This page provides supplementary material for an article that deals with the description of grain boundaries using octonions (8-D hypercomplex numbers).
T. Francis, I. Chesser, E.A. Holm and M. De Graef. "A New Octonion Metric for Grain Boundary Interpolations". Acta Materialia, 166:135-147 (2019). DOI: https://doi.org/10.1016/j.actamat.2018.12.034
The main result of the paper is a geodesic arc length metric on the 7-sphere that expresses the distance metric between two grain boundaries that are characterized by quaternions with respect to a reference frame attached to the grain boundary. This arc length can then be used in an interpolation formula to smoothly go from one grain boundary to a second one without ever leaving the 7-sphere.
The following links provide access to three different renderings corresponding to section 2 of the Supplementary Material:
- Geodesic arc surface
- Geodesic arcs
- All objects
The links below provide access to four different movies depicting the spherical linear interpolation or oSLERP between two octonions; each octonion represents particular grain boundary:
- Interpolation between two symmetric tilt boundaries, with the grains represented by nested spheres: Movie 1
- Interpolation between two symmetric tilt boundaries, with the grains represented by cubes on either side of the grain boundary plane: Movie 2
- Interpolation between two random grain boundaries without application of crystal symmetry: Movie 3
- Interpolation between the same two random grain boundaries, this time with application of cubic symmetry which results in a different (shorter) geodesic pathway: Movie 4
The complete Supplementary Materials document can be obtained from this link.
