tween the eyes-the line "stracks vor dem

ber of windows, doors, shelves, or other Aug" in the Schickhardt quotation.

perpendiculars, and while looking straight Thus, every line that coincides with the

ahead, wag the head vigorously from right horizon, that is, every horizontal line at eye

to left. While moving the head, pay special level, will appear absolutely straight. One

attention to the floor and ceiling lines, but may see any straight line without curvature

keep the head level. Thus are connected in by tilting or veering the head until the line

one continuous panorama the view to the passes through that imaginary perpendicu-

right, the view straight ahead, and the view lar line. Turn to look directly at the door

to the left, all of them familiar to us frame edge, and it resumes its straight

separately, those at right and left offering verticality. Look directly up at the line

radical foreshortening and inescapable di- formed by the meeting of ceiling and wall,

agonal convergence of the parallel horizon- and it becomes in a real sense your horizon,

tal edges of walls and windows, etc.

precisely straight. Look down at the floor

To see the verticals behave in the same line, and it in turn becomes the visibly

way, stand in a doorway, with your toes straight "horizon," all other horizontals

nearly touching the threshold, and nod the curving above and below it. From a sufficient

head up and down, keeping the door frame height the horizon itself is seen to curve.

and other verticals visible out of the corners From the astronaut's point of view, our fa-

of the eyes. In both examples the center miliar straight horizon has of course become

segment, the portion of the line nearest to a circle. And a wheel or a jig-saw shape when

your eye, will appear to contain the major seen edge-on presents a straight line, which

curvature. This head movement near the in turn may be seen as a straight line only

wall should serve primarily as an introduc- if it intersects that line projected straight

tion to curvature perception. After some forward from the point between the eyes,

practice, you should be able to see the the center of vision.

curving wall lines and door lines from a It has been suggested that physiological

comfortable distance, and without moving. factors may either account for or argue

It will always be easier at close range, and against the "reality" of these curves: bin-

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ROBERT

HANSEN

ocular vision, a spherical retinal surface, and the mind's interpretation of the retinal information.
Monocular vision of course presents a simpler case. For some individuals, closing one eye, if it were not somewhat uncomfortable, might facilitate these experiments. Also, the horizontal axis shared by the two eyes favors a clear measuring of horizontal lines, and somewhat confuses the observa-
tion of verticals. However, binocular and
stereoptical effects do not alter the phenomenon of parallel lines appearing to converge simultaneously in opposite directions. Convergence follows naturally from the more basic universal observation that every object appears smaller as it recedes from the eye.
Even in the case of a flat retina, or of a
compound eye, an object just larger than the eye must necessarily appear larger than the entire field of vision if the object is so close that it covers the eye. Only in a world in which receding objects and distances appear to retain their size on the retina could straight lines conceivably retain their straight appearance in all circumstances. As long as the height of a wall appears to diminish both to right and to left, straight lines must appear to curve. For this reason, I do not believe that the spherical cornea and the concave retina are at all relevant to the
question of whether we actually see straight lines as curves.
Some friends who are unable (or unwilling) to perceive these curves object that my mind has interpreted as curves the raw rectilinear data of the retina, in order to
satisfy my prior reasoning. I can only say that nearly everyone who has undertaken the head-wagging just described has quickly come to acknowledge the curving appearance of demonstrably straight lines. Furthermore, it seems to me that precisely the opposite occurs: the lines that appear to curve, with the exceptions noted, are the raw, uninterpreted sensory data; we have been persuaded by centuries of drawings, paintings, and photographs (by lenses selected to eliminate curvature) that our brain must reject what does not appear straight. Anyway, the curvature in the cen-

ter of our usual view is always slight and can be easily missed unless a special effort is made to attend to very long lines and not just to small segments near the center of vision.
I will discuss a widely read refutation of the curvature thesis that appeared in E. H. Gombrich's Art and Illusion.l Briefly, he claims that the eye sees the curvature only by turning the head, thus requiring the artist to construct "a compromise that does not represent one aspect but many.... What we call 'appearance' is always composed of such a succession of aspects ... which allows us to estimate distance and
size; it is obvious that this ... can be
imitated by the movie camera but not by the painter with his easel." But is it always necessary to separate into distinct "aspects" the single sensation I experience when I move my head? By simply revolving the eyes without moving the head, I can easily perceive the curves in a wall directly before me; and with a little practice, these curves can be seen as a single static pattern without moving as much as an eye-muscle, even at some distance from the wall. By recording a sufficiently wide view a painter or a still camera can then easily imitate this discrete visual image.
Gombrich disposes of other claims with disdain verging on contempt. "It may well
be ... that a taut string held very close to
our eyes...'looks like a curved string.' With strings held very close to our eye, judgment becomes uncertain and we may make mistakes. But to say that all straight lines in our field of vision look curved
seems to me a much more doubtful state-
ment. It would imply that all straight strings look like curved strings, and that is manifestly not the case." 2 I must say that straight lines appeared manifestly straight to me until I examined my vision closely. However, in attacking the claim that all straight strings look curved, he is misrepresenting the curvilinear proposition. He ignores exceptions that are generally acknowledged: the essentially straight appearance of all orthogonals, and the lines which pass "straight before the eyes" no matter at what distance from us. The

