### abstract ###
Some studies suggest that complex arm movements in humans and monkeys may optimize several objective functions, while others claim that arm movements satisfy geometric constraints and are composed of elementary components.
However, the ability to unify different constraints has remained an open question.
The criterion for a maximally smooth motion is satisfied for parabolic trajectories having constant equi-affine speed, which thus comply with the geometric constraint known as the two-thirds power law.
Here we empirically test the hypothesis that parabolic segments provide a compact representation of spontaneous drawing movements.
Monkey scribblings performed during a period of practice were recorded.
Practiced hand paths could be approximated well by relatively long parabolic segments.
Following practice, the orientations and spatial locations of the fitted parabolic segments could be drawn from only 2 4 clusters, and there was less discrepancy between the fitted parabolic segments and the executed paths.
This enabled us to show that well-practiced spontaneous scribbling movements can be represented as sequences of a small number of elementary parabolic primitives.
A movement primitive can be defined as a movement entity that cannot be intentionally stopped before its completion.
We found that in a well-trained monkey a movement was usually decelerated after receiving a reward, but it stopped only after the completion of a sequence composed of several parabolic segments.
Piece-wise parabolic segments can be generated by applying affine geometric transformations to a single parabolic template.
Thus, complex movements might be constructed by applying sequences of suitable geometric transformations to a few templates.
Our findings therefore suggest that the motor system aims at achieving more parsimonious internal representations through practice, that parabolas serve as geometric primitives and that non-Euclidean variables are employed in internal movement representations .
### introduction ###
Despite decades of research on the formation of human hand trajectories, the basic mechanisms of neuromotor control underlying the generation of even the simplest drawing movements remain poorly understood CITATION.
Various studies have proposed that human movement preparation aims at optimizing either kinematic CITATION CITATION or dynamic CITATION criteria, or minimizing movement variance CITATION CITATION.
Studies in vertebrates have suggested that voluntary movements are composed of basic movement elements combined in parallel or sequentially CITATION CITATION.
Such modular organization can account for the versatility of animal and human movements and for their ability to acquire new skills.
Geometrically invariant properties of drawing movements were formalized by the two-thirds power law CITATION.
These kinematic constraints were shown to hold both with respect to movement production CITATION and perception CITATION, CITATION.
Earlier studies also showed that the two-thirds power law is equivalent to moving at a constant equi-affine speed CITATION CITATION and there is psychophysical and neurophysiological evidence for the significant role of the invariance of human motion with respect to equi-affine transformations CITATION CITATION.
We argue that geometric invariance may provide a more compact representation of complex movements composed of geometric primitives.
Straight point-to-point movements show geometric invariance under dynamic perturbations involving the use of either elastic or viscous loads CITATION, CITATION.
Point-to-point movements retain the invariance of their geometric properties even when subjects are required to control the movements of a cursor on a computer screen by moving their fingers in an instrumented data glove CITATION.
Recent studies in monkeys CITATION, CITATION, CITATION and humans CITATION have indicated that repeatable geometric shapes used in the construction of complex trajectories emerge after extensive practice in the generation of drawing and sequential movements.
The ability to unify different kinds of movement constraints in the modeling of human and animal movements could lead to further insights CITATION, CITATION.
Parabolic movement primitives meet the demands of geometric invariance, kinematic optimality of movements and simplicity of movement representation, and may subserve as underlying building blocks in arm trajectory formation CITATION, CITATION.
Here, the hypothesis that parabolic segments are geometric primitives in practiced movements was experimentally tested using spontaneous scribbling movements made by two monkeys.
Our choice of the source of the data was motivated by the feasibility of subsequently analyzing the underlying activity of motor cortical neurons CITATION .
The predictions of both the two-thirds power law CITATION and the constrained minimum-jerk model CITATION are identical for a single parabolic stroke CITATION, CITATION.
The fit of the recorded trajectories to the predictions of these two models was assessed and is described in detail in Text S1.
Preliminary version of our findings was presented at the Tenth Biennial Conference of the International Graphonomics Society in 2001 and at the Computational Motor Control Workshops at Ben-Gurion University in 2005 and 2006.
