Evolute

An evolute is the locus of centers of curvature (the envelope) of a plane curve's normals. The original curve is then said to be the involute of its evolute. Given a plane curve represented parametrically by , the equation of the evolute is given by

			(1)
			(2)

where are the coordinates of the running point, is the radius of curvature

(3)

and is the angle between the unit tangent vector

(4)

and the x-axis,

			(5)
			(6)

Combining gives

			(7)
			(8)

The definition of the evolute of a curve is independent of parameterization for any differentiable function (Gray 1997). If is the evolute of a curve , then is said to be the involute of . The centers of the osculating circles to a curve form the evolute to that curve (Gray 1997, p. 111).

The following table lists the evolutes of some common curves, some of which are illustrated above.