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In geometry, an improper rotation, also called rotoreflection or rotary reflection is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane.

In 3D, equivalently it is the combination of a rotation and an inversion in a point on the axis. Therefore it is also called a rotoinversion or rotary inversion. A three-dimensional symmetry that has only one fixed point is necessarily an improper rotation.

In both cases the operations commute. Rotoreflection and rotoinversion are the same if they differ in angle of rotation by 180°, and the point of inversion is in the plane of reflection.

An improper rotation of an object thus produces a rotation of its mirror image. The axis is called the rotation-reflection axis. This is called an n-fold improper rotation if the angle of rotation is 360°/n. The notation Sn (S for "Spiegel", German for mirror) denotes the symmetry group generated by an n-fold improper rotation (not to be confused with the same notation for symmetric groups). The notation $$\bar{n}$$ is used for n-fold rotoinversion, i.e. rotation by an angle of rotation of 360°/n with inversion. The Coxeter notation for S2n is [2n+,2+], and orbifold notation is n×.

In a wider sense, an "improper rotation" may be defined as any indirect isometry, i.e., an element of E(3)\E+(3) (see Euclidean group): 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).

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).

Isometry
Orthogonal group

References

Morawiec, Adam (2004), Orientations and Rotations: Computations in Crystallographic Textures, Springer, p. 7, ISBN 9783540407348.
Kinsey, L. Christine; Moore, Teresa E. (2002), Symmetry, Shape, and Surfaces: An Introduction to Mathematics Through Geometry, Springer, p. 267, ISBN 9781930190092.
Salomon, David (1999), Computer Graphics and Geometric Modeling, Springer, p. 84, ISBN 9780387986821.
Bishop, David M. (1993), Group Theory and Chemistry, Courier Dover Publications, p. 13, ISBN 9780486673554.

Mathematics Encyclopedia