Rotations in Three Dimensions
A rotation in three dimensions is a linear map that preserves distances and handedness. The set of all such maps forms the special orthogonal group SO(3) — the group of 3×3 real matrices satisfying:
g ∈ SO(3): R·Rᵀ = I, det(R) = +1
In crystallography, orientations are elements of SO(3): they describe the rotation that maps the sample reference frame onto the crystal lattice frame. Multiple equivalent representations exist, each with advantages for different calculations in texture analysis, pole-figure simulation, and crystal plasticity modelling.
Parameterisations of SO(3)
The simulator supports four representations simultaneously, updating all of them whenever any one changes.
Euler angles (φ₁, Φ, φ₂) in Bunge convention decompose the rotation into three successive rotations about the Z, X′, and Z″ axes — the standard in electron backscatter diffraction. Quaternions (w, x, y, z) with unit norm avoid gimbal lock and compose rotations cheaply:
q = (cos(θ/2), n̂ sin(θ/2)), |q| = 1
Rodrigues–Frank vectors place the orientation in a bounded convex domain whose geometry directly reflects crystallographic symmetry — particularly useful for visualising the fundamental zone. Axis-angle (n̂, θ) provides the most intuitive geometric interpretation of a single rotation, specifying the unique axis left fixed and the angle rotated about it.
In the Simulation
The interactive viewer renders a set of reference directions (the crystal frame axes) rotated according to the chosen orientation. All four representations update in real time whenever any input changes:
Euler angle sliders — drag φ₁, Φ, φ₂ and watch the lattice rotate live ·
Quaternion input — enter (w, x, y, z) directly; auto-normalised on entry ·
Axis-angle control — pick an axis direction and rotation angle ·
Rodrigues vector — enter the Frank vector directly.
The equivalent rotation matrix R is always shown and updated consistently.
Orientation relationships between phases, misorientation between adjacent grains, and the boundaries of cubic, hexagonal, and tetragonal fundamental zones all follow from the basic rotation machinery explored here.