Definition:Translation Mapping/Euclidean Space

Definition

A translation $\tau_\mathbf x$ is an isometry on the real 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 $\R^n$.

As $\R^n$ is a vector space, $\struct {\R^n, +}$ is an abelian group.

Hence this definition is compatible with that of a translation in an abelian group.

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

• Translation Mapping is Isometry, demonstrating that $\tau_\mathbf x$ is indeed an isometry.

• Results about translation mappings can be found here.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: Exercise $2.6: 21$
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 34$. Examples of groups: $(5)$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): translation: 2.
• 2003: John H. Conway and Derek A. Smith: On Quaternions And Octonions ... (previous) ... (next): $\S 1$: The Complex Numbers: Introduction: $1.1$: The Algebra $\R$ of Real Numbers
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): translation: 2.