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Rotation composed with a reflection

In geometry, an improper rotation (also called rotation reflection, rotoreflection, or rotoinversion) 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. Reflection and inversion are each special case of improper rotation. Any improper rotation is an affine transformation and, in cases that keep the coordinate origin fixed, a linear transformation. It is used as a symmetry operation in the context of geometric symmetry, molecular symmetry and crystallography, where an object that is unchanged by a combination of rotation and reflection is said to have improper rotation symmetry. + Example polyhedra with rotoreflection symmetryGroup S4 S6 S8 S10 S12Subgroups C2 C3, S2 = Ci C4, C2 C5, S2 = Ci C6, S4, C3, C2Examplebeveled digonal antiprism triangular antiprism square antiprism pentagonal antiprism hexagonal antiprism Antiprisms with directed edges have rotoreflection symmetry. p antiprisms for odd p contain inversion symmetry, Ci.

The direct subgroup of S2n is Cn, order n, index 2, being the rotoreflection generator applied twice. For odd n, S2n contains an inversion, denoted Ci or S2. S2n is the direct product: S2n = Cn × S2, if n is odd. For any n, if odd p is a divisor of n, then S2n/p is a subgroup of S2n, index p. For example S4 is a subgroup of S12, index 3.

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. A proper rotation is an ordinary rotation. In the wider sense, a proper rotation is defined as a direct isometry; i. e., an element of E+(3): it can also be the identity, a rotation with a translation along the axis, or a pure translation. A direct isometry is an affine transformation with an orthogonal matrix that has a determinant of 1. In either the narrower or the wider senses, the composition of two improper rotations is a proper rotation, and the composition 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 (in 3 dimensions, pseudovectors are invariant under inversion).