On the minimal relative motion principle—the oscillating tilted bar

Beghi, Xausa, Tomat, and Zanforlin (J. Math. Psychol. 41(1997) 11) present a visual stereokinetic illusion. In the image plane, one end of an oblique bar moves horizontally back and forth, while the other end is stationary. Perceptually, this becomes a bar of a constant length rotating in depth around a vertical axis that passes through the stationary end of the bar. Beghi et al. (1997) provide a mathematical model of minimal relative motion to account for this percept. Here we show that the minimal relative motion principle cannot explain the perceptual phenomenon. Specifically, we raise two objections. (1) It is necessary to consider not only the length, but also the direction, of a vector when comparing vector fields. In fact, when directions are taken into consideration, Beghi et al.'s mathematical result diverges from their perceptual experimental result. (2) There is a mathematical inconsistency in Beghi et al. (1997): mixing absolute and relative velocities in their minimization is unwarranted, and does not lead to correct minimization.