# Evolute of a curve

From Diffgeom

## Definition

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

## 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.