[[Euclidean space]]
# Euclidean group

The **Euclidean group** $\mathrm{E}(n)$ is the [[isometry group]] of $n$-dimensional [[Euclidean space]] $\mathbb{E}^n$, #m/def/geo/affine 
consisting of transformations that preserve Euclidean distance.
It is equivalent to the [[semidirect product]] of the appropriate [[real orthogonal group]] with the translation group
$$
\begin{align*}
\mathrm{E}(n) = \mathrm{O}(n) \rtimes  \mathbb{R}^n
\end{align*}
$$

## Related groups

- The relativistic counterpart is the [[Poincaré group]]

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