# Definition:Isometry (Metric Spaces)

## Definition

### Definition 1

Let $M_1 = \tuple {A_1, d_1}$ and $M_2 = \tuple {A_2, d_2}$ be metric spaces or pseudometric spaces.

Let $\phi: A_1 \to A_2$ be a bijection such that:

$\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$

Then $\phi$ is called an isometry.

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

### Definition 2

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces or pseudometric spaces.

$M_1$ and $M_2$ are isometric if and only if there exist inverse mappings $\phi: A_1 \to A_2$ and $\phi^{-1}: A_2 \to A_1$ such that:
$\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$
and:
$\forall u, v \in A_2: \map {d_2} {u, v} = \map {d_1} {\map {\phi^{-1} } u, \map {\phi^{-1} } v}$

Such metric spaces $M_1$ and $M_2$ are defined as being isometric.

### Isometry Into

Let $\phi: A_1 \to A_2$ be an injection 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 (from $M_1$) into $M_2$.

## Also defined as

Some sources do not insist that an isometry be surjective.

Make sure to know which prerequisites are used when quoting results about isometries.

## 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

• Results about isometries can be found here.