Calculation of normal distributions on the sphere S2

Abstract

Abstract In quantitative texture analysis normal distributions (NDs) on SO(3) group and on the sphere S2 are frequently used to calculate pole figures and orientation distribution function of crystallites in polycrystalline materials. Different motivations are used to define NDs on SO(3) and S2. This article considers a CLT-based motivation (CLT here is the central limit theorem on SO(m) rotation group, m≥2). This article defines ND on the sphere S2 as a “projection” of CLT-motivated ND on SO(3) group onto the sphere S2. The induction from the SO(3) group is performed by the way of transitioning from the NDs represented as expansions into series, where terms are matrix elements of group representations, to corresponding Fourier series for the class of functions invariant under rotation subgroup characterized by a fixed axis. This article studies NDs on the sphere S2 induced from SO(3) rotation group using corresponding NDs on the group. These NDs on S2 satisfy CLT on S2 as projections of NDs on SO(3) group that satisfy the corresponding CLT there. For central ND the corresponding induced ND on S2 matches the ND obtained during the study of Brownian motion. Different numerical algorithms are considered in order to calculate induced NDs on S2: Fourier series method, analytical approximation method and Monte Carlo method. These methods were used to approximate pole figures of NDs on S2. Also this article considers smoothing for modeling results. This task presents itself at the time of processing EBSD experimental data (obtained in the form of sample of orientations) when approximating NDs of orientations.