Geometric purity, kinematic scaling and dynamic optimality in drawing movements beyond ellipses
Drawing movements have been shown to comply with a power law constraining local curvature and instantaneous speed. In particular, ellipses have been extensively studied, enjoying a 2/3 exponent. While the origin of such non-trivial relationship remains debated, it has been proposed to be an outcome of the least action principle whereby mechanical work is minimized along 2/3 power law trajectories. Here we demonstrate that such claim is flawed. We then study a wider range of curves beyond ellipses that can have 2/3 power law scaling. We show that all such geometries are quasi-pure with the same spectral frequency. We then numerically estimate that their dynamics produce minimum jerk. Finally, using variational calculus and simulations, we discover that equi-affine displacement is invariant across different kinematics, power law or otherwise. In sum, we deepen and clarify the relationship between geometric purity, kinematic scaling and dynamic optimality for trajectories beyond ellipses. It is enticing to realize that we still do not fully understand why we move our pen on a piece of paper the way we do.\n\nHighlightsO_LISeveral curves beyond ellipses have power-law kinematics with 2/3 exponent.\nC_LIO_LIThe curvature spectrum of each of such geometries is quasi-pure at frequency 2.\nC_LIO_LITheir dynamics are shown to comply with minimum of jerk.\nC_LIO_LIBut the 2/3 power law is not an outcome of minimizing mechanical work.\nC_LIO_LIYet, equi-affine displacement is invariant upon different kinematics.\nC_LI\n\nGraphical abstract\n\nO_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=182 SRC=\"FIGDIR/small/737460v1_ufig1.gif\" ALT=\"Figure 1\">\nView larger version (19K):\norg.highwire.dtl.DTLVardef@1a14c5corg.highwire.dtl.DTLVardef@9c6338org.highwire.dtl.DTLVardef@1360763org.highwire.dtl.DTLVardef@1ef534_HPS_FORMAT_FIGEXP M_FIG C_FIG \"We must represent any change, any movement, as absolutely indivisible.\" -- Henri Bergson