A geometric approach to binocular sensing

In this paper, our eventual goal is to study optimal control problems on human binoculomotor system as a simple mechanical control system, extending our earlier studies on monocular system. We assume that during the eye movement, each eye separately obeys a suitable form of Donders’ law which states that, moving away from the primary gaze direction, eye orientations lie in a subset consisting of rotation matrices for which the axes are restricted to a suitable Donders’ surface. When head is fixed, this surface is usually the Listing’s plane. Additionally we assume that the gaze direction vectors, of the two eyes, remain coplanar during the entire time of the eye movement. This is equivalent to asking that the eyes are always fixated in the visual space. We define a restricted configuration space for the two eye system as a subset of SO(3) × SO(3) and describe a binocular system as a simple mechanical control system. A Riemannian metric has been introduced and the corresponding Euler Lagrange dynamics is written out. Two special cases of the Donders’ surface is detailed in this paper. The first one is when the Donders’ surface is the Listing’s plane. The second one is when the Donders’ surface for each of the two eyes are derived from Fick Gimbal. We have displayed the geodesic curves for binocular eye movements satisfying the Fick Gimbal. Optimal trajectory plots are subject of future research.