SE(n) - Knowlpedia

For every integer $n\ge1$ , $\mathrm{SE}(n)$ is the group of orientation-preserving Euclidean isometries of $\mathbb R^{n}$ . Each element is $x\mapsto Ax+b$ with $A\in\mathrm{SO}(n,\mathbb R)$ and $b\in\mathbb R^{n}$ , and

(A,b)(A^{\prime},b^{\prime})=(AA^{\prime},Ab^{\prime}+b).

Dimensions and structure

This group has real dimension $n(n+1)/2$ . This gives the semidirect product of translations by the indicated orthogonal group.

Direct verification

The determinant-one condition is preserved by composition and inversion. Its Lie algebra has a skew-symmetric block and an arbitrary translation vector, giving n(n−1)/2+n dimensions.