Translation

Builds on Euclidean vector Euclidean isometry

Definition

Let $v$ be a vector of $\mathbb{R}^n$ . The translation by $v$ is the map $T_v \colon \mathbb{R}^n \to \mathbb{R}^n, \qquad T_v(p) = p + v.$

Remarks

Each translation is a Euclidean isometry, and it is direct. For $v \neq 0$ , it has no fixed point. $T_v \circ T_w = T_{v + w}$ and $T_v^{-1} = T_{-v}$ . So the translations form an abelian group, isomorphic to $\mathbb{R}^n$ . In the plane, the composite of the reflections in two parallel lines is a translation. Mathlib's AffineEquiv.constVAdd is the translation by a vector in an affine space.

Reading path

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Euclidean isometry concept
Euclidean vector concept
Translation definition

Tag	`0R5QPQ3AGJ`
URI	`m2://0R5QPQ3AGJ`
Kind	definition
Writing	W1 stated
MSC	51F20
Defines	Translation
Uses	Euclidean vector, Euclidean isometry

Translation

Definition

Remarks

Reading path

Further reading

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