---
tag: 0878DN3WCS
kind: definition
name: Isometry
msc: ["54E40"]
defines: 02HPP5VCZH
uses: [063W5JMY85]
sources:
  - url: https://en.wikipedia.org/wiki/Isometry
    locator: Definition
---

## Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]].
A function $f \colon X \to Y$ is an isometry when
$$d_Y\bigl(f(x), f(y)\bigr) = d_X(x, y) \quad \text{for all } x, y \in X.$$

## Remarks

Each isometry is [[injective|02J4G4J4TJ]] and [[Lipschitz|0TMZ21WKCN]] with constant $1$.
Some authors require an isometry to be surjective. Here a surjective isometry is an [[isometric isomorphism|0VRD8GC5BM]].
The map $x \mapsto (x, 0)$ from $\mathbb{R}$ to $\mathbb{R}^2$ with the Euclidean metric is an isometry, but it is not surjective.
Rotations, reflections and translations of $\mathbb{R}^2$ are isometries for the Euclidean metric.

## Explanations

### L1 human

An isometry moves points without stretching or shrinking any distance.
It can put a space into a bigger space, as the real line is put into the plane as the horizontal axis.