Category:Definitions/Translation Mappings
This category contains definitions related to Translation Mappings.
Related results can be found in Category:Translation Mappings.
In an Abelian Group
Let $\struct {G, +}$ be an abelian group.
Let $g \in G$.
Then translation by $g$ is the mapping $\tau_g: G \to G$ defined by:
- $\forall h \in G: \map {\tau_g} h = h + \paren {-g}$
where $-g$ is the inverse of $g$ with respect to $+$ in $G$.
In an Affine Space
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 $\vec \EE$.
In a Vector Space
Let $K$ be a field.
Let $X$ be a vector space over $K$.
Let $x \in X$.
The translation mapping $\tau_x : X \to X$ is defined as:
- $\forall y \in X: \map {\tau_x} y = y - x$
where $y - x$ denotes vector subtraction.
In Euclidean Space
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 $\R^n$.
Pages in category "Definitions/Translation Mappings"
The following 9 pages are in this category, out of 9 total.
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- Definition:Translation in Abelian Group
- Definition:Translation in Euclidean Space
- Definition:Translation in Vector Space
- Definition:Translation Mapping
- Definition:Translation Mapping/Abelian Group
- Definition:Translation Mapping/Affine Space
- Definition:Translation Mapping/Caution
- Definition:Translation Mapping/Euclidean Space
- Definition:Translation Mapping/Vector Space