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.

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 Rⁿ. 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.

Apollonius (c. 200 BC) discussed evolutes in Book V of his Conics. However, Huygens is sometimes credited with being the first to study them (1673). Huygens formulated his theory of evolutes sometime around 1659 to help solve the problem of finding the , which in turn helped him construct an isochronous pendulum. This was because the tautochrone curve is a cycloid, and the cycloid has the unique property that its evolute is also a cycloid. The theory of evolutes, in fact, allowed Huygens to achieve many results that would later be found using calculus.

Let γ(s) be a plane curve, parameterized by its arclength s. The unit tangent vector to the curve is, by virtue of the arclength parameterization,

and the unit normal to the curve is the unit vector N(s) perpendicular to T(s) chosen so that the pair (T,N) is positively oriented.

...
Wikipedia