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. Reflection and inversion are each a special case of improper rotation. Any improper rotation is an affine transformation and, in cases that keep the coordinate origin fixed, a linear transformation.[5] 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.
| Group | S4 | S6 | S8 | S10 | S12 |
|---|---|---|---|---|---|
| Subgroups | C2 | C3, S2 = Ci | C4, C2 | C5, S2 = Ci | C6, S4, C3, C2 |
| Example |  beveled 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. | |||||
- ^ a b Morawiec, Adam (2004), Orientations and Rotations: Computations in Crystallographic Textures, Springer, p. 7, ISBN 978-3-540-40734-8.
- ^ Miessler, Gary; Fischer, Paul; Tarr, Donald (2014), Inorganic Chemistry (5 ed.), Pearson, p. 78
- ^ Kinsey, L. Christine; Moore, Teresa E. (2002), Symmetry, Shape, and Surfaces: An Introduction to Mathematics Through Geometry, Springer, p. 267, ISBN 978-1-930190-09-2.
- ^ Klein, Philpotts (2013). Earth Materials. Cambridge University Press. pp. 89–90. ISBN 978-0-521-14521-3.
- ^ Salomon, David (1999), Computer Graphics and Geometric Modeling, Springer, p. 84, ISBN 978-0-387-98682-1.