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 "
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
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
that can be transformed using both the a-matrix
and alpha matrix