# On Binocular Vision: The Geometric Horopter and Cyclopean Eye

## TL;DR

This paper explores the geometric properties of horopters in binocular vision, revealing new insights into how the eyes perceive depth and alignment.

## Contribution

The paper introduces a novel geometric analysis of horopters, revealing a two-component structure and their relation to the Vieth-Müller circle.

## Key findings

- The isovergence circular arc forms an infinite family of horizontal horopters connecting nodal points.
- The 3D geometric horopter consists of two perpendicular components: horizontal horopters and vertical lines through symmetric convergence.
- The difference between geometric horopters and the Vieth-Müller circle may impact depth discrimination and disparity processing.

## Abstract

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.