The contents of this chapter serve two purposes:

- To forge a link between geometry and calculations;
- To provide the tools for coordinate changes.

The first point is described in the first two sections. The questions to be discussed are:

- (i)
- How can the matrix-column pair be obtained when the geometric meaning of the symmetry operation is known ?
- (ii)
- Given a matrix-column pair, what is its geometric meaning ?

The second point is a practical one. The complexity and amount of
calculations depend strongly on the coordinate system of reference
for the geometric actions. Therefore, it is
advantageous to be flexible and free to choose for each calculation
the optimal coordinate system. This means to change the coordinate
system if necessary and to know what happens with the coordinates
and the matrix-column pairs by such a change. In the last section
of this chapter, partitioned into 3 subsections, coordinate changes
will be treated in 3 steps: Origin shift, change of basis, and
change of both, *i.e.* general coordinate changes.

**Copyright © 2002 International Union of
Crystallography**