Smooth surfaces in three-dimensional Euclidean space

are locally described by classical differential geometry. In case some spatial direction (e.g., the direction of gravity in the natural landscape or the viewing direction in visual space) assumes a special role, this formalism has to be replaced by the special theory of “topographic surfaces’’ and one speaks of “surface relief ’’ (Liebmann, 1902/1927). Examples include topographic relief and— in case of pictures of smooth objects—pictorial relief. The mathematical description of relief was first fully developed by Rothe (1915; see also Koenderink & van Doorn 1993, 1994) in the early part of the 20th century, although much had already been guessed (often erroneously though) by mathematicians in the second half of the 19th century. In the relief formalism, the basic quantities are the depth and the depth gradient. The depth gradient is often specified by the surface attitude, in terms of local surface slant and tilt. Singular points are the Morse singular points (Milnor, 1963), that are the near and far points (which are local extremes of the depth) and saddles (or mountain passes). Singular curves are ridges and courses (named after the crests and water courses of topography) and contours. The contours are envelopes of the constant depth curves, and they form the boundaries of the visible parts of objects due to self-occlusion: they do not occur (at least on the large scale) in topography but are the natural boundaries of pictorial reliefs. The structure of pictorial relief as the essential element of the perception of solid shape in general was first discussed extensively by the sculptor Hildebrand (1893) in his influential book “On the Problem of Form.’’ Hildebrand (as many before him, notably Leonardo) saw no essential difference between relief in the “real’’ visual world and that of pictorial space, at least not for purely “visual,’’ as distinct from “motor,’’ images. (By this, Hildebrand meant vision with a stationary vantage point and limited field of view— say a viewing distance at least three times the diameter of the object.) Such general ideas provide one motivation for the present study. In geometry, many formal identities exist between the basic quantities and their spatial derivatives—that is, such trivialities as that the gradient is the derivative of the depth. Even the very fact that the relief describes a coherent surface in the first place is expressed by simple identities: for instance, the curl of the depth gradient vanishes identically (and vice versa: any vector field with vanishing curl is the gradient of some surface). Such identities are of a trivial nature (e.g., the vanishing curl expresses the fact that you won’t gain height if you make a tour in some landscape that returns you to your point of departure): hence, their high degree of certainty. Such (formal) facts may lead us to certain expectations in the study of the perception of surfaces, when we naively confuse perceptual entities with geometrical objects (e.g., depth with physical range). However, in the study of perception, the basic quantities depth and attitude are operationally defined, and hence quite independent quantities. Here the above-mentioned identities are not formal trivialities at all, but subject to empirical verification. There is no compelling reason why the curl of an empirically determined “gradient’’ should vanish. In an earlier study (Koenderink, van Doorn, & Kappers, 1992), we showed that pictorial surface coherence