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 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 formula_1 is the parametric representation of a regular curve in the plane with its curvature nowhere 0 and formula_2 its curvature radius and formula_3 the unit normal pointing to the curvature center, then formula_4 describes the evolute of the given curve. For formula_5 and formula_6 one gets formula_7  and formula_8 Properties of the evolute. In order to derive properties of a regular curve it is advantageous to use the arc length formula_9 of the given curve as its parameter, because of formula_10 and formula_11 (see Frenet–Serret formulas). Hence the tangent vector of the evolute formula_12 is: formula_13 From this equation one gets the following properties of the evolute: "Proof" of the last property: Let be formula_15 at the section of consideration. An involute of the evolute can be described as follows: formula_18 where formula_19 is a fixed string extension (see Involute of a parameterized curve ). With formula_20  and formula_21 one gets formula_22 That means: For the string extension formula_23 the given curve is reproduced. "Proof:" A parallel curve with distance formula_24 off the given curve has the parametric representation formula_25 and the radius of curvature formula_26 (see parallel curve). Hence the evolute of the parallel curve is formula_27 Examples. Evolute of a parabola. For the parabola with the parametric representation formula_28 one gets from the formulae above the equations: formula_29 formula_30 which describes a semicubic parabola Evolute of an ellipse. For the ellipse with the parametric representation formula_31 one gets: formula_32 formula_33 These are the equations of  a non symmetric astroid. Eliminating parameter formula_34 leads to the implicit representation formula_35 Evolute of a cycloid. For the cycloid with the parametric representation formula_36 the evolute will be: formula_37 formula_38 which describes a transposed replica of itself. Evolutes of some curves. The evolute Radial curve. A curve with a similar definition is the radial of a given curve. For each point on the curve take the vector from the point to the center of curvature and translate it so that it begins at the origin. Then the locus of points at the end of such vectors is called the  radial of the curve. The equation for the radial is obtained by removing the and terms from the equation of the evolute. This produces <math display="block">(X, Y)= \left(-y'\frac