# Definition:Evolute

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## Definition

Consider a curve $C$ embedded in a plane.

The **evolute** of $C$ is the locus of the centers of curvature of each point on $C$.

## Also see

- Definition:Involute
- Results about
**evolutes**can be found**here**.

## Historical Note

The concept of the evolute of a curve in the plane was first introduced by Apollonius of Perga in his *Conics*.

However, the first detailed study of the evolute was undertaken by Christiaan Huygens during his analysis of the cycloid in his $1673$ treatise *Horologium Oscillatorium sive de Motu Pendularium*.

Some sources fail to register Apollonius's interest.

## Linguistic Note

The word **evolute** derives from the Latin word **evolvere**, which means **to unwind**.

This stems from the geometric property Curve is Involute of Evolute:

- if a cord is wrapped around the evolute of a curve $C$, then the end of that cord will trace out $C$.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.23$: Evolutes and Involutes. The Evolute of a Cycloid