Geometry of saccades and saccadic cycles

TL;DR

This paper develops a differential geometric framework for understanding saccades and saccadic cycles, interpreting eye movement laws through the Hopf fibration and characterizing saccades as geodesic segments and polygons in a specialized configuration space.

Contribution

It introduces a novel geometric interpretation of saccades using differential geometry and characterizes saccadic cycles as geodesic polygons, linking eye movement laws to the geometry of the configuration space.

Findings

Saccades are characterized as geodesic segments in Listing's hemisphere.

Saccadic cycles correspond to geodesic polygons in the configuration space.

Conditions for axes of rotation are derived in terms of world and retinotopic coordinates.

Abstract

The paper is devoted to the development of the differential geometry of saccades and saccadic cycles. We recall an interpretation of Donder's and Listing's law in terms of the Hopf fibration of the $3$-sphere over the $2$-sphere. In particular, the configuration space of the eye ball (when the head is fixed) is the 2-dimensional hemisphere $S^+_L$, which is called Listing's hemisphere. We give three characterizations of saccades: as geodesic segment $ab$ in the Listing's hemisphere, as the gaze curve and as a piecewise geodesic curve of the orthogonal group. We study the geometry of saccadic cycle, which is represented by a geodesic polygon in the Listing hemisphere, and give necessary and sufficient conditions, when a system of lines through the center of eye ball is the system of axes of rotation for saccades of the saccadic cycle, described in terms of world coordinates and retinotopic coordinates. This gives an approach to the study the visual stability problem.