Uniform Grids on SO(3)/K

Reference implementation for
Approximately Uniform Grids over Crystal Orientations
Z. T. Varley · M. De Graef

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The short version

Sampling crystal orientations means sampling the quotient $\mathrm{SO}(3)/K$, where $K$ is the rotational point group of the crystal. The textbook recipe (build a cubochoric grid, then throw away every point outside the fundamental zone) is wasteful for low-symmetry groups and produces uneven sampling near zone boundaries.

This repo takes a different route. For each crystallographic point group it constructs a constant-Jacobian Knothe-Rosenblatt transport that pushes the homochoric ball directly onto the fundamental zone. Feed in any structured point set on the cube (Sobol, Halton, a regular SC/FCC/BCC lattice) to the orientation quotient.

What's in the box

The five $K$-specific transports live in mappings/ as drop-in ho2hoFZ_* functions:

• $C_k$ and $D_k$: fully analytic, derived from per-axis CDF inverses.
• $T$ and $O$: piecewise analytic over the cubic FZ; the octahedral map ships in both standard cubochoric-aligned and FCC-aligned variants.
• $I$: fitted azimuthal/polar CDFs with a post-rotation correction for the icosahedral FZ.

Numerical backbone in src/: orientation conversions, Laue-group operators, $\mathrm{SO}(3)$ baselines (super-Fibonacci, Marsaglia, Shoemake, cubochoric), Riesz energy, covering radius, and a quotient-metric Thomson relaxer for benchmark baselines.

Plotting and the paper figure pipeline live in figures/; the camera-ready PNG / PDF / EPS bundles are checked in under paper_figures/.

Stereogram

An octahedral KR grid lifted to all of $\mathrm{SO}(3)$ by the 24 elements of $O$, drawn in homochoric coordinates.

Show stereogram

Free-fuse. ~50 cm from the screen, two spheres about 10 cm apart, cross your eyes until they overlap.

Reproduce the paper

pip install -r requirements.txt
python -m publication.export_figures            # all twelve panels
python -m publication.export_figures --only 1 4 # a subset

Outputs land in paper_figures/ as PNG, PDF, and native vector EPS. Layout details are in publication/README.md.

Cached data behind the figures can be rebuilt from scratch:

python -m figures.generate_cubochoric_anisotropy
python -m figures.generate_nn_cdf_data
python -m figures.generate_grid_method_metrics
python -m figures.generate_thomson_relaxation

Any panel module under figures/ is also runnable on its own, which is handy for tweaking limits or colour mappings interactively:

python -m figures.cubochoric_anisotropy

Figure 1 benefits from a working pykeops install for the kernel evaluations.

Citation

coming soon...

License

MIT. See LICENSE.