Crystallographic Symmetry and Orientation Equivalence
A crystal with point group symmetry G is physically invariant under all proper rotations in G. Consequently, two orientations g and g′ = h·g (with h ∈ G) are crystallographically equivalent — they describe the same crystal state. The fundamental zone (or fundamental domain) is the smallest subset F ⊂ SO(3) that contains exactly one representative from every equivalence class:
SO(3) = ⋃_{h∈G} h·F (disjoint union)
Reducing orientations to their fundamental-zone representative removes symmetry-induced ambiguity and enables meaningful comparison, interpolation, and averaging of orientations from electron backscatter diffraction (EBSD) maps.
Rodrigues–Frank Parameterisation
Rodrigues–Frank (RF) vectors map each rotation by axis n̂ and angle θ to a point in ℝ³:
r = n̂ tan(θ/2)
This parameterisation is particularly well-suited to fundamental-zone visualisation because the fundamental domain of any crystal symmetry group maps to a convex polyhedron in RF space. Straight geodesics in SO(3) become straight lines in RF space, making misorientation paths geometrically intuitive. The cubic fundamental zone is a truncated cube; the hexagonal zone is a prism with hexagonal cross-section.
In the Simulation
Select a crystal system from the full list of Laue classes (triclinic through cubic). The simulator renders the corresponding fundamental zone as a transparent 3D polyhedron and populates it with sample orientation points, coloured by misorientation angle from a reference orientation. Features include:
Point-group selector: all 11 Laue classes from Ci to Oh ·
Orientation sampling: uniform random, fibre texture, or single-crystal ·
Misorientation colouring: colour map encodes angular distance from a chosen reference ·
Interactive rotation: drag to orbit, scroll to zoom, hover a point to read its Euler angles.
The simulation makes it visually clear why unreduced orientations cluster at symmetry-equivalent corners, and how reduction maps them to a single compact region.