This Demonstration shows how coordinate systems are transformed and how the a-matrix and alpha-matrix are formed.

There are four transformation options: rotation, rotation with inversion (roto-inversion), reflection, and inversion. The vector

defines the direction about which the rotation occurs or the direction normal to the plane of reflection (depending on the transformation type selected). Rotation operations are described as "

-fold", where

refers to the number of steps to complete a full rotation. For example: a 4-fold rotation means 4 steps of

for a full

rotation about the axis. The a-matrix is a

matrix and the alpha-matrix is

; the elements of the a-matrix are used to calculate the alpha-matrix. Both matrix types are used for coordinate system transformations. For example, a

matrix

can be transformed to a new coordinate system

by the a-matrix

with the following formula

. The alpha-matrix can be used in a similar manner for a

matrix

that can be transformed using both the a-matrix

and alpha matrix

by

.