New results in stereopsis and Listing’s law

Human eyes' optical components are misaligned. This study presents comprehensive geometric constructions in the binocular system, with the eye model incorporating the fovea that is displaced from the posterior pole and the lens that is tilted away from the eye's optical axis. It extends their previously considered horizontal misalignment with the vertical components. When the eyes' binocular posture changes, 3D spatial coordinates of the retinal disparity (iso-disparity curves), the subjective vertical horopter, and the eye's torsional orientation transformations are visualized in GeoGebra's simulations. The consequences and functional roles of vertical misalignment of the eye's optical components are explained in the following findings: (1) The classic Helmholtz theory, which states that the subjective vertical retinal meridian inclination to the retinal horizon explains the backward tilt of the perceived vertical horopter, is less relevant when the eye's optical components are misaligned. Instead, the lens vertical tilt provides the retinal vertical criterion that explains the experimentally measured vertical horopter inclination. (2) Listing's law, which originally restricts single-eye torsional positions and has imprecise binocular extensions, is formulated for binocular fixations using Euler's rotation theorem. It, however, replaces Listing's plane, which is defined for eyes looking at infinity, with the eyes muscles' natural tonus resting position corresponding to the abathic distance fixation of empirical straight frontal horopter. This new meaning of Listing's plane provides neurophysiological significance that has remained elusive.