Spatial constant equi-affine speed and motion perception

Abstract

The two-thirds power law, postulating an inverse local relation between the velocity and cubed root of curvature of planar trajectories, is a long-established simplifying principle of human hand movements. In perception, the motion of a dot along a planar elliptical path appears most uniform for speed profiles closer to those predicted by the power law than to constant Euclidean speed, a kinetic-visual illusion. Mathematically, complying with this law is equivalent to moving at constant planar equi-affine speed, while unconstrained three-dimensional drawing movements generally follow constant spatial equi-affine speed. Here we test the generalization of this illusion to visual perception of spatial motion for a dot moving along five differently shaped paths, using stereoscopic projection. The movements appeared most uniform for speed profiles closer to constant spatial equi-affine speed than to constant Euclidean speed, with path torsion (i.e., local deviation from planarity) directly affecting the speed profiles perceived as most uniform, as predicted for constant spatial equi-affine speed. This demonstrates the dominance of equi-affine geometry in spatial motion perception. However, constant equi-affine speed did not fully account for the variability among the speed profiles selected as most uniform for different shapes. Moreover, in a followup experiment, we found that viewing distance affected the speed profile reported as most uniform for the extensively studied planar elliptical motion paths. These findings provide evidence for the critical role of equi-affine geometry in spatial motion perception and contribute to the mounting evidence for the role of non-Euclidean geometries in motion perception and production.

Keywords: constant equi-affine speed; motion invariants; motion perception; non-Euclidean geometry; relation between motion perception and production.