Definition:Isometry (Metric Spaces)/Definition 1

From ProofWiki
Jump to navigation Jump to search


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.

Also known as

An isometry is also known as a metric equivalence.

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


Euclidean Plane is Isometric to Complex Plane

Let $\R^2$ be the real number plane with the Euclidean metric.

Let $\C$ denote the complex plane.

Let $f: \R^2 \to \C$ be the function defined as:

$\forall \tuple {x_1, x_2} \in \R^2: \map f {x_1, x_2} = x_1 + i x_2$

Then $f$ is an isometry from $\R^2$ to $\C$.

Also see

  • Results about isometries can be found here.