On the dynamics of the eye: geodesics on a configuration manifold, motions of the gaze direction and Helmholtz’s theorem

Abstract

The simplest model with which to examine the dynamics of the human eye consists of a rigid body which is free to rotate about a fixed point. Two classical laws governing monocular vision, which are known as Listing’s law and Donders’ law, can be enforced in this model using a single holonomic constraint. While there has been considerable attention paid to the kinematics of the eye, the dynamics of the eye predicted by rigid body models has not received the same level of attention. In the present paper, the unforced dynamics of the resulting rigid body model are examined with particular emphasis placed on the geodesics of the configuration manifold. A comprehensive portrait of these motions is presented, and the insight gained is related to the dynamics of the gaze direction and saccadic motions of the eye. Among our results, we find that modeling the eye as an asymmetric rigid body produces a non-integrable system of governing equations and that the geodesics on the configuration manifold provide a wealth of potential motions for the gaze direction.