= Euclidean group
{c}
{title2=$E(n)$}
{wiki}
The <Euclidean group> $E(n)=O(n)\ltimes\mathbb R^n$ is the <group> of all <isometries> $x\mapsto Ax+b$ of <Euclidean space>, with $A$ an <orthogonal matrix>. Its orientation-preserving <subgroup> is $SE(n)=SO(n)\ltimes\mathbb R^n$, the <semidirect product> of the <special orthogonal group> with translations. The <Euclidean motion of Euclidean three-space> and the <proper Euclidean motion of Euclidean three-space> are the respective three-dimensional instances.
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