Pollick Frank E, Maoz Uri, Handzel Amir A, Giblin Peter J, Sapiro Guillermo, Flash Tamar
Department of Psychology, University of Glasgow, United Kingdom.
Cortex. 2009 Mar;45(3):325-39. doi: 10.1016/j.cortex.2008.03.010. Epub 2008 Jun 28.
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.
长期以来，人们一直认为平面手绘动作符合运动速度与形状之间的一种关系，即运动速度与曲率的三分之一次方成反比。先前的文献已经详细阐述了幂律存在的潜在解释以及与它的系统性偏差。然而，三维（3D）绘图动作的速度-形状关系情况在很大程度上仍未得到研究。在本文中，我们首先基于平面幂律意味着以恒定等仿射速度运动这一原理，推导出平面幂律到3D动作的一种推广。这种推广产生了一个3D幂律，其中速度与曲率的三分之一次方乘以挠率的六分之一次方成反比。接下来，我们展示了来自人类3D涂鸦动作的数据，并将获得的速度-形状关系与3D幂律所预测的关系进行比较。我们的结果表明，将挠率项引入3D幂律能显著解释运动数据速度-形状关系中更多的方差，并且获得的指数非常接近预测值。
