Statistical representation and analysis of orientational data

This letter presents a generalization of the Fisher distribution for directions in three-dimensional space to a distribution of orientation relations comprising rotation axes and angles. The distribution is based on the unit quaternion representation of a rotation. A general form of the distribution is derived using maximum entropy arguments. The simplest form of the orientation distribution is equivalent to a normal distribution in the quaternion space. A formula is derived to measure the overlap between two normal distributions. The influence of crystal point group symmetry on the distribution is also discussed. These results are expected to find applications in the representation and comparison of sets of experimental data on grain orientations in polycrystals.