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Diffgeom β

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.