Definition:Translation Mapping

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In an Abelian Group

Let $\struct {G, +}$ be an abelian group.

Let $g \in G$.

Then translation by $g$ is the mapping $\tau_g: G \to G$ defined by:

$\forall h \in G: \map {\tau_g} h = h + \paren {-g}$

where $-g$ is the inverse of $g$ with respect to $+$ in $G$.

In a Euclidean Space

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 $\R^n$.

In an Affine Space

Let $\mathcal E$ and $\mathcal F$ be affine spaces.

Let $\mathcal T: \mathcal E \to \mathcal F$ be affine transformations.

Then $\mathcal T$ is a translation if and only if the tangent map $\vec {\mathcal T}$ is the identity on the tangent space $\vec {\mathcal E}$.


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

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