How is stereographic projection obtained?

Stereographic projection is a drawing on which the angles between planes can be accurately measured. It allows us to solve crystallographic problems by graphical methods. The stereographic projection is a very wide subject. To know more information on this subject, please refer following books:

However, it is still worthwhile to introduce briefly how stereographic projection is obtained. As it is shown below, a sample reference frame is defined first. A reference sphere, centered at the origin of the sample reference frame, is then defined. The stereographic projection plane is defined as the cross section plane between the reference sphere and the XY plane in the sample reference frame. In this system, a crystallographic direction, for example, [100], is represented by its pole P, which is its intersection with the reference sphere. The pole P is, then, represented by its stereographic projection P', which is the intersection of the projection plane with the line OP, where OP is the line from the point of projection O to the pole P.

The pole figure reference frame is just the stereographic projection plane shown in the figure. In X-ray diffraction measurement, the normal directions of the selected crystallographic plane in all grains are projected, in the way described above, to the pole figure reference frame.