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 $(f(t),g(t))$, the equation of the evolute is given by

$$x = f-R\sin\tau$$ (1)

$$y = g+R\cos\tau,$$ (2)

where $(x,y)$ are the coordinates of the running point, $R$ is the Radius of Curvature

$$R={(f'^2+g'^2)^{3/2}\over f'g''-f''g'},$$ (3)

and $\tau$ is the angle between the unit Tangent Vector

$$\hat{\bf T} = {{\bf x}'\over \vert{\bf x}'\vert} = {1\over \sqrt{f'^2+g'^2}} {\left[{\matrix{f'\cr g'\cr}}\right]}$$ (4)

and the x-Axis,

$$\cos\tau = \hat {\bf T}\cdot\hat{\bf x}$$ (5)

$$\sin\tau = \hat {\bf T}\cdot\hat{\bf y}.$$ (6)

Combining gives

$$x = f-{(f'^2+g'^2)g'\over f'g''-f''g'}$$ (7)

$$y = g+{(f'^2+g'^2)g'\over f'g''-f''g'}.$$ (8)

The definition of the evolute of a curve is independent of parameterization for any differentiable function (Gray 1993). If $E$ is the evolute of a curve $I$, then $I$ is said to be the Involute of $E$. The centers of the Osculating Circles to a curve form the evolute to that curve (Gray 1993, p. 90).

The following table lists the evolutes of some common curves.

Curve	Evolute
Astroid	Astroid 2 times as large
Cardioid	Cardioid 1/3 as large
Cayley's Sextic	Nephroid
Circle	point (0, 0)
Cycloid	equal Cycloid
Deltoid	Deltoid 3 times as large
Ellipse	Lamé Curve
Epicycloid	enlarged Epicycloid
Hypocycloid	similar Hypocycloid
Limaçon	Circle Catacaustic for a point source
Logarithmic Spiral	equal Logarithmic Spiral
Nephroid	Nephroid 1/2 as large
Parabola	Neile's Parabola
Tractrix	Catenary

See also Involute, Osculating Circle

References

Cayley, A. ``On Evolutes of Parallel Curves.'' Quart. J. Pure Appl. Math. 11, 183-199, 1871.

Dixon, R. ``String Drawings.'' Ch. 2 in Mathographics. New York: Dover, pp. 75-78, 1991.

Gray, A. ``Evolutes.'' §5.1 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 76-80, 1993.

Jeffrey, H. M. ``On the Evolutes of Cubic Curves.'' Quart. J. Pure Appl. Math. 11, 78-81 and 145-155, 1871.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 40 and 202, 1972.

Lee, X. ``Evolute.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/Evolute_dir/evolute.html.

Lockwood, E. H. ``Evolutes and Involutes.'' Ch. 21 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 166-171, 1967.

Yates, R. C. ``Evolutes.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 86-92, 1952.