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The stereographic projection is a projection of points from the surface of a sphere on to its equatorial plane. The projection is defined as shown in Fig. 1. If any point P on the surface of the sphere is joined to the south pole S and the line PS cuts the equatorial plane at p, then p is the stereographic projection of P. The importance of the stereographic projection in crystallography derives from the fact that a set of points on the surface of the sphere provides a complete representation of a set of directions in three- dimensional space, the directions being the set of lines from the centre of the sphere to the set of points. The stereographic projection of these points is then, for many purposes, the best way of representing on a plane piece of paper the inter-relationship of a set of directions.
The most common application is that of representing the angles between the faces
of a crystal, and the symmetry relations between them. On a `well-formed`
crystal all the faces that are related to one another by the symmetry of the
crystal structure are developed to an equal extent, and the shape of the crystal
therefore reveals its true symmetry. However, real crystals are very rarely
well-formed in this way; accidents of crystal growth such as unequal access of
the crystallising liquid, or interference by adjacent crystals or other objects,
may have impeded the growth of the crystal in certain directions in such a way
as to mask the true symmetry. However these accidents do not affect the angles
between the faces--only their relative sizes. If the crystal is imagined to
lie within a sphere that is centred on some arbitrary point inside the crystal,
then normals to the faces can be constructed from this point and extended to cut
the sphere (Fig. 2). The points of intersection with the sphere represent the
faces of the crystal in terms of direction, entirely uninfluenced by their
relative sizes, and the symmetry of the arrangement of these points on the
surface of the sphere reveals the true symmetry of the crystal, whether or not
it be well-formed. This symmetry can then also be recognised in the
stereographic projection of these points.
In this connection the point where the normal to a face cuts the sphere is called the pole of the face. Since the angle between the two faces of a crystal is conventionally defined as the angle between their normals, it is equivalent to the (angular) great circle distance between their poles.
A point on the southern hemisphere projects to a point outside the equator, and as such a point approaches the S-pole its projection recedes to infinity. For many purposes it is convenient to represent the whole sphere within the equator, and this can be done if points in the southern hemisphere are projected to the N-pole instead of the S-pole. It is then usual to distinguish such points by marking those projected to the S-pole as and those projected to the N-pole as .
A complete stereographic projection of some particular set of points is usually called a stereogram.
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