Angular speed should not be used to test the speed-curvature power law

The speed-curvature power law is commonly estimated in two forms: tangential speed regressed on curvature (VC) and angular speed regressed on curvature (AC). Since angular speed is the product of curvature and tangential speed, exponents of the two power laws when fitted to the same trajectory satisfy the exact relation ${\beta }_{\mathrm{VC}}={\beta }_{\mathrm{AC}}-1$ , and the two power laws are generally treated as equivalent and interchangeable. However, this algebraic identity does not make the two formulations equivalent as statistical goodness-of-fit tests. In the AC regression, curvature appears in both the predictor and the outcome ( $A=VC$ ). Because of this, the AC coefficient of determination can be inflated by curvature variance even when movement does not slow in curved segments. I demonstrate this mechanism analytically, in simulations, and in three empirical tasks: elliptical target tracking, self-paced ellipse tracing, and free-form scribbling. In the three datasets, mean AC $R^2$ was very high: .77, .83, and .85, compared with mean VC $R^2$ values of .30, .25, and .39. In a randomized-speed check across all Experiment 2 trials, changing the timing of speed reduced VC $R^2$ from .28 to .017 while AC $R^2$ remained high, moving from .84 to .79. Therefore, a high AC $R^2$ is not independent evidence of biological speed-curvature coupling. Researchers should estimate the law with tangential speed and curvature and reexamine conclusions based only on angular-speed goodness-of-fit.