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An analysis of the mathematical structure of the rotation function is presented. The effect of truncation of the expansions used in the fast rotation function is discussed and an alternative procedure of calculation which drastically reduces the errors is proposed. A method of sampling on spherical surfaces is developed. The rotation function can thus be obtained from the values it takes at the sampling points. The method can also be used to compute expansions in spherical harmonics of Patterson functions restricted to arbitrary domains. Topological properties of the rotation group are used to obtain distortion-free plots of the different sections of the rotation function.
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