  Neuroscientific studies of 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. Constant rate of accumulating equi-affine arc
corresponds to compliance with the two-thirds power-law model. Geometric
measurement is invariant under a class of geometric transformations and may be
chosen to be an arc in certain geometry. 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 enforces
consistency with desired (geometric) parametrization of curves on solutions of
the first part. Equations in different geometries in plane and in space and
their known solutions are presented. Connection between geometric invariance,
motion smoothness, compositionality and performance of the compromised motor
control system is discussed. The derived class of differential equations is a
novel tool for discovering candidates for geometric movement primitives.