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151

"string held very close to our eyes" is particularly unlikely to appear curved unless it is placed at the periphery of the visual field.
It is not enough, as Gombrich and others suggest, to allow the natural foreshortening of the rectangular picture itself to supply whatever foreshortening that might exist in the subject matter parallel to the picture plane. The variable viewing distance separating the spectator from the picture must alter the curvilinear foreshortening at every step he takes. It is true of both one-point perspective and of curvilinear systems that there is only one position at which the spectator can experience a near-replica of the artist's view of the subject. For a small conventional picture of large objects such as architecture, that situation in which the
straight lines might supply "natural" curvature places the spectator uncomfortably close to the picture surface, forcing him to turn and tilt his head in order to see the
entire rectangle. At another point Gombrich says, "one
cannot insist enough that the art of perspective aims at a correct equation: it wants the image to appear like the object and the
object like the image.... It does not claim to show how things appear to us, for it is hard to see what such a claim should
mean." But can "to appear like the object" have any other meaning than to appear as the object appears? Must the image, in
order to satisfy Gombrich, represent all measurements in the same scale and not
depict distant columns smaller than near columns? I believe that traditional perspective, in distinguishing sizes of objects as they recede from us, does indeed claim to describe appearance. It is appearance at any rate, and only appearance, that I have attempted to measure in this paper.3
Finally, while discussing Leonardo's three
columns, Gombrich acknowledges, "all this is no doubt a little confusing; if it is a consolation to the reader, let me state my conviction that many writers on perspective have also become confused at this point, not excluding myself, of course." As John White writes, "the straight lines of common

architectural usage ... are indeed all that is
seen by the average modern man."
II
The artist has for four hundred years learned the principles of diminishing size and foreshortening with vanishing points, a system that was discovered and "perfected" in the fifteenth and sixteenth centuries.
Moreover, whereas Oriental and medieval devices for depicting distance and volume were obvious metaphors (or were they mere conventions to the Chinese or Gothic
painter?), Renaissance and Baroque systems have been accepted, at least today, as imitative of nature, to judge from the nonchalance with which painters, architects, and illustrators have portrayed "one-point" and "two-point" cubes, which, I submit, are not to be observed "in nature." (As figures 4, 5, 6, and 7 indicate, one-point becomes fivepoint and two-point becomes four-point in curvilinear systems.) From Giotto through Masaccio to Claude Lorraine and Turner, the science of aerial perspective advanced with man's growing worship of the material universe. The work of Newton led to the optical color experiments of Monet and Seurat. But linear perspective today is assumed to operate (that is, we assume that our eyes see) according to fifteenth-century rules, the formula of Leon Battista Alberti.
The following review of the history of curvilinear perspective is taken largely from John White's The Birth and Rebirth of Pictorial Space.4 Quotations and pages cited are from this work.
"It is in Alberti's Della Pittura, which he wrote in 1435, that a theory of perspective first attains formal being outside the individual work of art. Theoretical dissertation
replaces practical demonstration" (p. 121). The practical demonstrations include fourteenth- and fifteenth-century paintings by Giotto, Lorenzetti, Maso di Banco, and others, which utilize in a groping, experimental fashion an oblique foreshortening depicting architecture in what we would characterize as two-point perspective, and to two panels that Brunelleschi evidently painted expressly to demonstrate geometric

