Rotenteoremet

Rotenteoremet is a theorem in geometry describing rotations in three-dimensional space. In Danish-language mathematical literature, the term is used for what is widely known in English as Euler's rotation theorem.

Statement: In three-dimensional Euclidean space, any orientation obtained by a single rotation about the origin is equivalent to a rotation by some angle theta about a fixed axis through the origin. Equivalently, any proper rotation can be described by an axis of rotation and a rotation angle. The rotation axis is the eigenvector corresponding to the eigenvalue 1 of the rotation matrix, and the angle satisfies cos theta = (trace(R) − 1)/2.

History: The result is attributed to Leonhard Euler, who described it in the 18th century. The Danish term Rotenteoremet appears in some regional texts and textbooks to denote this classical fact about 3D rotations.

Proof sketch: For a rotation matrix R ∈ SO(3), Rv = v for the axis direction v, giving eigenvalue 1. The fixed axis can be identified as the set of vectors unchanged by the rotation, and the remaining plane rotates by angle theta. The angle is obtained from the trace formula cos theta = (trace(R) − 1)/2.

Generalizations and limitations: In dimensions higher than three, a general rotation can be decomposed into independent rotations in orthogonal planes; a single axis rotation is a special case of 3D rotations. Rotations with determinant −1 include reflections and are not covered by the theorem.

Applications: Euler's rotation theorem underpins axis-angle representations, quaternions, rotation matrices, and practical tools in robotics, computer graphics, aerospace, and crystallography.

See also: Euler's rotation theorem, axis-angle representation, rotation matrix, quaternion.