---
tag: PK77W5K15V
kind: definition
name: Similarity transformation
msc: [51M05, 51N20]
defines: D3GYHF6NAX
uses: [0XPR1E85W0, 0NPPGHQNXQ, 02HPP5VCZH, 0ZG1PNEP35, 0TKYAEJ8N7]
sources:
  - url: https://en.wikipedia.org/wiki/Similarity_(geometry)
---

## Definition

Let $n \ge 1$ and consider the [[Euclidean space|0XPR1E85W0]] $\R^n$ with the [[Euclidean distance|0NPPGHQNXQ]] $\norm{x - y}$.
A *similarity transformation* of $\R^n$ is a bijection $f \colon \R^n \to \R^n$ for which there is a real number $r > 0$ such that
$$\norm{f(x) - f(y)} = r \norm{x - y} \quad \text{for all } x, y \in \R^n.$$
The number $r$ is the *ratio* of $f$.

## Remarks

A similarity of ratio $1$ is an [[isometry|02HPP5VCZH]].
Every similarity of ratio $r$ has the form $f(x) = rAx + b$ with $A$ in the [[orthogonal group|0TKYAEJ8N7]] $\mathrm{O}(n)$ and $b \in \R^n$.
So every similarity is a [[homothety|0ZG1PNEP35]] followed by an isometry, and the similarities of $\R^n$ form a group.
Similarities preserve angles and ratios of distances.

Example: $f(x) = 3x + b$ on $\R^n$ is a similarity of ratio $3$.
Check: $\norm{f(x) - f(y)} = \norm{3x - 3y} = 3 \norm{x - y}$.

Non-example: $g(x_1, x_2) = (2x_1, x_2)$ on $\R^2$ is a bijection.
It doubles the distance between $(0, 0)$ and $(1, 0)$ but keeps the distance between $(0, 0)$ and $(0, 1)$, so no single ratio $r$ works.