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HANSEN

perspective. These "painted manifestos" of Brunelleschi have not survived but contem-
porary descriptions of them indicate that one panel depicted the Palazzo della Signoria from the diagonally opposite corner of the Piazza in what may have been oblique twopoint perspective, in which the north and west walls of the Palazzo were seen in their
entirety (p. 118).5 Alberti's theory incorporated some of
Brunelleschi's space but was restricted to a single vanishing point. Moreover, Alberti's codified perspective dictated that "(a) there is no distortion of straight lines, and (b) there is no distortion, or foreshortening, of
objects or distances parallel to the picture plane" (p. 123). According to White, "Alberti shows no sign of any awareness ... of the points at which these achievements are made possible only by the acceptance of a geometrical convention which runs counter to the artist's visual experience.... The contrast between the mathematical and the
empirical must not, however, be taken too far. It is not a question of the replacement of a method which is all fidelity to experi-
ence by another which is all convention. It is a substitution of one aspect of truth and one convention, for a different convention
and another truth" (p. 125). Within a few years, Alberti's single cen-
tral vanishing point had conquered. His rule dictated that all horizontal and verti-
cal lines parallel to the picture plane must be theoretically and practically parallel and perpendicular in the picture. Only those "orthogonal" lines, receding toward or radiating from the single central point remained diagonal and converging. This ingenious system, developed by architects, had simplified the artist's craft. Oblique
two-point perspective virtually disappeared from painting and relief sculpture. Alberti's powerful treatise had apparently become dogma by 1500, and remained inviolate for more than a century. When seventeenthcentury northern artists freed the depiction of architecture from this strict frontality,
they returned to the more natural visual experiments of the fourteenth century, when several painters often implied the curved concave plane upon which we see

the real world projected. Giotto, Maso di Banco, and, later, Uccello had expressed in numerous paintings some of the diagonal elements of "this curving world." And in an unattributed fresco in the lower church
at Assisi, a complex building is seen in the center in horizontality, but on both flanks
in two-point obliquity. It is, however, only in the work of Jean
Fouquet (1420?-1481?) that these adjacent sections are allowed to fuse in a continuous
curve, in both ceiling and floor of the Annunciation of the Death of the Virgin (Musee Condd, Chantilly) and in the Arrival of the Emperor at St. Denis (Bibliothbque National, Paris) where pavement tiles behave in the visual way, curving up toward a horizon on both left and right,
upon which, however, a procession is seen marching from left to right in an unyielding flat horizontality. And in another Fouquet manuscript illustration, The Building of the Temple at Jerusalem (Biblioth&que Nationale, Paris), as well as in Donatello's relief rondo, The Assumption of St. John (S. Lorenzo, Florence), verticals converge noticeably toward a zenith vanishing point.6
It is the extreme rarity of vertical convergence that makes these last examples astounding. To our prejudiced eye, their curves and diagonals may look crude and unsure, but they must be seen as the application of a sensitive eye, not yet intimi-
dated by formula. It is in Fouquet's manuscript painting that the sloping upper sections of the temple imply a curve as they rise from the almost vertical ground-level section of piers and columnar sculpture of the Gothic fasade. Even the trompe l'oeil tours de force of the eighteenth century fall short of the vision of this fifteenth-century
Frenchman, who expressed the curvature he saw in architectural lines in spite of the inconvenient adjustments necessary on a two-dimensional surface.
There was one other notable exception to the rule. Although Leonardo da Vinci did not publish a systematic theory of "synthetic" perspective, and did not follow these principles in his own painting, we know something of his thoughts from his own notes and from Cellini's description of

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Hyperbolic Linear Perspective

153

c

b

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which observations on the paths of meteors are combined with general comments on

linear perspective which include the taunt

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addressed to artists that is quoted at the beginning of this paper. Scholars, mostly

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concerned with architecture or with Leo-

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nardo, not with the problem of representational drawing, have debated these optical
effects since the early nineteenth century, when curves were discovered in entabla-

tures and stylobates of Greek temples.8

FIG. 3. Adapted from Leonardo's Manuscript A, 1492, as reproduced in White.

But artists were not affected. Except for occasional hints of simultaneous vanishing points at right and left in separate details

a manuscript book by Leonardo, a treatise of interiors by Dutch painters in the seven-

on perspective that has not survived (pp. teenth century (see paintings by Steen, Jor207 ff.). In Manuscript A (1492), he depicts daens, and Brouwer), there exist almost no the intersection of the cone of vision by a indications that artists since Jean Fouquet

surface concave to the eye (g-f) where have ever found fault with straight line

flanking columns appear smaller than the formulas.9 Two-point perspective became

nearer, central column, rather than equal commonplace by the nineteenth century,

or larger, as in the conventional plane and melodramatic use has been made of

intersection (e-d).

three-point perspective, more often perhaps

Commenting on another section of the in twentieth-century illustration than in

same manuscript, White states, "There is painting. But Fouquet's curves have not

no escape from the conclusion that, in his been seen again. Even Piranesi's theatrical

definition of simple perspective, Leonardo Carceri retain strictly parallel verticals.

