Definition:Translation Mapping/Euclidean Space

Definition
A translation $\tau_\mathbf x$ is an isometry on the Euclidean Space $\Gamma = \R^n$ defined as:


 * $\forall \mathbf y \in \R^n: \map {\tau_\mathbf x} {\mathbf y} = \mathbf y - \mathbf x$

where $\mathbf x$ is a vector in $\R^n$.

As $\R^n$ is a vector space, $\struct {\R^n, +}$ is an abelian group.

Hence this definition is compatible with that of a translation in an abelian group.

Also see

 * Translation Mapping is Isometry, demonstrating that $\tau_\mathbf x$ is indeed an isometry.