Reflection makes sense of rotation of the eyes

Our 3-D percept of the world is constructed from the two-dimensional visual images on the retina of each eye, but these images and the relationships between them are affected by the 3-D rotations of each eye. These 3-D eye rotations are constrained to patterns such as Listing's law, or its generalisation 'L2', according to the context. Our understanding of the patterns of such three-dimensional eye rotations, and their effect on the retinal images, has been greatly advanced by the development of algebraic methods (Haustein, 1989; Tweed & Vilis, 1987; Westheimer, 1957) for calculating the effect of eye rotations. But not many would say, with Dirac, that they understand the equations describing the 3-D geometry in the sense that they have "a way of figuring out the characteristic of its solution without actually solving it" (Dirac, according to Feynman, Leighton, & Sands, 1964). I show here how the geometry of 3-D rotations of the eye and their visual effects can be made easier to understand by use of the principle that a rotation through angle alpha can be achieved by a pair of reflections in planes with an angular separation alpha/2, and a common line that is the rotation axis (Tweed, 1997b; Tweed, Cadera, & Vilis, 1990). Mathematically (see Appendix A), the method is equivalent to decomposing the unit quaternions so successfully used to study three-dimensional eye rotations (Tweed & Vilis, 1987; Westheimer, 1957) into pairs of pure quaternions (ones whose scalar part is zero) which represent the reflections (Coxeter, 1946).