---
tag: 7JQS6VKFEZ
kind: concept
name: Curvature of a curve
msc: [53A04]
sources:
  - url: https://en.wikipedia.org/wiki/Curvature
---

## Summary

The curvature of a curve measures how fast its direction turns per unit of length travelled.
For a unit-speed curve it is the length of the acceleration vector.
A straight line has curvature $0$ and a circle of radius $r$ has curvature $1/r$.
Curvature, together with torsion, determines a space curve up to a rigid motion.

## Explanations

### L1 human

Drive a car along a road at constant speed and watch the steering wheel.
On a straight road the wheel is centred, and on a sharp bend it is turned far.
Curvature is the number that records how far the wheel is turned at each point of the road.
A circle of radius $2$ metres has curvature $1/2$ per metre at every point, and a larger circle bends more gently and has smaller curvature.