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Any three intersecting arcs of great circles on the sphere constitute a
spherical triangle. Such a triangle has six elements: the three sides which are
specified by the angles which they subtend at the centre of the sphere; and the
three angles which are defined as the dihedral angles between the central planes
that intersect the sphere in the great circles. If any three elements of a
spherical triangle are known the remainder can be calculated by the methods of
spherical trigonometry. A full discussion of these methods is beyond the scope
of this booklet, but if the triangle has vertices *A*, *B*, *C* and the sides
opposite to these vertices are denoted *a*, *b*, *c* respectively, the two most
generally useful formulae are:

= =

and

.

The latter may be rewritten with appropriate inter-changes of the elements concerned.

By joining up the poles on a stereogram by projections of great circles it is always possible to find spherical triangles from which one can calculate by spherical trigonometry previously unmeasured angles between poles. It frequently happens that the directions of crystallographic axes are not marked by the poles of faces on a crystal, but they can nevertheless be found as the intersection of appropriate circles on the stereogram. Suitable spherical triangles can then be found to permit the calculation of inter-axial angles and axial ratios from the angles between available faces. A full treatment of these methods is given in books on morphological crystallography.

**Copyright © 1984, 1997 International Union of
Crystallography**