Four experiments related human perception of depth-order relations in structure-from-motion displays to current Euclidean and affme theories of depth recovery from motion. Discrimination between parallel and nonparallel lines and relative-depth judgments was observed for orthographic projections of rigidlyoscillatingrandom-dot surfaces. Wefound that (1) depth-order relations were perceived veridically for surfaces with the same slant magnitudes, but were systematically biased for surfaces with different slant magnitudes. (2) Parallel (virtual) lines defmed by probe dots on surfaces with different slant magnitudes were judged to be nonparallel. (3) Relative-depthjudgments were internally inconsistent for probe dots on surfaces with different slant magnitudes. It is argued that both veridical performance and systematic misperceptions may be accounted for by a heuristic analysis of the first-order optic flow.Appropriate 2-D motions produce phenomenal impressions of movement in depth (see, e.g., Miles, 1931;Musatti, 1924;Wallach & O'Connell, 1953). Certain types of these phenomena have been named structure from motion (SFM). The questions of how these impressions arise and what type ofgeometric structure is derived from these motions have led to both experimental and theoretical work on depth recovery from motion. The psychophysical research has evaluated the capabilities of the human visual system in light of the constraints and the scope of the algorithms devised to derive 3-D geometric properties from 2-D motions (for a review, see Braunstein, 1989;Norman & Todd, 1992). Euclidean and Affine Theories of SFMThe analysis of the 2-D motions compatible with the orthographic projection ofrigid 3-D motion allows the recovery of metric, affine, or purely local properties of the projected objects. Ullman's theorem, for example, demonstrates that three orthographic views of four points undergoing rigid motion are sufficient to determine the relative depths of the points uniquely (under the assumption ofrigid motion), and hence the full metric Euclidean structure thereof (Ullman, 1979). Consistent with this theorem, Ullman proposed that the perceptual recovery of depth should also be unique and veridical for multipleview display and that subjects' abilities to perform metric judgments in SFM displays should improve with increasing numbers of views (Ullman, 1984). We term this

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Abstract. Monocular observers perceive as three-dimensional (3D) many displays that depict three points rotating rigidly in space but rotating about an axis that is itself tumbling. No theory of structure from motion currently available can account for this ability. We propose a formal theory for this ability based on the constraint of Poinsot motion, i.e., rigid motion with constant angular momentum. In particular, we prove that three (or more) views of three (or more) points are sufficient to decide if the motion of the points conserves angular momentum and, if it does, to compute a unique 3D interpretation. Our proof relies on an upper semicontinuity theorem for finite morphisms of algebraic varieties. We discuss some psychophysical implications of the theory.