Improper rotation

In 3D 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 an inversion about the origin.

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.

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. In the wider sense, an improper rotation is an indirect isometry, i.e., an element of E(3)\E+(3) (see Euclidean group): 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 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 the wider senses, the composition of two improper rotations is a proper rotation, and the product 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 (pseudovectors are invariant under inversion).