Euclidean group

Builds on Euclidean isometry Translation

Description

The Euclidean group $\mathrm{E}(n)$ is the set of all Euclidean isometries of $\mathbb{R}^n$ , with the composition of maps as the operation. It is a group: the identity map is the neutral element, and the inverse of an isometry is an isometry. Each element has the form $x \mapsto A x + b$ , with $A$ in the orthogonal group $\mathrm{O}(n)$ and $b \in \mathbb{R}^n$ . The direct isometries, with $\det A = 1$ , form a subgroup $\mathrm{SE}(n)$ of index $2$ . Its elements are the rigid motions.

Remarks

The translations form an abelian normal subgroup $T \cong \mathbb{R}^n$ , and $\mathrm{E}(n) / T \cong \mathrm{O}(n)$ . The group $\mathrm{E}(n)$ has $n(n+1)/2$ degrees of freedom: $3$ for $n = 2$ and $6$ for $n = 3$ . The group is not abelian for $n \geq 2$ . For example, a rotation about the origin and a translation do not commute in general. The symmetry group of each figure in $\mathbb{R}^n$ is a subgroup of $\mathrm{E}(n)$ .

Reading path

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Tag	`0H3YY8CPKH`
URI	`m2://0H3YY8CPKH`
Kind	object
Writing	W1 stated
MSC	20H15
Instance of	Group
Uses	Euclidean isometry, Translation
Same as	Wikidata Q852195 Wikipedia Euclidean group

Euclidean group

Description

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

Special cases

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