---
tag: 2FQAKBVBX5
kind: concept
name: Conformal modulus
msc: [30C75, 30C62]
sources:
  - url: https://en.wikipedia.org/wiki/Extremal_length
---

## Summary

The conformal modulus of a family of curves in the plane is the reciprocal of its [[extremal length|HX72KT9D0G]].
It is a conformal invariant.
It gives the modulus of a quadrilateral and of a ring domain, which classify these domains up to conformal equivalence.
Moduli are the basic quantities in the geometric definition of [[quasiconformal maps|SB8KG46NHP]], which distort them by a bounded factor.

## Explanations

### L1 human

Two annuli $\set{r < \abs{z} < R}$ and $\set{r' < \abs{z} < R'}$ are conformally equivalent exactly when $R/r = R'/r'$.
So one number, essentially $\log(R/r)$, tells which annulus one has, and this is the modulus of the ring.
The conformal modulus of a curve family generalises this: many curves that are short give a large modulus, few long curves give a small one.
For the curves crossing a rectangle of width $a$ and height $b$ from left to right, the modulus is $b/a$.