cézanne and the average effect of foreshortening on shape

Leonardo

Cézanne and the Average Effect of Foreshortening on ShapeAuthor(s): Kenneth R. AdamsSource: Leonardo, Vol. 8, No. 1 (Winter, 1975), pp. 21-25Published by: The MIT PressStable URL: http://www.jstor.org/stable/1573183 .

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Leonardo, Vol. 8, pp. 21-25. Pergamon Press 1975. Printed in Great Britain

CEZANNE AND THE AVERAGE EFFECT OF FORESHORTENING ON SHAPE

Kenneth R. Adams* Abstract-Curves are either flattened or sharpened when they are foreshortened in perspective. The two kinds of foreshortening seem to be perceived differently. The author argues that in some respects we see the world flatter than it is, and that, for example, Cezanne's and Mondrian's figurative paintings reflect this fact.

I.

In a previous article [1] I showed how one's viewpoint in relation to a drawing can determine how one interprets a drawn image. The effects that I described are modified by varying the figurative content of the drawn image, and some of these variations have been described by E. H. Gom- brich [2]. In the present article I consider the effects of changing viewpoints on foreshortening in general, using a circle as my example through- out, on the assumption that any curve can be approximated by a series of circular arcs.

In the still life paintings by Cezanne, there are many examples of circles in perspective. No two circles are shown alike. Such paintings, as his 'Still Life with Apples' (Fig. 1) [3], remind one how different the various parts of a circular rim can appear.

There is a standard psychology of perception test in which a series of ellipses, ranging between a circle and a straight line, is shown frontally on a plane surface, alongside a circle on a tilted plane surface. A viewer is asked to select the ellipse that matches the tilted circle. Geometrical perspective can be used to define a particular ellipse as the 'correct' answer but most viewers choose an ellipse that is somewhat closer in form to a circle. The test forms the basis for the shape-constancy hypothesis. But it does not allow the viewer to do what Cezanne did, which was to consider arcs of the tilted circle separately and to choose a matching oval that is not one in the series of true ellipses. In short, the test is too crude to reveal the subtleties of perception that are evident in the paintings of Cezanne.

* 19 Dartmouth Park Road, London NW5 1SU, England. (Received 19 March 1973.)

One may be inclined to think of foreshortening as a simple transformation, such as the gradation of texture that appears when a plane is tilted. However, a curve drawn on a tilted plane is foreshortened in one, or both, of two opposite ways. It is flattened, as are the nearest and farthest portions of the edge of a round table top, or it is sharpened as are the portions of the edge to the right and to the left.

Now I come to a curiosity in visual perception: the flattened portions of a round table edge and sharpened portions do not contribute equally to the perception of the table in three dimensions. One is inclined to accept the curvature of the flattened arc as undistorted and the sharpened arc as dis- torted and to utilize the 'distortion' of the latter as a depth cue. Indeed, Cezanne in his painting of trees, such as in his 'Montagne Sainte Victoire' (Fig. 2) [3, p. 74], makes this distinction clear. The sharp bends in the branches just above the summit

Fig. 1. P. Cezanne, 'Still Life with Apples', oil on canvas, 42 x 54 cm., 1890-1894 [3]. (Private collection.)

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Kenneth R. Adams

of the mountain suggest of depth, while the gentle curves in the branches to left and right seem parallel to the picture plane, or nearly so. It is not necessary to use a painting to demonstrate this effect; all leafless trees in wintertime give the same effect in their silhouettes. But Cezanne, through his economy of line and his analytic approach to visual sensation, has made clear what one experiences in nature but often fails to notice. Mondrian in his earlier tree paintings made many sharp arcs, e.g. in his 'Red Tree' (Fig. 3). In later paintings of trees, e.g. the 'Grey Tree' (Fig. 4), there are mainly gently curving branches, which one accepts as parallel to the picture plane and which prefigure the pictorial structure of his later highly abstract, yet figurative paintings.

There is, then, a bias in visual perception that, it seems, has not been formulated before as a principle. Generally, the larger part of a linear network that makes up the silhouette of a tree represents flattened rather than sharpened arcs. One might have expected that if the growth forms of branches in a particular species of tree were sufficiently familiar to an observer, he would find

Fig. 2. P. Cezanne, 'La Montagne Sainte-Victoire', oil on canvas, 60 x 90 cm., 1886-1888. (Collection of and photo

by Courtauld Institute Galleries, London.)

Fig. 3. P. Mondrian, 'The Red Tree', oil on canvas, 70 x 99 cm., 1909-1910. (Collection of Haags Gemeente Museum,

The Hague.)

that the flattening of an arc constitutes as a good a cue to depth as does the sharpening of an arc. Yet, that is not the case. An explanation for this bias may well be found by the application of Weber's Law, which states that the increment that must be added to a stimulus in order to make it just notice- ably different is, for any given physical range, a constant fraction of its physical magnitude (a good first approximation to experimental results in the middle range of magnitudes) [4]. As far as I know, this has not been studied experimentally. The appropriate physical measure for curvature appears to be a difficult question, one that well may have deterred investigators.

