Euclidean Cues versus Affine or Projective Invariants in the Visual Perception of Three-Dimensional Distances

Abstract

Lappin and Fuqua (1983 Science221 480 – 482) reported accurate measurement of three-dimensional (3-D) distances and suggested that performance was based on structural invariance of patterns undergoing perspective transformations. We wanted to replicate their basic findings and test specific hypotheses about the use of Euclidean cues versus affine or projective invariants. The displays consisted of orthographic and perspective projections of three collinear dots rotating rigidly around a fixed centre in a plane slanted 45° in depth. Observers were asked to decide whether the middle of the three dots was exactly centred in 3-D space between the other two dots. The visible rotation segments were 120°, 160°, or 200° and the displacements were 2%, 4%, or 6%. Although our untrained observers performed more poorly overall than Lappin's well-practiced observers, two main results were replicated. First, there was no effect of segment size, which suggests that Euclidean cues were not used. Second, there was no difference between orthographic and perspective displays, which suggests that the invariants used must be projective, not affine. Additional experiments will be performed to investigate the relative importance of invariants such as cross ratios, which are based on the positions of the dots within each time frame, versus invariants of the conic sections, which are based on the elliptic traces of the dots formed over time.