Ab initio Binocular Formulation of Listing's Law

Human eyes do not have perfectly aligned optical components; the fovea is displaced from the posterior pole, and the crystalline lens is tilted away from the eye's optical axis. Although important in the study of vision quality, it is used here in binocular and oculomotor vision research. In the binocular system with misaligned optics, the decomposition into torsion-free (geodesic) and torsional changes in the eye’s posture is computed in the framework of Rodrigues’ vector. It leads to the first consistently formulated binocular version of Listing’s law, which is important for oculomotor control by constraining the eye’s redundant torsional degree of freedom. It is also supported by 3D simulations. The Listing plane is replaced with the binocular eyes’ posture corresponding to the eye muscles’ natural tonus resting position, which serves as a zero-reference level for convergence effort. Modern ophthalmology studies, along with the results presented here, support binocular modification of Helmholtz's 1867 formulation of Listing’s law. It includes revising the elusive neurophysiological significance of the Listing plane. Further, the binocular constraints couple 3D changes in the torsional positions of the eyes within the binocular extension of Listing’s law, which was previously formulated ad hoc. Lastly, a precise noncommutativity rule underlying Listing’s law and the half-angle rule is discussed by specifying the configuration space of sequences of binocularly constrained eyes’ fixations, which is visualized in GeoGebra simulations. The results obtained in this study should be a part of the answers to the questions posted in the literature on the relevance of Listing’s law to clinical practices.