WOLFRAM|DEMONSTRATIONS PROJECT

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
-
3
4
-
3
2
2
-
1
4
3
2
2
-
1
2
-
3
2
2
-
1
4
3
2
2
-
3
4
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
1
4
2
1+sin
3π
14
1
2
2
cos
3π
14
1
4
2
sin
3π
14
-1
-
sin
3π
14
-1cos
3π
14

2
1
2
2
cos
3π
14
1+sin
3π
14
cos
3π
14

2
1
2
2
cos
3π
14
2
sin
3π
14
1
2
2
cos
3π
14
cos
π
14

2
-
2
cos
3π
14
-
cos
π
14

2
1
4
2
sin
3π
14
-1
1
2
2
cos
3π
14
1
4
2
1+sin
3π
14
-
1+sin
3π
14
cos
3π
14

2
1
2
2
cos
3π
14
sin
3π
14
-1cos
3π
14

2
sin
3π
14
-1cos
3π
14

2
2
-
cos
π
14

2
2
1+sin
3π
14
cos
3π
14

2
2
1
2
sin
3π
14
-sin
π
14
-
cos
π
14

2
2
2sin
π
14
2
cos
π
14
1
4
2
cos
3π
14
-
1
2
2
cos
3π
14
1
4
2
cos
3π
14
cos
π
14

2
2
1
2
1+
2
sin
3π
14
-
cos
π
14

2
2
-
1+sin
3π
14
cos
3π
14

2
2
cos
π
14

2
2
-
sin
3π
14
-1cos
3π
14

2
2
2sin
π
14
2
cos
π
14
cos
π
14

2
2
1
2
sin
3π
14
-sin
π
14
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
33
matrix and the alpha-matrix is
66
; 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
33
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
36
matrix
U
that can be transformed using both the a-matrix
a
and alpha matrix
α
by
′
U
=a·U·
-1
α
.