In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). Equivalently, any improper rotation can also be decomposed into an ordinary rotation preceded or followed by a mirror reflection (e.g. x goes to −x or y goes to −y).

An improper rotation of an object thus produces a rotation of its mirror image.

Improper rotations can be represented by 3×3 orthogonal matrices with determinants of −1. A proper rotation is simply an ordinary rotation, which has a determinant of 1. The product (composition) of two improper rotations is a proper rotation, and the product of an improper and a proper rotation is an improper rotation.

When studying the symmetry of a physical system under an improper rotation (e.g. if a system has a mirror symmetry plane), it is important to distinguish between vectors and pseudovectors (as well as scalars and pseudoscalars, and in general; between tensors and pseudotensors), since the latter transform differently under proper and improper rotations (pseudovectors are invariant under inversion).

See also

Isometry, Orthogonal group