is visualizing a concave spherical surface, Taking into account his extremely wide-

the three-dimensional counterpart of the angle architectural views, it is difficult to

arcs centered on the eye that have already avoid the tug of simultaneous foreshorten-

appeared amongst his diagrams. No other ing from zenith and nadir in Piranesi's surface can be 'equally distant from the eye drawings.

in every part.' It is the first step towards the Ever since Fouquet and Donatello, verti-

theory of synthetic perspective" (p. 211). cals have remained conveniently sacrosanct,

But the die had been cast. One-point per- absolutely parallel. And there seems to have

spective was law, and the development of been no questioning of the propriety of the

synthetic perspective was never consum- Albertian rectilinear system. Worse, it has

mated. Leonardo himself appears to have been assumed in too many academies, by

counseled against depicting such close views too many artists, as well as by art historians

that curvature would be difficult to avoid. and laymen, that the inherited system rep-

He recognized the practical advantages of resents not only propriety and practical

reasonably distant views and of the one- metaphor, but essential fidelity to ordinary

point system.7

"natural" vision. Although common false

Other similar observations have been assumptions about vision were successfully

made from time to time. A didactic draw- challenged in the area of color by Impres-

ing manual dating from c. 1580, influenced sionism and more recently by Op, I know

by and possibly copied from Leonardo's of no proposal since Leonardo either to

work, had applied these principles to draw- revise Albertian rules or to consider Leo-

ings of statuary and the human figure. nardo's diagrams critically. Numerous schol-

Wilhelm Schickhardt, an obscure Tubingen ars, including John White, to whom I am

linguist, mathematician, and dilettante indebted for much of the foregoing histori-

etcher, published in 1624 a pamphlet in cal survey, have analyzed and expanded the

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154

ROBERT

HANSEN

that is in virtual agreement with the artist's visual experience:

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FIG. 4. 15th Century 1-point space (Albertian "artificial" perspective).

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FIG. 5. 19th Century 1-point, 2-point, and 3point space. Whenever three planes of an opaque cube are visible, three points are operative; two planes, two points; one plane, one point.
implications of Leonardo's theories. In doing so however, they have retained the same arcs and intersecting straight lines implied by Leonardo, and which have appeared in the diagrams of Schickhardt, Guido Hauck, and others interested in the problem.'0
III
I now suggest one fundamental change in these diagrams of synthetic perspective, a change which will for the first time, I believe, produce a perspective convention

All straight lines, except those that pass through a line projected straight ahead from a point between our eyes, appear curved, not as arcs, but as hyperbolas.
The hyperbolic curve achieves these advantages: first, it is the way the world appears to me and, I believe, to all of us; second, it is more effective than the arc diagram as a synthesis of the separate right and left, up and down, foreshortening of straight lines. Perhaps most important, the hyperbola permits all straight lines to share the same visual form. It reconciles a seeming contradiction in synthetic perspective involving the orthogonal lines. All curvilinear diagrams to date suggest two kinds of lines: those lines receding directly in front of the eye toward the central vanishing point are depicted as straight lines, while all other receding lines, to right, left, up, and down, are arcs.1' Why? The briefest answer is that an arc viewed edge-on, from a 90? angle, appears straight, just as a rightangle profile of a wheel is a straight line. But a more empirical analysis may be in order:
Your view of the walls of your room is one of lines that curve appreciably only where they are nearest you, and approach straightness as they recede in any direction; that is, toward any one of five vanishing points. One of these is, like Alberti's, in the center as you face a wall. The others are to be sensed at your extreme left, at your extreme right, directly above your head, and directly below your feet, all of course infinitely distant in space. Note that all points are located not somewhere diagonally in front of you, but, except for the center one, are on a plane with your forehead, so to speak, and at the infinitely distant circle where that plane converges with all other planes parallel to it.
There is of course a sixth point, directly behind you. This sixth point is not an unfleshed abstraction. All you need do to activate it is to turn your head. It is then immediately integrated into the panoramic six-point system within which all perpen-

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Hyperbolic Linear Perspective

155

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FIG. 6. Circular 5-point space, derived from Leonardo's writings. All lines are simple arcs except the orthogonals which are straight. This arrangement resembles the compressed view of architecture observed in convex mirrors and in extreme wide-angle photographs.

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FIG. 7. Hyperbolic "natural" perspective: 3-point, 4-point, and 5-point space. When three planes of a cube are visible, three points are operative; two planes, four points; one plane, five points.