One might now explain in detail why Cezanne chose to judge the parts of the tilted circle separately and why he set up his still lifes so that the perimeter of a circle is interrupted in a variety of ways. I wish, however, to present a more rigorous geometrical basis for the understanding of foreshortening.

II.

If a plane on which a circle is drawn is rotated on a diameter (horizontal) from frontal viewing (0?) to edge-on viewing (90?), the circle is seen to transform through a series of ellipses having the same major axis (horizontal) but progressively shorter minor axes. At the edge-on position, the minor axis is zero in length and the horizontal straight line (the circle's diameter) is the limiting form of the ellipse. Fig. 5 shows, by means of orthographic projection, the shape of one quadrant of the ellipse starting with 0? tilt (given by a quadrant of the circle), continuing with 15?, 30?, 45?, 60?, 75? and ending with 90? (given by the horizontal line).

The curvature of ellipses in Fig. 5 becomes flatter at the intersection of the ellipse with the minor axis and sharper at the intersection with the major

Fig. 4. P. Mondrian, 'The Grey Tree', oil on canvas, 78-5 x 107-5 cm., 1912. (Collection of Haags Gemeente

Museum, The Hague.)

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Cezanne and the Average Effect of Foreshortening on Shape

axis as the angle of tilt increases. There is a point on the perimeter separating the portion of the ellipse quadrant that has undergone flattening (solid line) and the portion of the ellipse that has undergone sharpening (dotted line). The separation point marks the location on the ellipse where its radius of curvature equals the radius of the circle. As the tilt angle is increased, the separation point is positioned along the ellipse closer to its intersection with the major axis. When the tilt angle is 90?, the point is located at the end of the major axis. The position of the separation point for various angles of tilt is described by the concave curved line in Fig. 5. Fig. 6 shows a plot of this curved line on rectangular coordinates, where the quadrant arcs are shown as horizontal lines.

The concave curved line was constructed by con- necting separation points calculated for the series of ellipses shown in Fig. 5. The calculation was made using the construction shown in Fig. 7 and the following equation:

1 - cos4/3 T sin D-==

1 - cos2 T

where T is the angle of tilt and D is the angle formed by the major axis of the ellipse and the normal that meets the ellipse at the separation point. For a given value of T and with the resulting value of D, the corresponding value for the altitude angle A, measured in degrees clockwise and locating the position of the separation point on the ellipse, was calculated using the following equation:

tan A = tan (90? - D) b

where a/b represents the ratio of the lengths of the major and minor axes.

In order to assess the probability of an arc taken at random being flattened or sharpened, I find the following model convenient. A small circle is

00 150

159

300 300

450 \ 450

I-

c 600 600

754

900 90? 0? 15? 300 45? 60 75

imagined at the center of the Earth drawn on the plane of the Earth's equator. Clearly, in the northern hemisphere, there is a frontal view of the circle only from the North Pole at 90? north (0? angle of tilt). From any point on the equator it is viewed as a straight line and from any inter- mediate latitude position on the Earth's surface, for example on the 45? latitude (45? angle of tilt), it is viewed as an ellipse. At 40? latitude (50? angle of tilt), a flatter ellipse is viewed. Since the length of the 40? parallel of latitude is greater than that of the 45? parallel of latitude, one can say that the probability of seeing a flatter ellipse is greater when one views a circle positioned randomly in space. The probability of appearance of a given ellipse can be calculated because it varies in direct pro- portion to the length of the parallel of latitude associated with the ellipse in the above model. The probability of seeing any of the range of ellipses of the model between two parallels of latitude is equal to the ratio of the area of the zone defined by the two parallels of latitude to the surface area of the sphere. The area of a zone of a sphere is equal to the area on the surface of an enclosing cylinder (where the cylinder and sphere have the same diameter and the axis of the cylinder falls along the polar axis of the sphere) bounded by the two planes that bound the zones on the sphere [5]. The intervals along the side of Fig. 6 representing

Distance along quadrant arc (clockwise)

VQ C0 c 0

4--

-

0? 15? 300 450 600 750 90?

Divisions of quadrant arc, degrees (clockwise)

Fig. 6. Locus of separation point on rectangular coordinates (cf. Fig. 5); probabilities of flattening and sharpening.

Divisions of quadrant arc, degrees (clockwise)

Fig. 5. Orthographic projection of one quadrant of a circle in various angles of tilt with respect to a viewer. Fig. 7. Construction used in calculating the separation point.

L -

23



Kenneth R. Adams

the tilt angle were determined using this calculation. These intervals are identical to those in the orthographic projection in Fig. 5. The separate calculation is necessary. The orthographic projection of the globe provides the graticule for Fig. 5. The cylindrical equal-area projection of Lambert provides the graticule for Fig. 6. It is somewhat puzzling to find that meridians and parallels are reversed in the two figures but it follows from the fact that two different globes are involved. The globe of Fig. 6 is a globe of viewpoints on which lines of equal tilt correspond to parallels. The globe of Fig. 5 results from the rotation of a circle and here the lines of equal tilt correspond to meridians. The identity of the intervals is perhaps a happy coincidence.

