Improper rotation

In geometry, an improper rotation[1] (also called rotation-reflection,[2] rotoreflection,[1] rotary reflection,[3] or rotoinversion[4]) is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicular to that axis.

A three-dimensional symmetry that has only one fixed point is necessarily an improper rotation.

[6] There are several different systems for naming individual improper rotations: In a wider sense, an improper rotation may be defined as any indirect isometry; i.e., an element of E(3)\E+(3): thus it can also be a pure reflection in a plane, or have a glide plane.

An indirect isometry is an affine transformation with an orthogonal matrix that has a determinant of −1.

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 (in 3 dimensions, pseudovectors are invariant under inversion).