# Definition:Distance-Preserving Mapping

## Definition

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

Let $\phi: M_1 \to M_2$ be a mapping such that:

$\forall a, b \in M_1: d_1 \left({a, b}\right) = d_2 \left({\phi \left({a}\right), \phi \left({b}\right)}\right)$

Then $\phi$ is called a distance-preserving mapping.