Geometric conventions

Particle orientation

• Particle orientations can be described using Euler angles, unit quaternions, axis-angle-representations or rotation matrices.

• Any of these representations can serve to initialize a scipy rotation object.

• By default, Euler angles are handled in the ‘zxz’ convention, referring to extrinsic rotations around the respective axes.

• In the ‘zxz’ convention, the angles have the following ranges:

  • The first angle ranges between -180° and 180° (including both).

  • The second angle ranges between 0° and 180° (including both).

  • The third angle ranges between -180° and 180° (including both).

• Unit quaternions are handled in the scalar-last order: \(q = xi + yj + zk + w\), \(\Vert q\Vert = 1\).

• In the axis-angle-representation of a rotation (or particle orientation), a unit vector \(v\) serves as rotation axis, whereas an angle \(\theta\) describes the amount of rotation. The axis-angle is then given by \(\theta v\).

• Rotation matrices in 3-dimensional space are real \(3\times 3\) matrices \(A\) satisfying \(\det(A) = 1\), \(A^T = A^{-1}\).