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156

ROBERT HANSEN

FIG. 8. Two points are illustrated:1. The framedpicture representsa relativelywide-angle view. Most of our attention in normal viewing is limited to a much smaller area, often no largerthan the smallestrectangular"window"in the center,where the degree of curvatureis negligible. 2. The portion of the hyperbola that appears to curve is very small in the lines
near the center of vision, and increasesat the peripheryof vision. The smaller circlesenclose the nearlystraightsegmentsof the hyperbolas.

dicular planes are comprehended. As the side walls and the ceiling and floor ap proach you, defined by apparently straight orthogonals, they increase in size, are at their widest and highest nearest your eye, but passing behind you, over your head and beneath you, they decrease in size as they approach that sixth point. In addition, all lines that define oblique planes curve sooner or later at the point nearest your eye, their extensions curving sometimes over or under your shoulders toward a vanishing point behind you on that infinite sphere where all planes meet. This spherical surface of which your eye is the center contains all of the vanishing points of all possible rectangular planes and straight lines. Of course, you have only to turn your head 90? up, down, right, or left, in order to make the erstwhile orthogonals into lines parallel to your new picture plane, which,
answering now to a new pair of polar

opposite vanishing points, exhibits the same curvature nearest your eye that you saw in the previous direction. As you turn, the curving horizontals that had been oriented left-right gradually become straight orthogonals. Your eye, then, in the center of that sphere, is on a plane with every orthogonal and therefore sees the arc edgeon as a straight line.
Now the question remains: are the curves
really arcs? The answer, I believe, is: probably. With-
out moving the head, we of course cannot see accurately the entire area between opposite vanishing points. But my guess, based on the curves that I can see clearly in front of me, is that these curves do continue as
arcs, and that convex mirrors and wide-
angle lenses give us a fair picture of the circular nature of all straight lines as they would exist if we could see the entire 180?
field.

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157

However, do we see arcs? If in fact we
must turn our head in order to see the
extremities of each line, and if as the head
turns the curving line gradually straightens, is it not this graduation from curve to straight line that characterizes our experience? I am not prepared to prove that we see mathematically defined hyperbolas, but I am convinced that every straight line appears to us like a hyperbola, a curving
ligament connecting two straight orthogonal asymptotes. The task of reducing these hyperbolic lines to mathematically exact coordinates I must leave to others.
Inasmuch as these irrational curves depend more on freehand skill than on compass and straightedge, quantification, if it is possible at all, may prove to serve no practical function beyond satisfying the scholar's urge to know. Variations in application resulting from subjective interpretations are not likely to exceed the variations observable when conventional straight line systems are used.
If a drawing, painting, or photograph is to reflect our dynamic visual experience, I propose that the hyperbolic system (figures 1 and 7) is more faithful and more effective than any other. It is as logically consistent as other systems and at the same time surpasses them in the empirical test-it looks like this curving world, whether we move through it panoramically, or view it at rest, in separate limited views.
Thus, even in a static view, all verticals
are precisely perpendicular to the horizon only at the horizon, and immediately bend toward upper and lower vanishing points. They curve only as they approach and cross the horizon. Similarly, all horizontals parallel to the picture plane curve as they approach and pass beyond the line from zenith to nadir, which I call the zenith-line,
and are perpendicular to the zenith-line only at the zenith-line. Except for areas near horizon and zenith-line, these gentle curves, moving to the visual periphery (beyond the frame of the picture) as we turn to follow them, lose all appreciable curvature and appear to lead straight to one of the four external vanishing points. The orthogonal lines, however, appearing perfectly

straight as they emerge from the central internal vanishing point, may seem to curve only as they pass out of my field of vision over my ears or under my elbows, on their way to the sixth point behind me.
But I am here incapable of a decision: which way do the orthogonals appear to curve? For I must turn my head in order to see these lines pass near me. Now my decision must depend on favoring either a vertical movement of my head, making the
ceiling and floor convex and the walls concave; or a lateral movement which produces convex walls and concave ceiling and floor. I must remain neutral. The curvature of these orthogonal planes at any rate occurs outside the normal cone of vision.
It would appear sheer fantasy to attempt to symbolize this sixth vanishing point and the wall behind me in a two-dimensional drawing. Still, because this panorama is undeniably empirical, some graphic projection of this visual continuum may appear desirable. A globe enclosing my head is the only solution I can conceive.
The visible world that I carry with me can be metaphorically located at any moment on a concave spherical surface at all points equidistant from my eye, the picture plane implied by Leonardo. It is as if I wore at all times such a transparent sphere on my shoulders enclosing my head (actually two globes, one for each eye, requiring constant focusing), so that wherever I turned my eyes or my head, my world could be graphically plotted on it.
To transfer such a three-dimensional graphic diagram to a two-dimensional surface would require either an elastic paper to be wrapped first like a balloon over the globe, and then somehow stretched flat on a drawing board or canvas, a kind of Mercator's projection of any momentary view. Or else one or another kind of "equal-area" projection of the sort that cartographers

