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
discussed evolutes in Book V of his Conics. However, Huygens is sometimes credited with being the first to study them. 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.
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 . 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 the evolute has cusps.
For any arc of the evolute that does not include a cusp, the length of the arc equals the difference between the radii of curvature at its endpoints. This fact leads to an easyproof of the Tait–Kneser theorem on nesting of osculating circles.
The normals of the given curve are tangents to the evolute. Hence: the evolute is the envelope of the normals of the given curve.
At sections of the curve with or the curve is an involute of its evolute.
Proof of the last property:
Let be at the section of consideration. An involute of the evolute can be described as follows: where is a fixed string extension.
With and one gets That means: For the string extension the given curve is reproduced.
Parallel curves have the same evolute.
Proof: A parallel curve with distance off the given curve has the parametric representation and the radius of curvature . Hence the evolute of the parallel curve is
For the ellipse with the parametric representation one gets: These are the equations of a non symmetric astroid. Eliminating parameter leads to the implicit representation
Evolute of a cycloid
For the cycloid with the parametric representation the evolute will be: which describes a transposed replica of itself.
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
