Definition:Translation Mapping

Definition in an Abelian Group
Let $\left({G, +}\right)$ be an abelian group.

Let $g \in G$.

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


 * $\forall h \in G: \tau_g \left({h}\right) = h + \left({- g}\right)$

where $-g$ is the inverse of $g$.

Definition in $\R^n$
As $\R^n$ is a vector space, $\left({\R^n, +}\right)$ is an abelian group.

Hence translation by $x$ is the mapping $\tau_x: \R^n \to \R^n$ defined by:


 * $\forall y \in \R^n: \tau_x \left({y}\right) = y - x$

Definition 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 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 $x$.