---
tag: 550P8PFMVC
kind: concept
name: Geodesic curvature
msc: [53A05]
sources:
  - url: https://en.wikipedia.org/wiki/Geodesic_curvature
same_as:
  wikidata: Q1460595
  wikipedia: Geodesic curvature
---

## Summary

The geodesic curvature of a curve on a surface measures how much the curve bends within the surface.
The acceleration of a unit-speed curve on a surface splits into a part normal to the surface and a part tangent to it.
The geodesic curvature is the size of the tangent part, with a sign.
Curves with geodesic curvature zero are the geodesics, the straightest curves of the surface.
It is intrinsic: it depends only on the first fundamental form.

## Explanations

### L1 human

Walk on a large ball along a path.
Some of the bending of your path is forced on you by the ball itself; the rest comes from how you steer.
Geodesic curvature measures only the steering.
Walking along the equator you never steer, so its geodesic curvature is zero, while a small circle around the north pole needs constant steering.