---
tag: QP9DY5K8QW
kind: concept
name: Total curvature
msc: [53A04]
sources:
  - url: https://en.wikipedia.org/wiki/Total_curvature
same_as:
  wikidata: Q4370338
  wikipedia: Total curvature
---

## Summary

The total curvature of a curve is the integral of its [[curvature|7JQS6VKFEZ]] with respect to arc length.
It measures how much the direction of the curve turns in all, from start to end.
For closed curves it is bounded below by $2\pi$, by Fenchel's theorem, and knotted curves need more than $4\pi$, by the Fáry–Milnor theorem.

## Explanations

### L1 human

Walk once around a closed path and add up all the turning you do, ignoring left and right.
That sum is the total curvature.
Walking once around a circle you turn by a full turn, so the total curvature is $2\pi$.
A wiggly path makes you turn back and forth, and the total grows.