# Definition:Evolute

## Definition

Consider a curve $\CC$ embedded in a plane.

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

## Examples

### Parabola

Let $P$ be a parabola embedded in the Cartesian plane whose equation is given by:

- $y^2 = 4 a x + 8 a^2$

The **evolute** of $P$ is the semicubical parabola whose equation is given by:

- $4 x^3 = 27 a y^2$

## Also see

- 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 $\CC$, then the end of that cord will trace out $\CC$.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.23$: Evolutes and Involutes. The Evolute of a Cycloid - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**evolute** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**evolute**