Riemannian Geometries of Visual Space: Variable Curvature and Horizon

Abstract

This 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). The iso-disparity conic's arcs in the Cyclopean direction are the frontal visual geodesics. 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 iso-disparity conics consist of families of the ellipses or hyperbolas depending on both the AE's parameters and the bifoveal fixation. 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 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 can be inferred from the global behavior of the simulated iso-disparity conics.

A Critical Study in Stereopsis and Listing’s Law

“…These results indicate that the curvature of visual space vanishes for those eyes postures, extending the 2D results in [Turski, 2023b] for the horizontal visual plane to the 3D results considered here. Moreover, they clearly show that the vertical disparity is not a binocular cue, agreeing with the general evidence provided by the random-dot-stereograms [Julesz, 1971] and Wheatstone stereoscope [Wheatstone, 1838].…”

“…This paper extends the studies in [Turski, 2018, Turski, 2020, Turski, 2023b from the 2D setting of the binocular system with the AE model to the fully 3D framework. Here, I visualize in GeoGebra's dynamic geometry simulations the 3D longitudinal disparity curves and vertical horopter integrated with the eyes' posture using the binocular system with the new 3D AE model augmented by the vertical asymmetry of optical components.…”

“…where n = 1, 2, ... For n and n + 1, we have consecutive disparity objective frontal straight lines of relative disparity δ. The simulations of disparity lines in GeoGebra revealed their invariance to the abathic distance fixations and, hence, to AE's parameters modeling asymmetry of the human eye optical components, see Figure 2 in [Turski, 2023b]. For all other binocular fixations on the points of the horizontal plane, the simulated disparity curves were either a family of ellipses or hyperbolas, depending on the AE's parameters and the point of fixation.…”

“…The disparity lines in physical space are first constructed in GeoGebra dynamic geometry environment for the ERP; see [Turski, 2023b]. The nδ-disparity line is the projection of the points Q r + nδ and Q l and −nδ-disparity line is the set of projected points Q r − nδ and Q l .…”

“…In contrast to Ogle's theory, empirical horopters modeled in [Turski, 2020] are supported by the eye's anatomy that contributes to the aplanatic design of the misaligned optical components mentioned above. This work was further extended to the horizontal iso-disparity (or disparity for short) conic sections, where zero-disparity conic is the horopter, and applied to study the global aspects of phenomenal spatial relations in the framework of Riemannian geometry [Turski, 2023b]. I note that the lack of a family of iso-disparity curves that would more richly describe the structure of binocular visual space was mentioned a long time ago in [Arditi, 1986].…”