Einheitsquaternions

Einheitsquaternionen, or unit quaternions, are quaternions with unit norm. A quaternion q = a + bi + cj + dk has norm sqrt(a^2 + b^2 + c^2 + d^2). When this norm equals 1, q lies on the 3-sphere S^3 and forms a group under quaternion multiplication. The set of all unit quaternions provides a compact representation of 3D rotations: any rotation of a vector v can be expressed as v' = q v q^{-1}, where v is treated as a pure imaginary quaternion (0 + v_x i + v_y j + v_z k).

Two distinct unit quaternions q and -q correspond to the same rotation, so unit quaternions form a double cover of the rotation group SO(3). The map from unit quaternions to rotation matrices is a surjective homomorphism with kernel {±1}. The group of unit quaternions is isomorphic to SU(2).

Because of their algebraic properties, unit quaternions are widely used to represent orientations in computer graphics, robotics, and aerospace. They allow smooth, stable interpolation between rotations (spherical linear interpolation, or slerp) and avoid singularities associated with Euler angles. Any unit quaternion can be written as q = cos(θ/2) + sin(θ/2) (n_x i + n_y j + n_z k) where θ is the rotation angle about the unit axis n.

Key operations: the inverse of a unit quaternion is its conjugate q̄ = a − bi − cj − dk, since q q̄ = 1. The multiplication of unit quaternions corresponds to the composition of rotations.