---
tag: GYEX4BH5WM
kind: concept
name: Conformal radius
msc: [30C35, 31A15]
sources:
  - url: https://en.wikipedia.org/wiki/Conformal_radius
same_as:
  wikidata: Q5160255
  wikipedia: Conformal radius
---

## Summary

The conformal radius of a simply connected domain at a point measures the size of the domain as seen from that point.
It is the factor by which the Riemann map from the unit disc stretches lengths at the centre.
It is comparable to the distance to the boundary, up to a factor $4$.
It also appears as the constant term of the Green's function at its pole.

## Explanations

### L1 human

Map the unit disc onto a region so that the centre goes to a chosen point and the map has positive derivative there.
The derivative at the centre is the conformal radius.
For a disc of radius $R$ and its centre, the map $w \mapsto Rw$ works, so the conformal radius is $R$.
From a point near the edge of a region, the region looks small, and the conformal radius is small too.