# Definition:Translation Mapping/Affine Space

## Definition

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 $x$.