Alleys on an apparent frontoparallel plane

Abstract

In a previous article in this journal (T. Indow, (1979). Journal of Mathematical Psychology, 19, 221–258), equations were presented for a plane appearing frontoparallel in the three-dimensional visual space and for two alleys on that plane which run horizontally in the following way: an alley that appears straight and parallel (Hh) and an alley that appears to be separated by a constant interval (Dh). This is an extension of traditional parallel and equidistant alleys which run toward the subject on the horizontal plane. As was done by Luneburg for these alleys, the equations were defined in the Euclidean map (EM) under the assumption that visual space is a Riemannian space of constant curvature K. In the present article, the general definition of perceptually straight lines in EM is discussed and, on the basis of this representation, new equations for Hh and Dh are given. Previous and present sets of equations give the same qualitative predictions as to the behavior of Hh and Dh, but two predictions are quantitatively different.