\begin{definition}[Definition:Distance-Preserving Mapping]
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces, pseudometric spaces, or quasimetric spaces.
Let $\phi: M_1 \to M_2$ be a mapping such that:
:$\forall a, b \in M_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$
Then $\phi$ is called a '''distance-preserving mapping'''.
\end{definition}