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Voronoi cells in lie groups and coset decompositions: Implications for optimization, integration, and fourier analysis

The rotation group and special Euclidean group both contain discrete subgroups. In the case of the rotation group, these subgroups are the chiral point groups, and in the case of the special Euclidean group, the discrete subgroups are the chiral crystallographic space groups. Taking the quotients of either of these two Lie groups by any of their respective co-compact discrete subgroups results in coset spaces that are compact orientable manifolds. In this paper we develop methods for sampling on these manifolds by partitioning them further using double-coset decompositions. Fundamental domains associated with the aforementioned coset- and double-coset decompositions can be defined as Voronoi cells in the original groups. Division of these groups into Voronoi cells facilitates almost-uniform sampling. We explicitly compute these cells and illustrate their use in optimization, integration, and Fourier analysis on these groups. Motivating applications from the fields of protein crystallography, robotics, and control are reviewed in the context of this theory.