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.

Definition

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

• the intersection of the sphere and the plane is a circle, called the "trace" ;
• the intersection of the normal line and the sphere is the pole.

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.

Stereographic projection

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

Let us consider the x,y the (x,y) plane is the stereographic projection of the pole, P'.

It is possible to choose any .

Definition

The pole figure is the stereographic projection of the poles used to represent the orientation of an object in space.

Geometry in the pole...
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