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Improper rotation

In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with an inversion (x goes to -x).

Improper rotations are described by 3-by-3 orthogonal matrices with a determinant 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.

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

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