Stereographic Projection
A pole figure is the stereographic projection of a set of crystallographic directions (poles) onto a plane. To project a unit vector d̂ from the upper hemisphere, draw a line from the south pole S = (0, 0, −1) through d̂ and find where it intersects the equatorial plane. The projected point (X, Y) is:
X = d_x / (1 + d_z), Y = d_y / (1 + d_z)
Stereographic projection is conformal — it preserves angles locally. Small circles on the sphere (e.g. the set of directions within a fixed angle of a pole) map to circles in the projection plane, making it easy to identify crystallographic symmetry visually.
Texture and the ODF
In a polycrystalline material, individual grains each have their own orientation. The statistical distribution of orientations is described by the orientation distribution function (ODF) f(g), defined such that the volume fraction of grains with orientation in a neighbourhood dg of g is:
dV/V = f(g) dg
A pole figure for a crystallographic direction [hkl] is the marginal of the ODF integrated over all orientations that map [hkl] to a given sample direction. Materials with a non-uniform ODF are said to have texture or preferred orientation. Common textures — fibre, rolling, cube — all produce characteristic pole figure patterns.
In the Simulation
The viewer renders live pole figures for any chosen {hkl} family and any orientation distribution:
Single crystal — plot the symmetry-equivalent poles of one orientation ·
Fibre texture — a rotation axis is fixed in the sample frame; the poles lie on a circle ·
Random texture — uniformly distributed poles (no preferred orientation) ·
Pole family selector — switch between {100}, {110}, {111}, {hkl} and observe the symmetry ·
Crystal system — cubic, hexagonal, orthorhombic, etc.; the symmetry of the pole figure reflects the crystal and sample symmetries combined.
Pole figures link the orientation concepts from earlier submodules to measurable diffraction data, forming the bridge into the Texture Tomography module.