Definition:Isometry (Metric Spaces)/Definition 1

Definition
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces or pseudometric spaces.

Let $\phi: A_1 \to A_2$ be a bijection such that:
 * $\forall a, b \in A_1: d_1 \left({a, b}\right) = d_2 \left({\phi \left({a}\right), \phi \left({b}\right)}\right)$

Then $\phi$ is called an isometry.

That is, an isometry is a distance-preserving bijection.

Also known as
An isometry is also known as a metric equivalence.

Two isometric spaces can also be referred to as metrically equivalent.

Also see

 * Equivalence of Definitions of Isometry of Metric Spaces