Angular speed should be avoided when assessing the speed-curvature power law of movement

Abstract

The speed-curvature power law is one of the most studied constraints in biological movement. In many types of movements, there is a strong relationship between instantaneous speed and local curvature. For example, in elliptical trajectories, tangential speed is proportional to curvature raised to the power -1/3, (V[~]C-1/3). This phenomenon is known as the "one-thirds power law" and is generally considered to be mathematically equivalent to the "two-thirds power law" that describes the relationship between angular speed and curvature (A[~]C2/3); the two formulations are used interchangeably. However, in this paper, analysis of empirical and synthetic data demonstrates that using angular speed instead of tangential speed to estimate the power law tends to result in much stronger correlations, impacting the interpretation of the strength of the relationship, and therefore the existence of the law. Further analysis shows that angular speed and curvature are often trivially correlated, since angular speed is not a purely kinematic variable and depends on curvature. In conclusion, two forms of the law are not equivalent, angular speed should be avoided when expressing the speed-curvature power law.