The theoretical horopter is an interesting qualitative tool for conceptualizing binocular correspondence, but its quantitative applications have been limited because they have ignored ocular kinematics and vertical binocular sensory fusion. Here we extend the mathematical definition of the horopter to a full surface over visual space, and we use this extended horopter to quantify binocular alignment and visualize its dependence on eye position. We reproduce the deformation of the theoretical horopter into a spiral shape in tertiary gaze as first described by Helmholtz (1867). We also describe a new effect of ocular torsion, where the Vieth–Müller circle rotates out of the visual plane for symmetric vergence conditions in elevated or depressed gaze. We demonstrate how these deformations are reduced or abolished when the eyes follow the modification of Listing's law during convergence called L2, which enlarges the extended horopter and keeps its location and shape constant across gaze directions.

*h*

_{r}−

*c*

_{r}and

*h*

_{l}−

*c*

_{l}shown in Figure 1). While the spatial location of the point on the extended horopter and the disparity vector thus defined are independent of the coordinate system used, the same is not true for the horizontal and vertical components of the vector. In our simulations, we use a retinal Helmholtz (1867) coordinate system, corresponding to the epipolar line set for infinite distance viewing.

*C*is the transformation producing the coordinates of retinal point

*R*in the right eye corresponding to point

*L*in the left eye, that is,

*R*=

*C*(

*L*). For an object

*O*in space that projects onto

*o*and

_{l}*o*in the left and right eyes, respectively, there then exist two equally appropriate retinal disparities relative to the correspondence mapping in the two eyes, namely,

_{r}*d*=

_{l}*o*−

_{l}*C*

^{−1}(

*o*) for the left eye and

_{r}*d*=

_{r}*o*−

_{r}*C*(

*o*) for the right eye. If

_{l}*C*is an identity mapping, that is, if corresponding points are identical points, these two definitions coincide except for sign and represent the common definition of retinal disparity as the difference in visual angle between the two projections. But for general correspondence functions

*C,*the two eyes' disparity vectors differ in size and direction. In other words, there is no unambiguous retinal disparity relative to nonidentical retinal mapping.

*The pseudoinverse*. To obtain the classical point horopter from a map of corresponding retinal locations, we intersect the projection rays from corresponding point pairs. The intersection, if it exists, is part of the horopter. Mathematically, this means solving the vector equation,

*a*

_{i,}where

*p*

_{i}are the two eyes' projection vectors and

*i*is the interocular vector. For our simulations, we assumed an interocular distance of 6 cm. This equation has no solution for many pairs of projection rays, reflecting the fact that most of them do not intersect at all. If we rewrite this equation in matrix form as

*Eye movements*. The effect of eye movements on correspondence and the horopter was modeled by obtaining the projection vectors for the corresponding rays for the eyes pointed straight ahead, and then rotating them according to the motor program used. These rotations were carried out in Helmholtz (1867) angles; thus, the projection vector with the eyes rotated on target was obtained from the one for straight gaze by rotating torsionally, horizontally, and vertically, in that order:

*Retinal area*. We first determined for which of the points in a regular Helmholtz (1867) grid of corresponding points the horopter point would fall within a fusable distance from the correspondence. The total retinal area covered by the horopter was then calculated from this subset of corresponding points by summing the areas of grid patches centered on these dots and subtending the grid spacing angle in the horizontal and vertical direction:

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