Evolute

In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curve. The evolute of a circle is therefore a single point at its center.^[1] Equivalently, an evolute is the envelope of the normals to a curve.

The evolute of a curve, a surface, or more generally a submanifold, is the caustic of the normal map. Let $M$ be a smooth, regular submanifold in $R n$ . For each point $p$ in $M$ and each vector $v$ , based at $p$ and normal to $M$ , we associate the point $p + v$ . This defines a Lagrangian map, called the normal map. The caustic of the normal map is the evolute of $M$ .^[2]

Evolutes are closely connected to involutes: A curve is the evolute of any of its involutes.

^ Weisstein, Eric W. "Circle Evolute". MathWorld.
^ Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S. M. (1985). The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1. Birkhäuser. ISBN 0-8176-3187-9.

[1] Weisstein, Eric W. "Circle Evolute". MathWorld.

[Arnold-2] Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S. M. (1985). The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1. Birkhäuser. ISBN 0-8176-3187-9.

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