AchsenWinkelPaar

AchsenwinkelPaar is a concept in geometry and kinematics that denotes a pair of rotation angles associated with two axes, used to describe the orientation of a rigid body or a coordinate frame relative to a reference frame. The term combines Achsenwinkel (axis angle) with Paar (pair) and is used in contexts where the orientation is decomposed into two consecutive axis-based rotations.

A typical formulation represents the orientation as a product of two elementary rotations: R = R_axis1(alpha) · R_axis2(beta). Each rotation is defined by an axis (a unit vector) and an angle about that axis. In many practical cases the axes are fixed in the world frame or in the moving frame, and the pair (alpha, beta) is not unique; different axis choices or decomposition orders can produce the same overall orientation.

From a computational standpoint, one can compute a rotation matrix, quaternion, or Euler-like parameters from a given AchsenwinkelPaar by composing the two elementary rotations and then extracting the resulting orientation. Conversely, given a target rotation, solving for a specific pair (alpha, beta) can be non-unique and may require additional constraints, such as fixing one axis or limiting the range of angles.

Relation to other representations: AchsenwinkelPaar is closely related to Euler angles and Tait–Bryan angles, and it can be converted to and from axis–angle representations with appropriate formulas. It is useful in robotics, attitude estimation, and 3D graphics where a two-step rotation is natural or convenient.

Limitations include non-uniqueness, singular configurations, and sensitivity to axis choice. See also Euler angles, axis–angle representation, rotation matrices, quaternions.