# Improper rotation

In geometry, an **improper rotation**,^{[1]} also called **rotoreflection**^{[1]} or **rotary reflection**^{[2]} 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.^{[3]}

In 3D, equivalently it is the combination of a rotation and an inversion in a point on the axis.^{[1]} 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.^{[2]}

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**.^{[4]} This is called an ** n-fold improper rotation** if the angle of rotation is 360°/

*n*.

^{[4]}The notation

*(*

**S**_{n}*S*for "Spiegel{{#invoke:Category handler|main}}", 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).

^{[4]}The notation is used for

**, i.e. rotation by an angle of rotation of 360°/**

*n*-fold rotoinversion*n*with inversion. The Coxeter notation for S

_{2n}is [2

*n*

^{+},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).

## See also

## References

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