Definition:Translation Mapping/Euclidean Space

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A translation $\tau_\mathbf x$ is an isometry on the real 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.


It is easy to confuse the mappings $\tau_{\mathbf x}$ and $\tau_{-\mathbf x}$, and the choice made here is arbitrary.

The map $\tau_{\mathbf x}$ can be understood (appealing to our planar $\R^2$ intuition) as translating the coordinate axes by $\mathbf x$.

Also see

  • Results about translation mappings can be found here.