On Riemannian Geometries of Visual Space

Abstract

AbstractThis is a study of the phenomenal geometries constructed in the Riemannian geometry framework from simulated iso-disparity conics in the horizontal visual plane of the binocular system with the asymmetric eyes (AEs). For the eyes’ resting vergence posture, which depends on parameters of the AE, the iso-disparity conics are frontal straight lines in physical space. For all other fixations, the simulated iso-disparity conics consist of families of the ellipses or hyperbolas depending on both the AE’s parameters and the bifoveal fixation. However, the iso-disparity conic’s arcs are perceived in the gaze direction as the frontal lines. Thus, geometries of physical and visual spaces are different. An assumption underlying the relevant architecture of the human visual system is combined with results from simulated iso-disparity straight lines, giving the relative depth as a function of the AE’s parameters and the distance. This establishes the metric tensor in binocular space of fixations for the eyes’ resting vergence posture. The resulting geodesics in the gaze direction, give the distance to the horizon and zero curvature. For all other fixations, only the sign of the curvature may be inferred from the global behavior of the simulated iso-disparity conics.