Knowlpedia

Definition

E(2)

E(2) as a separately identified real Lie group, with its defining operations and dimension.

E(2)\mathrm{E}(2) is the group of Euclidean isometries of R2\mathbb R^{2}. Each element is x↦Ax+bx\mapsto Ax+b with A∈O(2,R)A\in\mathrm{O}(2,\mathbb R) and b∈R2b\in\mathbb R^{2}, and

(A,b)(A′,b′)=(AA′,Ab′+b).(A,b)(A^{\prime},b^{\prime})=(AA^{\prime},Ab^{\prime}+b).
Dimensions and structure

This group has real dimension 33. This gives the of translations by the indicated .

Direct verification

The displayed product is composition of affine maps. Orthogonal matrices preserve the Euclidean distance, and any Euclidean isometry is a translation followed by an orthogonal map. Adding the translation dimension to n(n−1)/2 gives n(n+1)/2.