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Aug 18

Cubochoric coordinates: a new parameterization of 3D rotations

The contents of this page were extracted from a manuscript in press in Modeling and Simulations in Materials Science and Engineering (MSMSE).  Contact M. De Graef for additional information.

Fortran-90 source code that implements this rotation parameterization is available for download (zip file containing so3-sampler.f90; compiles with gfortran).  This program asks the user for a sampling rate and a point group number, and generates a vtk file of the RFZ that can be displayed with Paraview.

In texture analysis, it is common practice to represent orientations in a way that reflects the symmetry of the material. One way of achieving this is to employ the Rodrigues-Frank parameterization of orientations. In this neo-Eulerian parameterization, each orientation is represented by a rotation that is decomposed into a rotation axis, represented by a unit vector n, and a rotation angle w. The Rodrigues-Frank vector is then defined by R = n tan(w/2). Note that rotations by 180° are thus represented by vectors of infinite length, which is a numerical inconvenience. Every point R corresponds to a unique rotation/orientation in the absence of object or material symmetry.

As a consequence of rotational crystal symmetry, different points R in Rodrigues space will describe symmetrically equivalent rotations, so that it is sufficient to consider only a sub-region of the full Rodrigues space; such a sub-region is known as a Fundamental Zone. By definition, the Rodrigues Fundamental Zone (RFZ) contains all rotations that lie closer to the origin than to any of the points that are symmetrically equivalent to the origin (note that the word "closer" implies the use of the appropriate metric tensor for the Rodrigues space, not just a Euclidean metric).

One distinguishes between Cyclic, Dihedral, Tetrahedral, and Octahedral rotational symmetries, depending on the point group symmetry of the material. One can show (see A. Morawiec and D. P. Field (1996) Rodrigues parameterization for orientation and misorientation distributions, Philosophical Magazine A, 73:4, 1113-1130), that for the cyclic point groups (groups with a single rotation axis of order n along the z axis), the Fundamental Zone becomes a region of space bounded by two planes at z = +/- tan(pi/2n). For the dihedral groups, with two-fold rotation axis normal to the principal rotation axis, the RFZ becomes a polygonal disk with the same top and bottom planes as for the cyclic case, and with 2n square vertical facets at unit distance (Euclidean) from the origin. The tetrahedral RFZ is bounded by eight octahedral planes at distance tan(pi/6), and for the octahedral case this octahedron is further truncated by six cube planes at distance tan(pi/4), creating an RFZ shaped as a cuboctahedron, as shown in the figure on the right.

The question we addressed in this research project is the generation of a set of rotations that uniformly samples the space of all orientations, SO(3); SO(3) is the space of Special Orthogonal 3x3 matrices, which represent 3D rotations. Once such a uniform sample has been realized, it should be straightforward to restrict the sampling to any of the above illustrated RFZs. Our approach starts from a uniform grid inside a cube (easily generated from a computational point of view), and generates an equal-area map between the cube and the unit quaternion hemisphere, which is isomorphous with SO(3). The new mapping takes a point of the cubic grid, transforms it into the homochoric representation, which is then mapped onto the unit quaternion sphere using a separate equal-area mapping. Our new contribution lies in the creation of the first equal-area mapping, from the cube to the homochoric sphere.