Definition:Translation Mapping/Euclidean Space
< Definition:Translation Mapping(Redirected from Definition:Translation in Euclidean Space)
Jump to navigation
Jump to search
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.