The Geometry of Eye Rotations and Listing's Law

Various parameterizations of rotations are related through a unifying mathematical treatment, and transformations between coordinate systems are computed using the Campbell-Baker Hausdorff formula. Next, we describe Listing's law by means of the Lie algebra so(3). This enables us to demonstrate a direct connection to Donders' law, by showing that eye orientations are restricted to the quotient space 80(3)/80(2). The latter is equivalent to the sphere S2, which is exactly the space of gaze directions. Our analysis provides a mathematical framework for studying the oculomotor system and could also be extended to investigate the geometry of mUlti-joint arm movements.