SE(1) - Knowlpedia

$\mathrm{SE}(1)$ is the group of orientation-preserving Euclidean isometries of $\mathbb R^{1}$ . Each element is $x\mapsto Ax+b$ with $A\in\mathrm{SO}(1,\mathbb R)$ and $b\in\mathbb R^{1}$ , and

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

Dimensions and structure

This group has real dimension $1$ . This gives the semidirect product of translations by the indicated orthogonal group. In dimension one this is just the additive real line.

An orientation-preserving Euclidean isometry of the real line is exactly a translation. See $(\mathbb R^{1},+)$ .

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.