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.

Implied Dynamics Biases the Visual Perception of Velocity

“…The selected profiles were actually highly non-uniform, mimicking closely the kinematics of drawing movements in which instantaneous velocity and curvature covary according to the 2/3 Power Law [8]. This connection between visual perception and the velocity-curvature covariance has been confirmed for 2D [9] and 3D trajectories (where torsion also plays a role, [10]). Moreover, it has been shown that the 2/3 Power Law can be derived from the assumption that the equi-affine velocity of movements is constant [11]–[12].…”

Motor primitives—new data and future questions

“…Kinematic MPs are seen in a range of animal forms from soft-bodied muscular hydrostat effectors, to exoskeletons and endoskeletons. The commonalities in the kinematic operations of these very different effectors are likely to be geometric factors in visuomotor integration, see [49,50]. Analyses of the geometric factors continue to be explored, focusing on unifying features of affine geometries and links to dynamics.…”

Gesture segmentation and classification using affine speed and energy

“…where a is a gain factor and _ x, _ y, € x, and € y are the first and second derivatives of x, y with respect to time. Important research from around the world 25,24 has demonstrated that in drawing trajectories, the gain factor a is approximately constant for simple elliptical shapes and piecewise constant for more complex patterns. 26,27 In addition, the affine speed v a for 2D movements, which is equal to the time derivative of the equi-affine arc-length, is defined by…”

“…22,23 Based on these results, it was demonstrated, that for 3D movements, the one-sixth power law explains the data better than the two-thirds power law. 22,24 In this article, affine speed is proposed as a feature to segment hand trajectories associated to medical training. In addition, energy calculation is introduced as a correlation measure to characterize hand skills.…”