# Definition:Translation Mapping

## Definition

### 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 an Affine Space

Let $\EE$ and $\FF$ be affine spaces.

Let $\TT: \EE \to \FF$ be affine transformations.

Then $\TT$ is a translation if and only if the tangent map $\vec \TT$ is the identity on the tangent space $\vec \EE$.

### In a Vector Space

Let $K$ be a field.

Let $X$ be a vector space over $K$.

Let $x \in X$.

The translation mapping $\tau_x : X \to X$ is defined as:

$\forall y \in X: \map {\tau_x} y = y - x$

where $y - x$ denotes vector subtraction.

## Caution

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.