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ROBERT HANSEN

t

devise by cutting gores and flat patterns which simulate ovoid sections of the earth's surface, leaving gaps at the corners of a rectangular flat picture. It seems to me that such a procedure offers the artist a symbol closer to our visual perceptions than any other heretofore proposed. I believe that this hyperbolic system is both more logical and more faithful to raw sensory data than either conventional straight-line systems (whose most glaring lapse is the sanctity of all parallel verticals) or curvilinear systems employing arcs.
IV
Photography and twentieth-century painting have of course cast suspicion on the traditional attitude that the artist must be acquainted with his eyes' picture of objective shapes, proportion, and color. In surrealist painting, where an illusion of infinite physical distances might seem ap-

propriate, Dali, Tanguy, and de Chirico manipulated academic formulas in creating their super-real fantasies, producing eloquent metaphors, but not real replicas in the same sense as Baroque landscape. Developing under the influence of the academy, even photography, which began to reveal "this curving world," has been carefully limited by lenses that maintain "true" rectilinearity. Of course a wide-angle, "fisheye" lens is expected to produce this swollen effect, which is then considered an
oddity and a distortion. Wide-angle views, and reflections seen in spoons or in convex mirrors (like that in Van Eyck's Arnolfini Wedding Portrait) do indeed distort our retinal image, but they depict only an exaggeration, a difference in degree, not in kind.
Most twentieth-century figurative painters have attempted no substantial replica-
tion of a three-dimensional world, paying
more attention either to existential and

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Hyperbolic Linear Perspective

159

intuitional visual patterns (notably figura- kinetic sculpture and by film. Motion pic-

tive Diebenkorn) or to third-hand cliches tures have increasingly focused our atten-

(Pop and Common Object), most of them tion on the fleeting, the images given us

insistently flat. For a hundred years artists when the camera pans quickly. We have

have shown little interest in imitating deep begun to accept as real these vague, blurred

space, apparent exceptions like Hopper and patterns, which were also forcefully pre-

Sheeler notwithstanding. The current sented by certain action paintings, and less

Photo-realist movement shows signs of re- directly by Futurism.

viving artists' portrayal of "natural" vision, Outside of Photo-realism and Op, there

but the majority of its practitioners pre- seems no good reason for us to consider at

serve the flatness of the photographic sur- this time in art history any amendments to

face, a subtle but important distinction the laws of linear perspective. Who cares?

when the original subject matter implies What does it profit the artist or student of

space, modeling, and foreshortening. The art if he becomes aware of a curving world,

subject matter of many examples of the when his friends, patrons, and critics live in

new realism is really the photographic a perfectly adequate rectilinear one? Even

print, or the printed page that reproduces filmmakers, photographers, illustrators, and

the photograph, and not the three-dimen- painters of camera images may find a

sional subject that the camera first re- theory of appearance amusing but irrele-

corded. Linear perspective in most cases vant to their craft. However, beyond verisi-

must follow the parallax-corrected rectilin- militude, it is the subjective personal bias

ear pattern provided by the normal (ap- and the implied respect for primary infor-

proximately 55 mm.) lens.

mation afforded us by our visual sensations

Moreover, the visible world seems to that may be of particularly contemporary

many of us less objective than it used to relevance. Moreover, the sense of a voluptu-

be-less sharable-and more directly de- ous experience conveyed by these swinging

rived from each observer's retina-more irrational curves seems to me to better

private. Anyway, each artist's unique express the vital kinetic world we are com-

psychic and social associations make every ing to acknowledge than do rectilinear and

realist painting an interpretation, not a arc diagrams. In spite of the limitations of

copy. Only a selection of available small- an admittedly metaphoric convention, the

scale sense data are noticed, and then trans- hyperbolic system may be considered "true"

lated into the language of the medium: in more than one sense of the word.

