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.
Evolutes are closely connected to involutes: A curve is the evolute of any of its involutes.
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 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
Properties of the evolute
The normal at point P is the tangent at the curvature center C.
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 (s. parabola, ellipse, nephroid).
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 easy proof of the Tait–Kneser theorem on nesting of osculating circles.
The normals of the given curve at points of nonzero curvature are tangents to the evolute, and the normals of the curve at points of zero curvature are asymptotes 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. (In the diagram: The blue parabola is an involute of the red semicubic parabola, which is actually the evolute of the blue parabola.)
Proof of the last property:
Let be at the section of consideration. An involute of the evolute can be described as follows:
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 x and y terms from the equation of the evolute. This produces
^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.