Pole figure

A pole figure is a graphical representation of the orientation of objects in space. For example, pole figures in the form of stereographic projections are used to represent the orientation distribution of crystallographic lattice planes in crystallography and texture analysis in materials science.

Let us consider an object with a basis attached to it. The orientation of the object in space can be determined by three rotations to transform the reference basis of space to the basis attached to the object; these are the Euler angles.

If we consider a plane of the object, the orientation of the plane can be given by its normal line. If we draw a sphere with the center on the plane, then

A single pole is not enough to fully determine the orientation of an object: the pole stays the same if we apply a rotation around the normal line. The orientation of the object is fully determined by the use of poles of two planes that are not parallel.

The upper sphere is projected on a plane using the stereographic projection.

Let us consider the (x,y) plane of the reference basis; its trace on the sphere is the equator of the sphere. We draw a line joining the South pole with the pole of interest P.

It is possible to choose any projection plane parallel to the equator (except the South pole): the figures will be proportional (property of similar triangles). It is usual to place the projection plane at the North pole.

A Wulff net is used to read a pole figure.

The stereographic projection of a trace is an arc. The Wulff net is arcs corresponding to planes which share a common axis in the (x,y) plane.

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