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 be a smooth, regular submanifold in . For each point in and each vector , based at and normal to , we associate the point . This defines a Lagrangian map, called the normal map. The caustic of the normal map is the evolute of . Evolutes are closely connected to involutes: A curve is the evolute of any of its involutes. History Apollonius ( 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 tautochrone curve, 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. Evolute of a parametric curve If is the parametric representation of a regular curve in the plane with its curvature nowhere 0 and its curvature radius and the unit normal pointing to the curvature center, then describes the evolute of the given curve. For and one gets and Properties of the evolute In order to derive properties of a regular curve it is advantageous to use the arc length of the given curve as its parameter, because of and (see Frenet–Serret formulas). Hence the tangent vector of the evolute is: From this equation one gets the following properties of the evolute: At points with the evolute is not regular. That means: at points with maximal or minimal curvature (vertices of the given curve) the evolute has cusps. (See the diagrams of the evolutes of the parabola, the ellipse, the cycloid and the nephroid.) For any arc of the evolute