Euclidean isometry

Builds on Euclidean space Euclidean distance Affine map

Definition

Let $d$ be the Euclidean distance on the Euclidean space $\mathbb{R}^n$ . A Euclidean isometry of $\mathbb{R}^n$ is a map $f \colon \mathbb{R}^n \to \mathbb{R}^n$ with $d(f(p), f(q)) = d(p, q) \quad \text{for all } p, q \in \mathbb{R}^n.$

Remarks

Each Euclidean isometry is an affine map $f(x) = A x + b$ , with an orthogonal $n \times n$ matrix $A$ and a vector $b \in \mathbb{R}^n$ . Thus $f$ is a bijection, and its inverse is also an isometry. The isometry is direct if $\det A = 1$ and indirect if $\det A = -1$ . The direct isometries keep the orientation of figures. Mathlib's AffineIsometryEquiv is an affine bijection that keeps distances, between normed affine spaces.

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Tag	`0NXJM01QX1`
URI	`m2://0NXJM01QX1`
Kind	definition
Writing	W0 stub
MSC	51F20
Defines	Euclidean isometry
Uses	Euclidean space, Euclidean distance, Affine map

Euclidean isometry

Definition

Remarks

Reading path

Further reading

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