A Mathematical Model of Depth Displacement with a Contracting Bar

When a vertical line segment contracts at both ends according to a (decelerating) time law of hyperbolic type, it appears to rotate around its midpoint. The phenomenon is quite surprising from the point of view of projective geometry as the segment should rather appear to recede along the sagittal plane. Apparent displacement in depth is however obtained when the bar simultaneously contracts and is displaced laterally on the frontal plane. But, here again, projective expectations are contradicted, because apparent displacement in depth occurs whether movement of the bar in two dimensions is hyperbolic (decelerating), uniform, harmonic (accelerating), or a mixture of the three. It is suggested here that the visual system operates in such a way as to minimize the differences between the lengths of the velocity vectors of all points of a moving configuration. The mathematical model derived from this hypothesis allows qualitative and quantitative predictions that are in good agreement with experimental results. Copyright 2001 Academic Press.