= Special Euclidean group
{title2=$\operatorname{SE}(n)$}
The <group> of <Euclidean isometries> preserving <orientation> of $\mathbb R^n$. Its elements have the form $x\mapsto Ax+b$, with $A$ in the <special orthogonal group> and $b\in\mathbb R^n$. The <function composition> law is $(A,b)(B,c)=(AB,Ac+b)$, the <identity element> is $(I,0)$, and the <inverse element> is $(A^{-1},-A^{-1}b)$. These formulas prove closure and inverses; <function composition> supplies <associativity>. For $n=2$, $A$ is a <planar rotation>.
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