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

Neuroscientific studies of the drawing-like movements usually analyze neural representation of either geometric (eg. direction, shape) or temporal (eg. speed) features of trajectories rather than trajectory's representation as a whole. This work is about empirically supported mathematical ideas behind splitting and merging geometric and temporal features which characterize biological movements. Movement primitives supposedly facilitate the efficiency of movements' representation in the brain and comply with different criteria for biological movements, among them kinematic smoothness and geometric constraint. Criterion for trajectories' maximal smoothness of arbitrary order $n$ is employed, $n = 3$ is the case of the minimum-jerk model. I derive a class of differential equations obeyed by movement paths for which $n$th order maximally smooth trajectories have constant rate of accumulating geometric measurement along the drawn path. The geometric measurement is invariant under a class of geometric transformations and may be chosen to be an arc in certain geometry. For example the two-thirds power-law model corresponds to piece-wise constant speed of accumulating equi-affine arc. Equations' solutions presumably serve as candidates for geometric movement primitives. The derived class of differential equations consists of two parts. The first part is identical for all geometric parameterizations of the path. The second part is parametrization specific and is needed to identify whether a solution of the first part indeed represents a curve. Counter-examples are provided. Equations in different geometries in plane and in space and their known solutions are presented. A method for constructing trajectories based on primitives in different geometries is proposed. The derived class of differential equations is a novel tool for discovering candidates for geometric movement primitives.