---
tag: FA8QE1SGP4
kind: concept
name: Principal curvature
msc: [53A05]
sources:
  - url: https://en.wikipedia.org/wiki/Principal_curvature
same_as:
  wikidata: Q1589551
  wikipedia: Principal curvature
---

## Summary

The principal curvatures of a surface at a point are the largest and the smallest bending of the surface over all tangent directions.
They are the eigenvalues of the [[shape operator|DQCD11X1VM]], and the corresponding eigenvectors are the principal directions.
Their product is the Gaussian curvature and their average is the mean curvature.

## Explanations

### L1 human

Stand on a saddle-shaped surface and look in different directions.
In one direction the surface curves up the most, in another it curves down the most.
These two extreme amounts of bending are the principal curvatures.
When they differ, the two directions are at right angles.
On a cylinder of radius $r$, one principal curvature is $1/r$, around the cylinder, and the other is $0$, along it.