kvaternioner

Kvaternioner, or quaternions, are a number system that extends complex numbers. A quaternion is written as q = a + b i + c j + d k, where a, b, c, d are real numbers and i, j, k are imaginary units with i^2 = j^2 = k^2 = ijk = -1. Quaternions multiply according to these rules and are noncommutative: in general ab ≠ ba. The norm is N(q) = a^2 + b^2 + c^2 + d^2, the conjugate q = a - b i - c j - d k, and the inverse q^-1 = q/N(q) for q ≠ 0.

History and structure: Hamilton introduced quaternions in 1843 as a framework for representing 3D rotations. Quaternions form a real division algebra; the subset of unit quaternions (N(q) = 1) forms a group under multiplication and is isomorphic to the Lie group SU(2), which serves as a double cover of the rotation group SO(3).

Rotation representation: A rotation can be represented by a unit quaternion q, corresponding to a rotation by angle theta around a unit axis u. The rotated vector v is obtained by the quaternion product v' = q v q^-1, where v is treated as a pure quaternion (0, v). This is equivalent to axis-angle form: q = (cos(theta/2), sin(theta/2) u). Interpolating between orientations uses SLERP (spherical linear interpolation).

Applications and considerations: Kvaternioner are widely used in computer graphics, robotics, aerospace, and animation for compact rotation representation, efficient concatenation, and avoidance of gimbal lock. They require normalization to unit length for rotations; numerical drift and the noncommutative nature of multiplication are important considerations in implementations.