Group of Euclidean motions

Definition

The group of Euclidean motions in $n$ dimensions is defined in the following equivalent ways:

• It is the group of isometries of Euclidean space $\mathbb {R} ^{n}$, where the metric is the Euclidean metric.
• It is the affine orthogonal group of order $n$ over the field $\mathbb {R}$. In other words, it is the semidirect product of $\mathbb {R} ^{n}$ with the orthogonal group $O(n,\mathbb {R} )$, viewed as a subgroup of the general affine group $GA(n,\mathbb {R} )$.