Three-dimensional arm movements at constant equi-affine speed

Abstract

It has long been acknowledged that planar hand drawing movements conform to a relationship between movement speed and shape, such that movement speed is inversely proportional to the curvature to the power of one-third. Previous literature has detailed potential explanations for the power law's existence as well as systematic deviations from it. However, the case of speed-shape relations for three-dimensional (3D) drawing movements has remained largely unstudied. In this paper we first derive a generalization of the planar power law to 3D movements, which is based on the principle that this power law implies motion at constant equi-affine speed. This generalization results in a 3D power law where speed is inversely related to the one-third power of the curvature multiplied by the one-sixth power of the torsion. Next, we present data from human 3D scribbling movements, and compare the obtained speed-shape relation to that predicted by the 3D power law. Our results indicate that the introduction of the torsion term into the 3D power law accounts for significantly more of the variance in speed-shape relations of the movement data and that the obtained exponents are very close to the predicted values.

Drawing ellipses in water: evidence for dynamic constraints in the relation between velocity and path curvature

“…Another kinematic constraint that can predict the power law is provided by the principle of constant equi-affine speed (Pollick and Sapiro 1997;Flash and Handzel 2007;Pollick et al 2009) or a combination of Euclidean, affine, and equi-affine geometries (Bennequin et al 2009). Interestingly, the principle of constant equi-affine speed leads to a generalization of the power law to three-dimensional (3D) movements Pollick et al 2009).…”

“…Interestingly, the principle of constant equi-affine speed leads to a generalization of the power law to three-dimensional (3D) movements Pollick et al 2009). Still another kinematic constraint that has been shown to be compatible with the 2/3-PL for ellipses is represented by the composition of simple harmonic oscillations with the same frequency and a phase offset, either involving the Cartesian coordinates of endpoint motion (Lacquaniti et al 1983) or the angular coordinates of upper limb segments motion (Soechting and Terzuolo 1986;Schaal and Sternad 2001;Dounskaia 2007).…”

Spatial constant equi-affine speed and motion perception

“…We have recently shown that constant equi-affine speed generalizes to 3D and that 3D movement at constant equi-affine speed entails a power law different from the two-thirds power law. Moreover, general self-paced unconstrained scribbling and shape-tracing hand movements in 3D both seem to be produced at roughly constant spatial (3D) equi-affine speed (Pollick et al , 2009Maoz et al 2009).…”

“…Following a rationale similar to that used to prove the equivalence of the two-thirds power law to moving at planar constant equi-affine speed, we derived the formula for motion at constant spatial equi-affine speed (see Pollick et al 2009). Geometrically, spatial equi-affine transformations preserve the volume (rather than the area) enclosed by a 3D shape.…”

“…Furthermore, planar motion according to this power law is perceived as most uniform Stucchi 1989, 1992). The equivalence of the power law to planar motion with constant equi-affine speed has led to its successful generalization to 3D motion generation (Pollick et al 2009;Maoz et al 2009). Here we first and foremost tested whether constant equi-affine speed for 3D visual motion is also perceived as more uniform.…”