Appendix 3.1. The stereographic projection

In many texts on crystallography you will come across the stereographic projection. Because of time limitations we will pobably not have a formal practical session on the stereographic projection, however the following notes are included here for the more adventurous of you who really want to come to grips with crystallography. The stereographic projection provides a useful way of conveying information about the orientation of lines and planes in 3-dimensional space (the principal problem with this being the limitations imposed by trying to portray 3-dimensional information on 2-dimensional sheets of paper).

About any line X-X' we can construct a sphere with its origin centred on the line. The line must intersect this sphere at two points, one in the upper hemisphere and one in the lower hemisphere (unless of course it is a horizontal line in which case it intersects the equator twice). A new line constructed from the intercept on upper hemisphere (P in the figure below) to the lower hemisphere pole , will intercept a horizontal plane passing through origin of the sphere, in a unique position (P') dependent only on the orientation of the line X-X'.This point P' represents the stereographic projection of the line X-X'.

The projection plane viewed from directly above looks like this:

Planes can be treated in a similar fashion, although in this case the projection is a line (termed a great circle).

The projection of the plane X-X'-Y-Y' looks like this:

So far we have discussed the sterographic projection from the point of view of geographical coordinates (you will use stereographic projections based on geographical coordinate systems in structural geology, utilising a lower hemisphere projection scheme rather than the upper hemisphere projection scheme shown here). What use, you may ask, is all this in crystallography where we are largely concerned with lattice planes whose orientation is defined with respect to a coordinate set dictated by the unit-cell and which therefore is not necessarily an orthogonal coordinate set. Well, the stereographic projection provides a powerful graphic method for conveying quantitiative information about the orientation of crystal faces as well as symmetry elements, for example the triads and tetrads in a cube can be illustrated in the following way (try to plot all the diads and mirror planes in the cube):

When dealing with crystal faces it is convenient to plot the orientation of the pole to the crystal face rather than the plane itself; the pole to a plane simply being the line perpendicular to that plane.