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) |

(4) |

(5) | |||

(6) |

Combining gives

(7) | |||

(8) |

The definition of the evolute of a curve is independent of parameterization for any differentiable function (Gray 1993). 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 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 |

**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.

© 1996-9

1999-05-25