---
tag: Y3ZNV24V2C
kind: concept
name: Evolute
msc: [53A04]
sources:
  - url: https://en.wikipedia.org/wiki/Evolute
same_as:
  wikidata: Q658654
  wikipedia: Evolute
---

## Summary

The evolute of a plane curve is the curve traced by its centres of curvature.
Equivalently, it is the envelope of the normal lines of the curve.
The evolute stops moving, and typically has a cusp, at the vertices of the curve, the points where the derivative of the curvature vanishes.

## Explanations

### L1 human

At each point of a curve, draw the [[osculating circle|NWSFMCRWHJ]], the circle that hugs the curve best.
As the point moves, the centre of that circle moves too, and its path is the evolute.
For a circle, the centre never moves, so the evolute is a single point.
For the parabola $y = x^2$, the evolute is a curve with a sharp cusp, called a semicubical parabola.