In geometry, an improper rotation is the combination of an ordinary rotation of three-dimensional Euclidean space, that keeps the origin fixed, with a coordinate inversion (a vector x goes to −x). Equivalently, any improper rotation can also be decomposed into an ordinary rotation preceded or followed by a mirror reflection (e.g. x goes to −x or y goes to −y).
An improper rotation of an object thus produces a rotation of its mirror image.
Improper rotations can be represented by 3×3 orthogonal matrices with determinants of −1. A proper rotation is simply an ordinary rotation, which has a determinant of 1. The product (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).
Such a rotation is a linear transformation that preserves the length of vectors, and also preserves the orientation, or handedness, of space.
The composition of two rotations is a rotation, and every rotation has a unique inverse which is again a rotation.
The group of all proper and improperrotations in n dimensions is called the orthogonal group, O(n), and the subgroup of proper rotations is called the special orthogonal group, SO(n).
In 3D geometry, an improperrotation, 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 perpendicular plane.
Equivalently it is the combination of a rotation and an inversion in a point on the axis.
In the wider sense, an improperrotation is an indirect isometry, i.e., an element of E(3)\E
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