brush strokes, glazes, spray-dots, inevitably The five-point hyperbolic system here

conferring a fresh and personal meaning on proposed is then offered as a contribution

the physical characteristics of the thing to the science of perception, and only inci-

described.l2 For a sensing of the real third dentally as a practical formula which artists

dimension, we must nowadays participate. might employ. It is presented first to con-

While Op seldom referred to the "objective vince skeptics that curvilinear perspective is

world," it often required us to perceive natural and consistent with experience, and

volume on a flat surface. Films, dance, hap- further that hyperbolic curves, not arcs of

penings, light shows, sound sculpture, con- circles, most accurately represent the ap-

ceptual art, and political art gesture all pearance of architecture in perspective.

seem to obviate the artist's need to examine For my part, if "often" implies exceptions

visual space.

to the general rule, I would maintain that

We continue to discover aspects that the optical information that remains open

enlarge our perceptions or contradict our to analytic introspection near the center of assumptions. The macroscopic and micro- vision, even as we move, is sufficient to

scopic have had their analogies in non- merit the effort to indentify and character-

objective painting. The assumption that ize those visual data. I do not believe that

reality can be measured only when it or the Gibson's writings justify the extreme judg-

observer is static has been cast in doubt by ment that "sensations are not entailed in

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ROBERT HANSEN

perception at all," even when he demonstrates that many additional factors must be considered in order to understand our com-
plex perceptual systems.
1E. H. Gombrich, Art and Illusion (London, New York, 1959), pp. 250-60.
'The same unthinking rejection occurs in William G. Lycan, "Gombrich, Wittgenstein, and The Duck-Rabbit" JAAC 30 (Winter 1971): 235.
8Psychologist James Gibson makes a useful distinction between the objective "visual world" and the subjective "visual field" and argues that other factors, such as the decision to attend to this or that aspect, are more basic to perception than are subjective sensory data. Several of his observations bear upon Gombrich's contentions.
From Gibson's The Perception of the Visual World (Boston, 1950):
If human beings had a visual field whose width included the entire horizon-if they could see all the way around at the same time like a rabbitthe field during locomotion would appear to open up ahead and close in behind in a rather astonishing manner. Such characteristics of the visual field created a great deal of difficulty for the early students of perspective and for painters who wished to represent a large sector of the visual world on a picture-plane.
Actually, of course, no rabbits and relatively few men have ever adopted the peculiar attitude of psychologists, artists, and geometers which enables them to see their visual field. They are, with good reason, perfectly content with the visual world as it is normally perceived, conforming to the rules of Euclidean geometry (p. 122).
Ordinary visual perception is not delimited by an oval-shaped boundary, nor does it have a clear center and a vague fringe. These are the characteristics of that unusual kind of visual experience, the visual field, which we get when we fixate a point and take note of the experience, concentrating on how it feels to see.... Can we find an explanation in the facts of ocular movement for the absence of the above characteristics in the visual world-its lack of boundaries, its more nearly uniform clarity and its possession of what we might call a panoramic character?... Unquestionably the panoramic visual world depends on a temporal series of excitations and just as unquestionably the succession of the excitations is not represented in the final experience (pp. 155, 157).
The visual world, it will be remembered, differs from the visual field in a number of ways. First, it has depth or distance, and it includes the experience of solid objects which lie behind one another. Second, it is Euclidean in the sense that neither the objects nor the spaces between them appear to change their dimensions in perception when the observer moves about. This is a general

