The relative orientation between two orthogonal righthanded 3D cartesian coordinate systems xyz and XYZ is described by a real orthogonal 3x3 rotation matrix, which is commonly parameterized by three so-called Euler angles α, β and γ.
The angles define the relative orientation between xyz and XYZ by three successive rotations of one coordinate system xyz that align it with the other one XYZ.
This is called the zyz or y convention, for obvious reasons. Is used by many textbooks:
The z axis is called the vertical, the y' axis (same as the y'' axis) is called the line of nodes, and z'' (same as Z) is termed the figure axis.
The matrix describing this rotation is a product of 3 matrices describing one single-axis rotation each.
where c and s mean cos() and sin(). Rotation matrices are real, symmetric and orthogonal, i.e. their inverse is equal their transpose. The determinant of R is +1.
The rows and columns of R have simple geometric meaning:
The same rotation matrix is obtained, if the rotations are carried out in reverse order around other axes: First by γ around the z axis, then by β around the original y axis, and finally by α again around the original z axis. In other words
To obtain the Euler angles for the rotation R2:XYZ -> xyz from the Euler angles of the rotation R1:xyz -> XYZ (the one described above), use
i.e., interchange and
and
invert all the signs.
Of course, the addition to any angle of an arbitrary multiple of 2
has no effect on the rotation matrix.
There are, however, other sets of Euler angles which give the same rotation matrix
So, if you invert the sign of β, you have to add
(or subtract) to α and γ. If you invert
the sign of α and γ,you have to add (or subtract)
to β.
As a consequence, there is a one-to-one correspondence between Euler angles and rotation matrices only if the Euler angle domains are restricted, e.g. to
In addition to the set of three Euler angles and the rotation matrix, a rotation can also be represented by a vector specifying the rotation axis and the angle of rotation around this axis.
This representation is rarely used in EPR, but is visually very simple to understand.
The labelling of the principal axes of a tensor is completely arbitrary. There are in total 24 possible xyz arrangements that describe the principal axes frame of a tensor. Correspondingly, there are 24 different sets of Euler angles. Why 24? It's easy to enumerate them: The z axis can point in any of the six principal axis direction of an ellipsoid, and the xy axis pair can have 4 possible orientations for each z orientation, giving 24 in total.
If the z axis is required to points along the axes with the largest eigenvalue of the tensor, then it can only have 2 orientations, and there are only 8 different coordinate systems and 8 sets of Euler angles.
Rotations can be active or passive. In active ("alibi") rotations, the object (vector, tensor) is rotated and the coordinate system is left unchanged. In passive ("alias") rotations, the object is left unchanged and the coordinate axes system is rotation.
For a vector v defined in the xyz frame and a rotation matrix R defined as above, w = R*v is the same vector as v, but represented in the XYZ frame instead of the xyz frame.
For a tensor defined in the xyz frame and a rotation matrix R defined as above, T2 = R*T*RT is the same tensor as T, but represented in the XYZ frame instead of the xyz frame.