# Isometry

An isometry is a map which preserves distances between points. Isometries exist in any space in which a distance function is defined, i.e. an arbitrary abstract metric space. In the particular case where we take our space to be the usual Euclidean plane or Euclidean 3-space ( $\mathbb{R}^2$ or $\mathbb{R}^3$ with the standard Euclidean metric), the isometries are known as rigid motions and two sets which can be transformed onto each other by an isometry are said to be congruent.

## Isometries are injective

Since for any metric we have $d(x, y) > 0$ whenever $x \neq y$, it follows that every isometry must be an injection.

### Proof

Suppose otherwise. Then there is some non-injective isometry $\phi: S \to T$. Since $\phi$ is not injective, we know $\exists x, y \in S$ such that $x \neq y$ and $\phi(x) = \phi(y)$. But $x \neq y \Longrightarrow d_S(x, y) > 0$ while $\phi(x) = \phi(y) \Longrightarrow d_T(\phi(x), \phi(y)) = 0$, and this contradicts the fact that $\phi$ is an isometry.

Note that this does not mean that isometries are necessarily bijections. Consider, for example, the discrete metric on the integers, $(\mathbb{Z}, d)$ such that $d(x, x) = 0$ and $x\neq y \Longrightarrow d(x, y) = 1$. It is simple to verify that this is a metric space. The map $\phi: \mathbb{Z} \to \mathbb{Z}$ given by $\phi(n) = 2n$ is an isometry but it is not surjective.