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.
Pollick, F. E., Maoz, U., Giblin, P. J., Handzel, A. A., Giblin, P. J., Sapiro, G., & Flash, T. (2009). Three-dimensional arm movements at constant equi-affine speed. Cortex, 45(3): 325-339. doi: 10.1016/j.cortex.2008.03.010
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NOTICE: this is the author’s version of a work that was accepted for publication in Cortex. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Cortex, volume 45, issue 3, in 2009. DOI: 10.1016/j.cortex.2008.03.010
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