# Definition:Translation Mapping

## Contents

## 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 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 $\vec {\mathcal E}$.

## Caution

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 $x$.