a pole figure

How do you draw, on one flat page, which way thousands of grains are pointing? A pole figure is the standard map. Pick one kind of crystal plane — say the {100} cube faces — and for every grain mark the direction its plane-normal (its pole) points, relative to fixed sample axes (for a sheet: rolling direction, transverse direction, sheet normal). Project all those directions onto a disc using a stereographic projection. Where the poles cluster, the map is dense; where grains avoid an orientation, it is blank. The pattern of clustering is the texture, seen at a glance.

Concretely, imagine standing each grain's chosen plane-normal as an arrow from the center of a globe; it pierces the globe's surface at a point, and stereographic projection flattens that point onto the disc. A random (untextured) sample scatters poles evenly and the pole figure is a uniform grey disc. A textured sample shows bright spots or arcs where many grains share the same pole direction. Pole figures are measured directly by X-ray diffraction (rotating the sample and recording where a chosen {hkl} reflection is strong) or computed from grain orientations mapped by electron backscatter diffraction (EBSD).

Pole figures are the everyday working language of texture — quick to read, and directly measurable. But they carry an honest limitation: a single pole figure shows only one plane's poles and, being a 2D projection, does not fully pin down each grain's full 3D orientation (a rotation about the plane-normal is not resolved). That is why several pole figures for different {hkl}, or the full orientation distribution function computed from them, are needed to specify a texture completely. A pole figure is a projection of the texture, not the whole of it.