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Coordinate Transformation of a-Matrix and alpha-Matrix

​
type
rotation
rotoinversion
reflection
inversion
[h k l]
[1 0 0]
n-fold
2
a-matrix =
1
0
0
0
-1
0
0
0
-1
alpha-matrix =
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
-1
0
0
0
0
0
0
-1
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
[hkl]
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 "
n
-fold", where
n
refers to the number of steps to complete a full rotation. For example: a 4-fold rotation means 4 steps of
π/2
for a full
2π
rotation about the axis. The a-matrix is a
33
matrix and the alpha-matrix is
66
; 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
33
matrix
T
can be transformed to a new coordinate system
′
T
by the a-matrix
a
with the following formula
′
T
=a·T·
T
a
. The alpha-matrix can be used in a similar manner for a
36
matrix
U
that can be transformed using both the a-matrix
a
and alpha matrix
α
by
′
U
=a·U·
-1
α
.
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