Definition:Translation Mapping/Affine Space

Definition

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 $\vec \EE$.

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.

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