On the role of ocular torsion in binocular visual matching

When an observer scans the visual surround, the images cast on the two retinae are slightly different due to the different viewpoints of the two eyes. Objects in the horizontal plane of regard can be seen single by aligning the lines of sight without changing the torsional stance of the eyes. Due to the peculiar ocular kinematics this is not possible for objects above or below the horizontal plane of regard. We provide evidence that binocular fusion can be achieved independently of viewing direction by adjusting the mutual torsional orientation of the eyes in the frontal plane. We characterize the fusion positions of the eyes across the oculomotor range by deriving simple trigonometric equations for the required torsion as a function of gaze direction and compute the iso-torsion contours yielding binocular fusion. Finally, we provide experimental evidence that eye positions in far-to-near re-fixation saccades indeed converge towards the predicted positions by adjusting the torsion of the eyes. This is the first report that describes the three-dimensional orientation of the eyes at binocular fusion positions based on the three-dimensional ocular kinematics. It closes a gap between the sensory and the motor side of binocular vision and stereoscopy.


Supplementary information
To evaluate rotations, we used the Clifford algebra of the 3D-Euclidean space, which is generated by three numbers, labelled 1  , 2  , 3  and a unity denoted I. These numbers are defined by the properties  

General eye positions
A general eye position can be described by the following compounded rotation The result (1) now follows from the observation that (3)

Ocular torsion enables single binocular vision in general eye positions
We now determined the set of single binocular fixation positions based on equation (3) that describes the general gaze direction of the right and left eye. We have earlier found that the distances from the rotation centers to a common fixation point are for the left eye. In the horizontal plane of regard, the angles α and β coincide with the horizontal rotation angles ϑa and ϑb (see Fig. 1). To distinguish the formulas for the right and left eye, we use subscripts 'a' and 'b'. For the gaze-line coordinates, we shall use superscripts 'a' and 'b' in parentheses. We obtained the following set of equations of the gaze vectors ga and gb, expressed in Cartesian coordinates.
A first geometric condition for target fusion is that   This relation implies that the ratio between the elevations is unity in conjugate eye movements as well as in symmetric convergence. Notice that such a segregation of the elevation as a function of the eyes' azimuths is not possible in Helmholtz coordinates.
A second constraint is that the torsional rotation angles of the right and left eye must fulfill the following equations. cos sin cos sin 1 cos sin cos sin 0 On one hand, we derived from these linear equations Finally, we obtained from equation (7a Several conclusions can be derived from the equations 5 to 8. Firstly, the torsion of each eye during fusion depends on the azimuth and elevation of the other eye. Secondly, the torsional angles ξa and ξb of equations 8a and 8b solve the linear equations 6 that express the geometric constraint for aligning the gaze lines on a single object in secondary vertical and tertiary eye positions. Thirdly, in conjugate gaze (α =β) the elevation of the eyes must be equal ( ab   ) that is the eyes must move in conjugation in parallel planes. Moreover, it follows also that the torsion of the eyes must be equal and remain invariant, independently of the conjugate motion.

Gaze lines are co-planar during fusion
Equations (4), (5) and (7) describe the conditions for the intersection of the gaze lines at a single point when the eyes are in general positions. We now check that these equations imply coplanarity of the gaze lines, which is a necessary condition for intersection of the gaze lines during fusion. Firstly, we notice that the Donders-Listing positions are in general not coplanar except for version movements (α =β) or symmetric convergence of the eyes (α =-β) where the ratio     Introducing equation (7a) on the left side and using the definition of Λ-matrix we obtain 1 sin sin sin cos cos cos Hereby we used equation (5) and the property that the inverse of unitary operator Λ is its adjoint: