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. 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.

Example polyhedra with rotoreflection symmetry
Group	S₄	S₆	S₈	S₁₀	S₁₂
Subgroups	C₂	C₃, S₂ = C_i	C₄, C₂	C₅, S₂ = C_i	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, C_i.

^ ^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.