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 | S_{4} | S_{6} | S_{8} | S_{10} | S_{12} |
---|---|---|---|---|---|
Subgroups | C_{2} | C_{3}, S_{2} = C_{i} | C_{4}, C_{2} | C_{5}, S_{2} = C_{i} | C_{6}, S_{4}, C_{3}, C_{2} |
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}. |
In 3 dimensions, improper rotation is equivalently defined as a combination of rotation about an axis and inversion in a point on the axis.^{[1]} For this reason it is also called a rotoinversion or rotary inversion. The two definitions are equivalent because rotation by an angle θ followed by reflection is the same transformation as rotation by θ + 180° followed by inversion (taking the point of inversion to be in the plane of reflection). In both definitions, the operations commute.
A three-dimensional symmetry that has only one fixed point is necessarily an improper rotation.^{[3]}
An improper rotation of an object thus produces a rotation of its mirror image. The axis is called the rotation-reflection axis.^{[6]} This is called an n-fold improper rotation if the angle of rotation, before or after reflexion, is 360°/n (where n must be even).^{[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.
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).