World Library  
Flag as Inappropriate
Email this Article

Improper rotation

Article Id: WHEBN0000244324
Reproduction Date:

Title: Improper rotation  
Author: World Heritage Encyclopedia
Language: English
Subject: Point groups in three dimensions, Coxeter notation, Pseudovector, Hermann–Mauguin notation, Schoenflies notation
Collection: Euclidean Symmetries, Lie Groups
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Improper rotation

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 including that axis.[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

References

  1. ^ a b c Morawiec, Adam (2004), Orientations and Rotations: Computations in Crystallographic Textures, Springer, p. 7,  .
  2. ^ a b Kinsey, L. Christine; Moore, Teresa E. (2002), Symmetry, Shape, and Surfaces: An Introduction to Mathematics Through Geometry, Springer, p. 267,  .
  3. ^ Salomon, David (1999), Computer Graphics and Geometric Modeling, Springer, p. 84,  .
  4. ^ a b c Bishop, David M. (1993), Group Theory and Chemistry, Courier Dover Publications, p. 13,  .
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.