The Geometry of Eye Movement Dynamics

Eye movements consist of spherical rotations, with orientation generally constrained by Listing’s law. Our main result is a complete explicit formulation of ballistic eye movement under the Listing constraint. We present a conceptual framework for eye movement bas-ing the dynamics of Listing motion on the equator of the sphere of unit quaternions, which we call the Listing sphere. Analytical dynamics shows that ballistic Listing motion corre-sponds to free particle motion on the Listing sphere. Thus, ballistic Listing movement is greatly simplified by transposition to the Listing sphere, where it consists of shortest dis-tance trajectories. This proves that ballistic eye motion consists of constant speed rotation along circles passing through the occipital point. The relevance of the occipital point in eye movement was already noted by Helmholtz, which we explain by the fact that it cor-responds to the equator of the Listing sphere. We designed a physical mechanism produc-ing the correspondence between eye movement and particle motion on the Listing sphere. Our straightforward description of ballistic eye motion under the pure Listing kinematic constraint serves as a useful idealized benchmark in the study of actual physiological eye movements, whose orientations are known to deviate slightly from the Listing constraint.