3D Visualization of neo-Eulerian and other rotation representations

[This page provides Supplementary Material for the following paper: (DOI:10.1107/S1600576717001157) P. G. Callahan, M. Echlin, T. M. Pollock, S. Singh and M. De Graef, "Three-dimensional texture visualization approaches: theoretical analysis and examples", J. Appl. Cryst. (2017) 50.]

3D rotations are typically represented in terms of 3x3 rotation matrices that are based on an Euler angle triplet; the set of rotation matrices is then represented by SO(3). While Euler angles form the basis for the study of material textures, they suffer from well known issues, such as the lack of a unique representation of the identity rotation and gimbal lock. Visual representations of texture orientation distribution functions typically involve contour plots on multiple planar sections through Euler space, and it takes a bit of experience to be able to correctly interpret those; in addition, discovering crystal symmetry in Euler space plots is also not trivial. In this research project, we start from a number of alternative rotation/orientation representations, including the neo-Eulerian representation and the recently introduced cubochoric representation and combine them with modern 3D visualization tools to generate alternative ways to visualize orientation distribution functions.

In the tables below, links are provided to mp4 animations of the fundamental zones (FZs) for four different orientation representations: Rodrigues, homochoric, cubochoric, and stereographic. Two animations are available for each representation and each rotational point group symmetry; one is a solid (S) representation in which the FZ is filled with small solid spheres, the other is a wireframe (W) representation of the outline of the FZ. In the Rodrigues case, the space is infinite, so for the cyclic rotation groups the FZ stretches to infinity; for all other representations the FZ has a finite volume. It should be noted that all representations use the same length scale, so that the FZs in each representation have the correct relative size.