### 3D visualization of rotations and crystallographic fundamental zones

[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.

Representation |
2 |
3 |
4 |
6 |
222 |
32 |
422 |
622 |
23 |
432 |
---|---|---|---|---|---|---|---|---|---|---|

Rodrigues |
S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W |

Homochoric |
S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W |

Cubochoric |
S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W |

Stereographic |
S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W |

Representation |
2 |
3 |
4 |
6 |
222 |
32 |
422 |
622 |
23 |
432 |
---|---|---|---|---|---|---|---|---|---|---|

Rodrigues |
S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W |

Homochoric |
S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W |

Cubochoric |
S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W |

Stereographic |
S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W | S,W |

The movies in the tables above all have "empty" fundamental zones. In the next table, we provide a few links to 3D stereographic projections of different sets of rotations around a given axis for both the cubic (octahedral **432**) and hexagonal (dihedral **622**) fundamental zones. Each orientation is represented by a colored sphere; spheres with identical colors are related to each other by the symmetry operations of the rotational point group. Note that the equivalent FZs have also been drawn. Each movie is available in standard and anaglyph versions.

Fundamental Zone |
[100] |
[110] |
[111] |
[123] |
---|---|---|---|---|

Cubic 432 |
A,S | A,S | A,S | A,S |

Hexagonal 622 |
A,S | A,S | A,S | A,S |

*x-y*plane and a rotation angle of 45°; they are represented by quaternions of the type

*[cos(pi/8), sin(pi/8)cos(theta), sin(pi/8)sin(theta), 0]*, where theta is the angle of the rotation axis with respect to the

*x*-axis. In the stereographic projection, which is an equal-angle projection, those rotations are represented by a circle of 180 spheres (theta is incremented from 0° to 358° in steps of 2°). A stereographic projection of all the orientations that are equivalent to the initial set is available here in standard rendering, and here in red-blue anaglyph mode. Superposition of the cubic fundamental zone outlines results in the following standard and anaglyph representations.

- EquivalentRotations [451 Mb]
- FZmovies-solid [551 Mb]
- FZmovies-wireframe [834 Mb]
- RandomRotations [54 Mb]
- RFZinEulerSpace [46 Mb]
- POVfiles [34 Mb]