The affine group

Nebe, G.

doi:10.1107/97809553602060000541

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. A1. ch. 1.5, pp. 30-31 | 1 | 2 |

Section 1.5.2.4. The affine group

Gabriele Nebe^a ^*

^a Abteilung Reine Mathematik, Universität Ulm, D-89069 Ulm, Germany
Correspondence e-mail: nebe@mathematik.uni-ulm.de

1.5.2.4. The affine group

| top | pdf |

The affine group of geometry is the set of all mappings of the point space which fulfil the conditions

(1) parallel straight lines are mapped onto parallel straight lines;
(2) collinear points are mapped onto collinear points and the ratio of distances between them remains constant.

In the mathematical model, the affine group is the automorphism group of the affine space and can be viewed as the set of all linear mappings of $[{\bf V}_{n+1}]$ that preserve $[{\bb A}_n]$ .

Definition 1.5.2.4.1. The affine group $[{\cal A}_{n}]$ is the subset of the set of all linear mappings $[\varphi:]$ $[{\bf V}_{n+1} \rightarrow]$ $[{\bf V} _{n+1}]$ with $[\varphi ({\bb A}_n) =]$ $[{\bb A}_n]$ . The elements of $[{\cal A}_{n}]$ are called affine mappings.

Since $[\varphi]$ is linear, it holds that $[\varphi (\overrightarrow{P\,Q}) = \varphi (Q-P) = \varphi (Q) - \varphi (P) = \overrightarrow{\varphi(P)\varphi(Q)}.]$ Hence an affine mapping also maps $[\tau ({\bb A}_n)]$ into itself.

Since the first n basis vectors of the chosen basis lie in $[\tau ({\bb A}_n)]$ and the last one in $[{\bb A}_n]$ , it is clear that with respect to this basis the affine mappings correspond to matrices of the form $[\specialfonts{\bbsf W} = \left(\matrix{{\bi W}\,\vphantom{(^2{\big(_2}}&\vrule\, &{\bi w} \cr \noalign{\vskip-1pt\hrule} \cr {\bi o}^{\rm T}\,\vphantom{{\big(^2}(_2}&\vrule\, &1 } \right).]$ The linear mapping induced by $[\varphi]$ on $[\tau ({\bb A}_n)]$ which is represented by the matrix $[{\bi W}]$ will be referred to as the linear part $[\overline{\varphi }]$ of $[\varphi ]$ . The image $[\varphi (P)]$ of a point P with coordinates $[\specialfonts{\bbsf x} = \left(\matrix{ \, {\bi p} \, \cr\noalign{\vskip4pt\hrule} \cr 1 } \right) \in {\bb A}_n]$ can easily be found as $[\specialfonts{\bbsf W}{\bbsf x} = \left(\matrix{ \, {\bi Wp} + {\bi w}\, \cr\noalign{\vskip4pt\hrule} \cr 1 } \right).]$

If one has a way to measure lengths and angles (i.e. a Euclidean metric) on the underlying vector space $[\tau ({\bb A}_n)]$ , one can compute the distance between P and Q $[\in {\bb A}_n]$ as the length of the vector $[{\overrightarrow {P\,Q}}]$ and the angle determined by P, Q and R $[\in {\bb A}_n]$ with vertex Q is obtained from $[\cos(P,Q,R) = \cos ({\overrightarrow {Q\,P}}, {\overrightarrow {Q\,R}})]$ . In this case, $[{\bb A}_n]$ is the Euclidean affine space, $[{\bb E}_n]$ .

An affine mapping of the Euclidean affine space is called an isometry if its linear part is an orthogonal mapping of the Euclidean space $[\tau ({\bb A}_n)]$ . The set of all isometries in $[{\cal A}_{n}]$ is called the Euclidean group and denoted by $[{\cal E}_{n}]$ . Hence $[{\cal E}_{n}]$ is the set of all distance-preserving mappings of $[{\bb E}_n]$ onto itself. The isometries are the affine mappings with matrices of the form $[\specialfonts{\bbsf W} = \left(\matrix{{\bi W}\,\vphantom{(^2{\big(_2}}&\vrule\, &{\bi w} \cr \noalign{\vskip-1pt\hrule} \cr {\bi o}^{\rm T}\,\vphantom{{\big(^2}(_2}&\vrule\, &1 } \right).]$ where the linear part W belongs to the orthogonal group of $[\tau ({\bb A}_n)]$ .

Special isometries are the translations, the isometries where the linear part is $[{\bi I}]$ , with matrix $[\specialfonts{\bbsf T}_{\bi w} = \left(\matrix{ {\bi I}\,\vphantom{(^2{\big(_2}}&\vrule\, &{\bi w} \cr \noalign{\vskip-1pt\hrule} \cr{\bi o}^{\rm T}\,\vphantom{{\big(^2}(_2}&\vrule\, &1 } \right).]$ The group of all translations in $[{\cal E}_{n}]$ is the translation subgroup of $[{\cal E}_{n}]$ and is denoted by $[{\cal T}_{n}]$ . Note that composition of two translations means addition of the translation vectors and $[{\cal T}_{n}]$ is isomorphic to the translation vector space $[\tau ({\bb E}_{n})]$ .

References

International Tables for Crystallography (2006). Vol. A1. ch. 1.5, pp. 30-31