Symmetry studies of refraction data

In this paper we describe the additive and multiplicative decomposition of refraction matrices with respect to a dihedral basis and show the effectiveness of that basis in developing new statistical summaries of refraction data. We also show that under certain projective restrictions the refraction matrices are homeomorphic to an open solid torus and discuss the many potential data-analytic consequences of that representation. A local analysis based on a class of Mobius transformations is introduced leading to a set of new summary statistics of potential use in comparative and correlational studies of refraction data.