---
tag: D3GYHF6NAX
kind: concept
name: Similarity transformation
msc: [51M05, 51N20]
sources:
  - url: https://en.wikipedia.org/wiki/Similarity_(geometry)
---

## Summary

A similarity transformation of Euclidean space is a bijection that multiplies all distances by the same positive factor, its ratio.
Similarities are the compositions of [[isometries|02HPP5VCZH]] and [[homotheties|0ZG1PNEP35]], and they preserve angles, shapes and ratios of lengths.
Two figures are similar when a similarity maps one onto the other.
The theory of similar figures, such as similar triangles, rests on this notion.

## Explanations

### L1 human

A similarity is a change of scale followed by a rigid motion.
Photographs enlarged or reduced, maps drawn at a scale, and models of buildings all use similarities.
The map that doubles every coordinate, $(x, y) \mapsto (2x, 2y)$, is a similarity of ratio $2$: every distance doubles.
Angles do not change, so a triangle is mapped to a triangle with the same angles.