way of saying that they tend to remain constant.
Third, it is stable and upright; things as seen have constant directions-from-here when the ob-
server moves his eyes and the perceived ground remains horizontal when the observer tilts his
head. Fourth, it is unbounded; our experience of the world does not have any visible margins or limits such as the visual field or a picture has. Finally, it has a characteristic to which we have scarcely referred but which, in a way, is the most important of all: it is composed of phenomenal things which have meaning (p. 164). And from Gibson's The Senses Considered as
Perceptual Systems (Boston, 1966); italics mine: A man, if he tries, can almost see the world as it
would project on a glass plate in front of his face -the inverse of his retinal projection, or a socalled retinal "image." He can never quite do so, for there is always some compromise with natural perception. If it were easy to detect pure sensations, we could all be representational painters without training.... I would now maintain that the optical (not retinal) gradients and the other invariants that carry the information for perception are often not open to analytic introspection, and that perception is therefore, in principle, not reducible to sensations.... The problem of depth perception considered as the conversion of two-dimensional experience into three-dimensional experience seems to me quite insoluble. In this book, and implicitly in my earlier book (Gibson, 1950), the problem disappears. If sensations are not entailed in perception at all, why speculate about how they might be changed into
perceptions? (pp. 237, 238) 'John White, The Birth and Rebirth of Pictorial
Space (London, 1957), esp. chapt. 8, 13, 14, 15. 6 For another interpretation, see Decio Giosefi,
"Perspective," Encyclopedia of World Art (New York 1966), p. 204.
e The four examples cited are illustrated in White. 7 Carlo Pedretti, in "Leonardo on Curvilinear
Perspective," Bibliotheque d'Humanisme et Renaissance, XXV (Geneva, 1963), pp. 69-87, argues that Leonardo alluded only to the lateral foreshortening of the panel picture itself and never advocated depicting such foreshortening.
8Goodyear, Greek Refinements (New Haven, 1912), pp. 55-57, 75-76.
9As Patrick Heelan has recently pointed out in "Toward a New Analysis of the Pictorial Space of Vincent Van Gogh," Art Bulletin (Dec., 1972), Van Gogh appears to have used segments of hyperbolic curves in a few paintings, although inconsistently and unsystematically even within a given painting. Heelan is mistaken, I believe, when he implies, following Rudolf Luneburg, that hyperbolic curves especially characterize binocular vision and not monocular vision, which he says is rectangular. I believe that he may also be misapplying Luneburg's metric when he segregates zones of convex and concave curvature in natural vision.
1A rare example of applied curvilinear perspective is the drawing accompanying the following ex-

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Hyperbolic Linear Perspective

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cerpt from Jan Gordon, Modern French Painting
(New York, 1926): ... We are, of course, well aware that horizontal
lines going away from us run into perspective, and we have accepted this fact of vision as pictorial possibility, although the Chinese have not. We must at the same time be aware that vertical lines
are also subject to the same laws, and this the photograph shows clearly. [He refers, I take it, to views with the camera pointed up or down.] But this law of nature we are disinclined to adopt as an artistic fact. We are aware also that all lines
sufficiently long are subjected to this perspective effect, there being in fact only two lines in nature which are visibly straight, the lines passing vertically and horizontally through the centre of vision. Thus, if we are facing a long building we get the following effect, the long lines curving in obedience to the laws of perspective:-

-IL ,B-C^E3u^B/|i:B

fpS^l--;]

Since these facts of perspective are ignored in art, and since our eyes are not trained to look for them, we ignore them in nature, and, indeed, in
spite of their existence, they are difficult to perceive. It is difficult to realize that when we are
sitting in a room the walls visually slope inward, or that when we face a piece of architecture such as fHampton Court it is usually barrel shaped in outline. The steps of St. Paul's are built with a slight horizontal curve to defeat this phenomenon.
If we claim that the difference between the
camera and our eye is that the camera has a fixed centre of vision, whereas we swing our vertical centre of sight as we turn round, we do not make nature any the more stable by our attempt to escape from error. As we swing about, each verti-

cal line as we look at it becomes upright, the other
lines going off into perspective, and so in fact all
of them swing about in the most confusing way as we move our eyes-the Greeks, with the good
taste that inspired their architecture, avoided all
vertical straight lines in their best work (pp. 14,
15). U Orthogonals appear straight, for instance, in
White's 1957 book. But in an earlier article by him, a similar diagram indicated the orthogonals as curves. The question, which way must they curve? was avoided by the 1957 version. "Developments in Renaissance Perspective," Journal of the Warburg and Courtauld Institutes, no. 12 (London 1949), 59.
In the light of Marcus Hester's "Are Paintings and Photographs Inherently Interpretative?" JAAC
32 (Winter 1972): 235.
Perhaps I am using a meaning of interpretation more normal in scholarship than in everyday usage, as in the interpretation of data. Certainly a scientist or scholar does not limit his critique of meaning to "persons and actions" but in addition examines ab-
stract facts and patterns large and small. It is in this sense, I believe, that Goodman and others apply the word to aesthetics.
Relativity of vision is most forcefully demonstrated by simply closing each eye alternately, especially if one eye is in sun and the other in shade.
When my two eyes disagree, I am forced to interpret the objective color and perspective of the scene before me as something close to the average of the two images, in order to depict it in two dimensions.
The new realism appears to avoid interpretation of data. But anyone who compares the actual paintings will recognize the distinct differences in style (texture, color, etc.) between such skillful Photorealists as McLain, Goings, Estes, and Eddy, for instance. The magazine reproductions of the paintings, on the other hand, three or four generations removed from the original subject, are extremely misleading in this respect. (See the special issue of Art in Amnerica,Nov.-Dec. 1972.)

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