Isometry

Definition

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. 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 and Lipschitz with constant $1$. Some authors require an isometry to be surjective. Here a surjective isometry is an isometric isomorphism. 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.