Ideal Patterns of Crystallographic Preferred Orientation and Their Representation by the von Mises - Fisher Matrix or Bingham Quaternion Distribution

摘要

Spherical geometry of quaternions representing rotations is employed to provide a unique correspondence between distinguished cases of the Bingham distribution on the 3-dimensional sphere S~3 of rotations and a classification of ideal textures, i.e. patterns of preferred Crystallographic orientations. It is shown that the Bingham distribution can represent most common types of ideal preferred orientation patterns; in particular single component, fibre and surface textures are represented by bipolar, circular and spherical distributions, respectively. The spherical Radon transform of the Bingham probability density function of rotations, which provides the probability density function of statistical coincidence of a given direction subjected to these random rotations with another given direction, is derived and displayed for the general and special cases. We also refer to the one-one correspondence of the Bingham distributions for quaternions and the von Mises-Fisher distribution for matrices in SO(3). It is also shown that the Bingham distribution cannot represent cone or ring fibre textures, and that a model representation of those types of ideal textures requires second order elements of the Crystallographic exponential family.