The horopter: Old and new.

The horopter's history may partly be responsible for its ambiguous psychophysical definitions and obscured physiological significance. However, the horopter is a useful clinical tool integrating physiological optics and binocular vision. This article aims to help understand how it could come to such different attitudes toward the horopter. After the basic concepts underlying binocular space perception and stereopsis are presented, the horopter's old ideas that influence today's research show their inconsistencies with the conceptualized binocular vision. Two recent geometric theories of the horopter with progressively higher eye model fidelity that resolve the inconsistencies are reviewed. The first theory corrects the 200-year-old Vieth-Müller circle still used as a geometric horopter. The second theory advances Ogle's classical work by modeling empirical horopters as conic sections in the binocular system with the asymmetric eye model that accounts for the observed misalignment of optical components in human eyes. Its extension to iso-disparity conics is discussed.

A Geometric Theory Integrating Human Binocular Vision With Eye Movement.

A theory of the binocular system with asymmetric eyes (AEs) is developed in the framework of bicentric perspective projections. The AE accounts for the eyeball's global asymmetry produced by the foveal displacement from the posterior pole, the main source of the eye's optical aberrations, and the crystalline lens' tilt countering some of these aberrations. In this theory, the horopter curves, which specify retinal correspondence of binocular single vision, are conic sections resembling empirical horopters. This advances the classic model of empirical horopters as conic sections introduced in an ad hoc way by Ogle in 1932. In contrast to Ogle's theory, here, anatomically supported horopteric conics vary with the AEs' position in the visual plane of bifoveal fixations and their transformations are visualized in a computer simulation. Integrating horopteric conics with eye movements can help design algorithms for maintaining a stable perceptual world from visual information captured by a mobile robot's camera head. Further, this paper proposes a neurophysiologically meaningful definition for the eyes' primary position, a concept which has remained elusive despite its theoretical importance to oculomotor research. Finally, because the horopteric conic's shape is dependent on the AE's parameters, this theory allows for changes in retinal correspondence, which is usually considered preformed and stable.

Binocular system with asymmetric eyes: erratum.

Corrections are given for misprints in J. Opt. Soc. Am. A35, 1180 (2018)JOAOD60740-323210.1364/JOSAA.35.001180.

Binocular system with asymmetric eyes.

I elaborate binocular geometry with a novel eye model that incorporates the fovea's temporalward displacement and the cornea and the lens' misalignment. The formulated binocular correspondence results in longitudinal horopters that are conic sections resembling empirical horopters. When the eye model's asymmetry parameters' range is that which is observed in healthy eyes, abathic distance also falls within its experimentally observed range. This range in abathic distance is similar to that of the vergence resting position distance. Further, the conic's orientation is specified by the eyes' version angle, integrating binocular geometry with eye movement. This integration presents the possibility for modeling 3D perceptual stability during physiological eye movements.

On binocular vision: The geometric horopter and Cyclopean eye.

We study geometric properties of horopters defined by the criterion of equality of angle. Our primary goal is to derive the precise geometry for anatomically correct horopters. When eyes fixate on points along a curve in the horizontal visual plane for which the vergence remains constant, this curve is the larger arc of a circle connecting the eyes' rotation centers. This isovergence circle is known as the Vieth-Müller circle. We show that, along the isovergence circular arc, there is an infinite family of horizontal horopters formed by circular arcs connecting the nodal points. These horopters intersect at the point of symmetric convergence. We prove that the family of 3D geometric horopters consists of two perpendicular components. The first component consists of the horizontal horopters parametrized by vergence, the point of the isovergence circle, and the choice of the nodal point location. The second component is formed by straight lines parametrized by vergence. Each of these straight lines is perpendicular to the visual plane and passes through the point of symmetric convergence. Finally, we evaluate the difference between the geometric horopter and the Vieth-Müller circle for typical near fixation distances and discuss its possible significance for depth discrimination and other related functions of vision that make use of disparity processing.