The area to the left of the concave curve in Figs. 5 and 6 contains the series of flattened portions of the quadrant arc, while the area to the right contains the sharpened portion series. The two areas in Fig. 6 constitute 55% and 45%, respectively, of the total. Consequently, these are the probabilities of an arc taken at random being flattened or sharpened. However, sharpened arc lengths are shortened more than flattened arc lengths are. This can be seen in Fig. 8 where the length of the projection of a circle quadrant is shown (solid curved line) to vary from -ra/2 (one-quarter the circumference) at 0? tilt to a (the circle's radius) at 90? tilt and where the concave line indicates the length of the projection of the flattened portion of the circle quadrant as function of tilt angle. The length of the projection of the sharpened portion is indicated by the horizontal distance between the concave line and the solid S-curve line. If one selects at random an arc of an ellipse, there is a chance of about 63 % that it represents a flattened portion of the circle from which the ellipse was generated.

The last observation noted above seems to give support for the assertion made earlier-that the larger part of a linear network making up the silhouette of a tree is flattened rather than sharpened. But before accepting it as a valid

Distance along projected quadrant arc measured clockwise

of305____?__ ______fa

1 -?\ (P ) 45? _ _

60? ' _.

V~%0

90O

Fig. 8. The lengths of the projection of a circle quadrant; probabilities of flattening and sharpening.

support, one should examine the implications of identifying all the major axes of the series of ellipses with the diameter on which the original circle was tilted. In normal geometrical perspective this is not the case, for the major axis of a perceived ellipse is a chord lying parallel to and somewhat in front of the diameter on which the original circle turned. Only if the viewpoint is supposed to be at infinity do they coincide. However, by removing the viewpoint to infinity one can separate the two logically distinct effects of perspective: foreshortening, which is of interest here, and the diminution of size with increasing distance, which is conveniently ignored. The complications that result from vatying the distance of the viewpoint are certainly worth studying but I do not think that they alter the general conclusions.

III.

The considerations about the tilted circle can be extended to a general statement about the appearance of curves in space. One can assume a universe containing only short circular arcs, randomly dis- tributed and orientated, and of randomly different radius of curvature. When all the arcs are projected onto a plane, how are they transformed? The answer is clear. About 45 % of the arcs are sharpened and about 55 % are flattened. When an arc is sharpened on foreshortening, its length is decreased more than it is when the arc is flattened. The greater part of the total arc length projected represents flattened arcs and an arc taken at random from the visual field is more likely to have been flattened than sharpened. This argument requires qualification along the lines noted in the previous paragraph. Plane projection with a finite sightline would be rendered useless because the greater part of the picture plane would be seen obliquely and the shapes on it would be subject to Leonardo's paradox: in a row of cylindrical pillars, parallel to the picture plane, the more distant pillars have larger projective images than the nearer ones. The standard way of avoiding Leonardo's paradox is to use 'natural' perspective, projecting onto a sphere, centered on the viewer's eye. Every part of the sphere is then orthogonal to the viewer's line of sight. By making the radius of the sphere equal to infinity one avoids the des- criptive difficulties that shapes drawn on spherical surfaces would introduce.

These probabilities could provide viewers with baseline expectancies, which the interpretation of specific depth cues would modify rather than replace. As mentioned above, this does not appear to happen, for the flattened arc is less subject to interpretation in depth than is the sharpened arc. A surprising conclusion follows: The world is seen

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Cezanne and the A verage Effect of Foreshortening on Shape

flatter than it is. This conclusion should be treated with philosophical care, since it derives from the study of only a limited set of visual features. The proviso should be added, 'other things being equal', since, they are not often so. Nevertheless, it does provide support for the view that Cezanne and Mondrian found it necessary to invent new pictorial structures, not simply to come to terms with a flat canvas but in order to reflect a feature of perception.

I wish to acknowledge the assistance of Professor E. H. Thompson of University College, London, in supplying the first equation.

REFERENCES

1. K. R. Adams, Perspective and the Viewpoint, Leonardo 5, 209 (1972).

2. E. H. Gombrich, The 'What' and 'How': Perspective Representation and the Phenomenal World, in Logic and Art: Essays in Honor of Nelson Goodman, R. Rudner and I. Scheffler, eds. (New York: Bobbs- Merrill, 1972).

3. M. Shapiro, Cezanne (London: Thames & Hudson, 1952), p. 100.

4. E. G. Boring, Sensation and Perception in the History of Experimental Psychology (New York: Appleton- Century-Crofts, 1942), pp. 34-45.

5. J. S. Hails and E. J. Hopkins, Solid Geometry (London: Oxford Univ. Press, 1957), p. 114.

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cézanne and the average effect of foreshortening on shape

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