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.

[1] 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.[2] 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.

The theory of evolutes, in fact, allowed Huygens to achieve many results that would later be found using calculus.

is the parametric representation of a regular curve in the plane with its curvature nowhere 0 and

the unit normal pointing to the curvature center, then

In order to derive properties of a regular curve it is advantageous to use the arc length

Hence the tangent vector of the evolute

An involute of the evolute can be described as follows:

is a fixed string extension (see Involute of a parameterized curve ).

That means: For the string extension

Proof: A parallel curve with distance

off the given curve has the parametric representation

For the parabola with the parametric representation

which describes a semicubic parabola For the ellipse with the parametric representation

These are the equations of a non symmetric astroid.

leads to the implicit representation

For the cycloid with the parametric representation

which describes a transposed replica of itself.

[7] One instance of this relation is that the evolute of an Euler spiral is a spiral with Cesàro equation

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.