# Evolute of a curve

## Definition

Let $\gamma$ be a smooth planar curve, space curve, or more generally, a smooth curve in a Riemannian manifold. The evolute of $\gamma$ is defined as the locus of the center of curvature for each point on $\gamma$, with respect to $\gamma$.

## Facts

### For a planar curve

• The length of the evolute between any two points is the total variation in the radius of curvature between those points.
• The radial line to a point on the curve from its center of curvature, is tangent to the evolute at the center of curvature.