A polar coordinate system for describing binocular disparity

When a meridional magnifier is introduced in front of one eye, a planar surface is perceived as slanted about a vertical axis. If the horizontal meridian is magnified, the perceived slant is away from the eye with the magnifier (geometric effect). If the vertical meridian is magnified, the slant is towards the eye with the magnifier (induced effect). While the geometric effect can be explained by the binocular horizontal disparities introduced by the horizontal magnifier, the induced effect has to be explained differently. Various models have been developed and the induced effect has generally been explained as a reinterpretation of horizontal disparity under a new reference of stereoscopic localization which is resultant from the vertical positional disparity introduced by the vertical magnifier. In this paper we describe binocular disparity in a polar coordinate system. Under this system, horizontal and vertical disparities are combined into a single stimulus variable, polar angle disparity. We show that the spatial distribution of polar angle disparity can faithfully describe the three-dimensional slant and inclination of a planar surface relative to the gaze normal plane. Both geometric and induced effects can be explained as direct responses to the polar angle disparity map distorted by the magnifier. Theoretical predictions based on the polar angle disparity are compared with experimental findings.