Improper rotation

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A pentagonal antiprism with marked edges shows rotoreflectional symmetry, with an order of 10.

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 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).[4] 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).

See also[edit]


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