Complex Unconstrained Three-Dimensional Hand Movement and Constant Equi-Affine Speed

Maoz, Uri; Berthoz, Alain; Flash, Tamar

doi:10.1152/jn.90702.2008

Cited by 40 publications

(53 citation statements)

References 53 publications

(75 reference statements)

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“…However, this Euclidean interpretation does not hold, and the earlier results of Viviani and Stucchi (1992) and LevitBinnun et al (2006) and recent findings on 3D movement production (Pollick et al 2009;Maoz et al 2009) together with our results here suggest that the most parsimonious interpretation of planar and spatial results may be found in the nonEuclidean equi-affine framework, or even more general and abstract geometries (Bennequin et al 2009). As was demonstrated over the last decade and a half, the many phenomena apparently obeying the two-thirds power law, in both motion perception and production, can all be alternatively explained as stemming from the simplest equi-affine-geometric invariant: movement at constant equi-affine speed Handzel 1996, 2007;Handzel and Flash 1999;Bennequin et al 2009).…”

Section: Discussioncontrasting

confidence: 80%

“…We have recently shown that constant equi-affine speed generalizes to 3D and that 3D movement at constant equi-affine speed entails a power law different from the two-thirds power law. Moreover, general self-paced unconstrained scribbling and shape-tracing hand movements in 3D both seem to be produced at roughly constant spatial (3D) equi-affine speed (Pollick et al , 2009Maoz et al 2009).…”

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confidence: 97%

“…Furthermore, planar motion according to this power law is perceived as most uniform Stucchi 1989, 1992). The equivalence of the power law to planar motion with constant equi-affine speed has led to its successful generalization to 3D motion generation (Pollick et al 2009;Maoz et al 2009). Here we first and foremost tested whether constant equi-affine speed for 3D visual motion is also perceived as more uniform.…”

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confidence: 99%

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Spatial constant equi-affine speed and motion perception

Maoz

Flash

2014

Journal of Neurophysiology

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The two-thirds power law, postulating an inverse local relation between the velocity and cubed root of curvature of planar trajectories, is a long-established simplifying principle of human hand movements. In perception, the motion of a dot along a planar elliptical path appears most uniform for speed profiles closer to those predicted by the power law than to constant Euclidean speed, a kinetic-visual illusion. Mathematically, complying with this law is equivalent to moving at constant planar equi-affine speed, while unconstrained three-dimensional drawing movements generally follow constant spatial equi-affine speed. Here we test the generalization of this illusion to visual perception of spatial motion for a dot moving along five differently shaped paths, using stereoscopic projection. The movements appeared most uniform for speed profiles closer to constant spatial equi-affine speed than to constant Euclidean speed, with path torsion (i.e., local deviation from planarity) directly affecting the speed profiles perceived as most uniform, as predicted for constant spatial equi-affine speed. This demonstrates the dominance of equi-affine geometry in spatial motion perception. However, constant equi-affine speed did not fully account for the variability among the speed profiles selected as most uniform for different shapes. Moreover, in a followup experiment, we found that viewing distance affected the speed profile reported as most uniform for the extensively studied planar elliptical motion paths. These findings provide evidence for the critical role of equi-affine geometry in spatial motion perception and contribute to the mounting evidence for the role of non-Euclidean geometries in motion perception and production.

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Section: Discussioncontrasting

confidence: 80%

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confidence: 97%

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confidence: 99%

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Spatial constant equi-affine speed and motion perception

Maoz

Flash

2014

Journal of Neurophysiology

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“…The same dimensionless representation may also be useful in analyzing other scale-invariant features and reveal spectrum of power laws in other motor contexts. Power laws have indeed been observed in many types of movements, including 3D hand movements (16), smooth pursuit eye movements (17), speech (18), and walking (19), and also appear in visual motion perception (20).…”

Section: Discussionmentioning

confidence: 92%

“…e* = a 1 a 2 = I 1 ðνÞ 2I 2 ðνÞ , [16] which is a decreasing function of ν with a minimum lim ν→∞ e * = 32=17 (Fig. 9).…”

Section: Appendix B: Euler-lagrange Equation In Angle Coordinatementioning

confidence: 94%

Spectrum of power laws for curved hand movements

Huh

Sejnowski

2015

Proc. Natl. Acad. Sci. U.S.A.

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In a planar free-hand drawing of an ellipse, the speed of movement is proportional to the −1/3 power of the local curvature, which is widely thought to hold for general curved shapes. We investigated this phenomenon for general curved hand movements by analyzing an optimal control model that maximizes a smoothness cost and exhibits the −1/3 power for ellipses. For the analysis, we introduced a new representation for curved movements based on a moving reference frame and a dimensionless angle coordinate that revealed scale-invariant features of curved movements. The analysis confirmed the power law for drawing ellipses but also predicted a spectrum of power laws with exponents ranging between 0 and −2/3 for simple movements that can be characterized by a single angular frequency. Moreover, it predicted mixtures of power laws for more complex, multifrequency movements that were confirmed with human drawing experiments. The speed profiles of arbitrary doodling movements that exhibit broadband curvature profiles were accurately predicted as well. These findings have implications for motor planning and predict that movements only depend on one radian of angle coordinate in the past and only need to be planned one radian ahead.optimal control | motor system | free-hand drawing | power law N atural body movements have surprising regularities despite the complexity of the movements and the many degrees of freedom that are involved. These regularities provide insights into principles underlying the mechanisms that generate the movements. We investigate here conditions when a known regularity fails to adequately describe motor behavior, which adds new insights into how movements are generated by the nervous system.One of the best-studied movement regularities is the inverse relationship between the speed and curvature of 2D hand movements observed when subjects are instructed to freely draw curved shapes such as ellipses, which follow a remarkably simple power law (1):This empirical relationship is called the one-third power law between speed and curvature, or equivalently, the two-thirds power law between angular speed and curvature. On a log curvature vs. log speed plot of hand movement, the power law appears as a single straight line with a slope of −1=3 (Fig. 1A). However, deviations from Eq. 1 have been observed for some movement trajectories. One class of deviation occurs when the curvature changes sign: The power law predicts the speed should diverge to infinity as the curvature approaches zero, which is physically implausible, whereas actual movements exhibit smooth, finite speed profiles at such inflection points. Deviations have also been observed for movements without inflection points (1-4). These noninflectional deviations occur in complex movements and are characterized by fragmentation of the log speed vs. log curvature plot into multiple line segments (Fig. 1B). These observations led to the segmented control hypothesis that complex movements could be generated by concatenating smaller and simpler movement...

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Embodied Language in Brains and Robots: The Question of Geometrical Reference Frames

Berthoz

2019

Wording Robotics

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Complex Unconstrained Three-Dimensional Hand Movement and Constant Equi-Affine Speed

Cited by 40 publications

References 53 publications

Spatial constant equi-affine speed and motion perception

Spatial constant equi-affine speed and motion perception

Spectrum of power laws for curved hand movements

Embodied Language in Brains and Robots: The Question of Geometrical Reference Frames

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