Definition:Evolute

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 $C$.

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

• 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
• 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