Affine differential geometry and smoothness maximization as tools for identifying geometric movement primitives

Neuroscientific studies of drawing-like movements usually analyze neural representation of either geometric (e.g., direction, shape) or temporal (e.g., speed) parameters of trajectories rather than trajectory's representation as a whole. This work is about identifying geometric building blocks of movements by unifying different empirically supported mathematical descriptions that characterize relationship between geometric and temporal aspects of biological motion. Movement primitives supposedly facilitate the efficiency of movements' representation in the brain and comply with such criteria for biological movements as kinematic smoothness and geometric constraint. The minimum-jerk model formalizes criterion for trajectories' maximal smoothness of order 3. I derive a class of differential equations obeyed by movement paths whose nth-order maximally smooth trajectories accumulate path measurement with constant rate. Constant rate of accumulating equi-affine arc complies with the 2/3 power-law model. Candidate primitive shapes identified as equations' solutions for arcs in different geometries in plane and in space are presented. Connection between geometric invariance, motion smoothness, compositionality and performance of the compromised motor control system is proposed within single invariance-smoothness framework. The derived class of differential equations is a novel tool for discovering candidates for geometric movement primitives.