Evolute

An evolute is, in differential geometry, the locus of centers of curvature of a curve. Equivalently, it can be described as the envelope of the normals to the original curve. For a plane curve parameterized by r(t) with nonzero curvature kappa(t), the evolute is given by E(t) = r(t) + (1/kappa(t)) N(t), where N(t) is the unit principal normal. The radius of curvature at each point is the distance from the point on the curve to its center of curvature on the evolute.

Key properties include that the distance from a point on the original curve to its evolute equals the local radius of curvature, and that E'(t) is proportional to the normal direction and vanishes precisely where the curvature has an extremum (kappa'(t) = 0). Consequently, cusps appear on the evolute at points corresponding to vertices of the original curve, where curvature is extremal. A classic example is a circle, whose curvature is constant; its evolute collapses to the circle’s center.

Examples and related concepts:

- Ellipse: The evolute of an ellipse is a four-cusped curve, often described as an astroid-like figure aligned with the ellipse’s axes.

- Involutes and evolutes are closely related: the involute of a curve has the original curve as its evolute.

- In three dimensions, the concept generalizes to the locus of centers of curvature of a space curve, forming a space curve that lies in the curve’s osculating plane when curvature is nonzero.

Overview: The evolute encodes how a curve bends, serving as a geometric summary of its curvature profile and providing a basis for constructions in classical geometry and computer graphics.