In geometry, an improper rotation, also called rotation-reflection, rotoreflection,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 perpendicular to that axis.
Three dimensions
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. There are several different systems for naming individual improper rotations:
Sn denotes the symmetry group generated by an n-fold improper rotation. For example, the symmetry operation S6 is the combination of a rotation of =60° and a mirror plane reflection..
The direct subgroup of S2n, of index 2, is Cn, +, or, of order n, being the rotoreflection generator applied twice. S2n for oddn contains an inversion, denoted Ci
In a wider sense, an improper rotation may be defined as any indirect isometry; i.e., an element ofE\E+: 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+: 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, it is important to distinguish between vectors and pseudovectors, since the latter transform differently under proper and improper rotations.
