---
tag: S4BZ697DSH
kind: concept
name: Torsion of a curve
msc: [53A04]
sources:
  - url: https://en.wikipedia.org/wiki/Torsion_of_a_curve
same_as:
  wikidata: Q1892644
  wikipedia: Torsion of a curve
---

## Summary

The torsion of a space curve measures how fast the curve twists out of its osculating plane.
This is the plane that best fits the curve at a point.
A plane curve has torsion $0$ everywhere.
With [[curvature|7JQS6VKFEZ]], torsion appears in the Frenet–Serret formulas and determines a space curve up to a rigid motion.

## Explanations

### L1 human

Curvature tells how a curve bends inside a plane, and torsion tells how that plane itself turns.
Picture the thread of a screw: it bends around the axis and also climbs, so it leaves every plane.
A circle lies flat in one plane and has torsion $0$.
The helix $t \mapsto (\cos t, \sin t, t)$ climbs at a steady rate and has constant torsion $1/2$.