---
tag: ADZAXRA6QT
kind: concept
name: Quasicircle
msc: [30C62]
sources:
  - url: https://en.wikipedia.org/wiki/Quasicircle
same_as:
  wikidata: Q7269488
  wikipedia: Quasicircle
---

## Summary

A quasicircle is the image of a circle under a [[quasiconformal mapping|SB8KG46NHP]] of the plane.
Quasicircles are Jordan curves, but they can be very rough, such as the Koch snowflake.
Ahlfors characterized them geometrically: a Jordan curve is a quasicircle exactly when it has no narrow necks or cusps, in a precise metric sense.
They appear as limit sets of quasi-Fuchsian groups and as Julia sets of some polynomials.

## Explanations

### L1 human

Take a round circle and deform the whole plane by a map that never squashes tiny circles too much.
The image of the circle is a quasicircle.
Such a curve may wiggle a lot at every scale, but it never comes back close to itself after a long detour.
For example, the boundary of a square is a quasicircle, while a curve with a sharp inward cusp is not.