Rotations and Euler angles

The relative orientation between two orthogonal righthanded 3D cartesian coordinate systems, let's call them xyz and ABC, is described by a real orthogonal 3x3 rotation matrix R, which is commonly parameterized by three so-called Euler angles α, β and γ.

Euler angles

The angles define the relative orientation between xyz and ABC. The angles describe three successive rotations of one coordinate system xyz that align it with the other one, ABC.

  1. Rotate xyz counterclockwise around its z axis by α to give x'y'z'.
  2. Rotate x'y'z' counterclockwise around its y' axis by β to give x''y''z''.
  3. Rotate x''y''z'' counterclockwise around its z'' axis by γ to give the final ABC.

This is called the zyz or y convention, for obvious reasons. It is used in many textbooks:

There exists also a zxz or x convention that is used in fields unrelated to EPR (e.g., classical mechanics).

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 C) is termed the figure axis.

The matrix describing this rotation is a product of 3 matrices describing one single-axis rotation each.

[eqn]

where c and s mean cos() and sin(). Rotation matrices are real, symmetric and orthogonal, i.e. their inverse is equal to 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

[eqn]

To obtain the Euler angles for the rotation R2: ABC → xyz from the Euler angles of the rotation R1:xyz → ABC (the one described above), use

[eqn]

i.e., interchange [eqn] and [eqn] and invert all the signs.

Equivalent sets of Euler angles

Of course, the addition to any angle of an arbitrary multiple of 2[eqn] has no effect on the rotation matrix.

[eqn]

There are, however, other sets of Euler angles which give the same rotation matrix

[eqn]

So, if you invert the sign of β, you have to add (or subtract) [eqn] to both α and γ.

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

[eqn]
Rotation axis and angle

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.

Choices for tensor eigenframes

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.

Applying rotations

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 rotated.

Here is how passive rotations can be done using the rotation matrix R as defined above: For a vector v_xyz defined in the xyz frame, v_ABC = R*v_xyz is the same vector as v_xyz, but represented in the ABC frame instead of the xyz frame. For a tensor T_xyz defined in the xyz frame, T_ABC = R*T_xyz*RT is the same tensor as T, but represented in the ABC frame instead of the xyz frame.