EulerWinkeln

EulerWinkeln, or Euler angles, are a way to describe the orientation of a rigid body in three-dimensional space using three angular parameters. They are named after the Swiss mathematician Leonhard Euler, who introduced the concept in the 18th century.

A sequence of rotations about coordinate axes defines the orientation. The angles are typically denoted φ (roll), θ (pitch), ψ (yaw). There are multiple conventions, such as intrinsic rotations about body-fixed axes or extrinsic rotations about fixed axes. Common sequences include ZYX (yaw-pitch-roll) and ZXZ (a proper Euler angle sequence).

A rotation is represented as the product of three elementary rotations; the same three angles can produce different final orientations depending on the chosen sequence. Conversion to and from rotation matrices or to quaternions is standard. For some sequences, particularly when pitch is ±90°, the three parameters lose one degree of freedom (gimbal lock).

Applications encompass aerospace (aircraft attitude), robotics (pose representation), computer graphics (camera orientation), motion capture, and animation.

Limitations include non-uniqueness, range ambiguities, and near singularities. They are often supplemented or replaced by quaternions for smooth interpolation and robust optimization, though EulerWinkeln remain common in engineering and for human-readable orientation.