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Sampling Sets and Quadrature Formulae on the Rotation Group

In this paper, we construct sampling sets over the rotation group SO(3). The proposed construction is based on a parameterization, which reflects the product nature 𝕊2 × 𝕊1 of SO(3) very well, and leads to a spherical Pythagorean-like formula in the parameter domain. We prove that by using uniformly distributed points on 𝕊2 and 𝕊1, we obtain uniformly sampling nodes on the rotation group SO(3). Furthermore, quadrature formulae on 𝕊2 and 𝕊1 lead to quadratures on SO(3), as well. For scattered data on SO(3), we give a necessary condition on the mesh norm such that the sampling nodes possess nonnegative quadrature weights. We propose an algorithm for computing the quadrature weights for scattered data on SO(3) based on fast algorithms. We confirm our theoretical results with examples and numerical tests.