On one hand, human movements show several prominent features, such as movement duration being nearly independent of movement size (the isochrony principle), the dependency of instantaneous speed on the movement's curvature (captured by the 2/3 power law) and movement compositionality (complex movements are composed of simpler elements). On the other hand, several types of optimization principles, such as the minimum jerk, the minimum variance and the optimal feedback control principles have successfully accounted for many observed characteristics, but no existing theory can successfully account for all of the above features. Here we present a new theory of trajectory formation inspired by geometrical invariance. The theory proposes that movement duration and compositionality arise from cooperation among Euclidian, equi-affine and full affine geometries, but also that the choice of the specific combination of the geometries can be accounted for by optimization principles. Each geometry possesses a canonical measure of distance along curves, i.e., an invariant arc-length parameter. We suggest that the actual movement duration reflects a particular tensorial mixture of these parameters. The theory predictions were tested on four data sets: drawings of elliptical curves, locomotion and drawing trajectories of complex figural forms (cloverleaves, lemniscates and limacons) and drawing of curves through via points. Our theory succeeded in accounting for the kinematic and temporal features of the recorded locomotion and drawing movements. During both tasks equi-affine geometry was found to be the most dominant geometry. Affine geometry was the second most important geometry during drawing, while Euclidian geometry was the second most important during locomotion. We have calculated the jerk of the velocity profiles of the different possible geometrical combinations and found an agreement between the combinations that resulted in lower jerk and those employed by the subjects during drawing experiments. The underlying strategy used in selecting particular combinations and segmentations for different types of optimization principles and the connection between segmentation and the learning of new tasks should be studied further.;The study described in Chapter 2, Chapter 3 and Chapter 4 is a joint work with Daniel Bennequin, which has been published in [3]. The author of this thesis was a significant contributor to this publication. She contributed to the development of the mathematical model, analyzed the data, wrote algorithms, and wrote part of the paper.
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