# Evolute

Mathematica Notebook for This Page.

## History

Studied by Huygens in 1673.

## Description

Evolute is a method of deriving a new curve based on a given curve. It is the locus of the centers of tangent circles of the given curve.

Evolute of a ellipse.

## Formula

Given a curve in parametric form {x[t], y[t]}, its evolute is

{x + (y'*(x'^2 + y'^2)) / ( y'*x'' - x'*y''), y + (x'*(x'^2 + y'^2)) / (-(y'*x'') + x'*y'')}

## Properties

### Parallels and Evolute

Theorem: The locus of Cusps of a curve C's parallel curves is the evolute of C. This is a alternative definition of evolute. See the Parallel page.

### Evolute and Involute

If curve A is the involute of curve B, then curve B is the evolute of curve A. The converse is true locally, that is: If curve B is the evolute of curve A, then any part of curve A is the involute of some parts of B.

### Curves relations by evolute and involute

Base Curve | Evolute |
---|---|

cardioid | cardioid scaled by 1/3 |

nephroid | nephroid 1/2 |

astroid | astroid 2 |

deltoid | deltoid 3 |

epicycloid | epicycloid |

hypocycloid | hypocycloid |

cycloid | cycloid |

Cayley's sextic | nephroid |

parabola | semicubic parabola |

limacon of Pascal | catacaustic of a circle |

equiangular spiral | equiangular spiral |

tractrix | catenary |

## Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

Robert Yates: Curves and Their Properties.

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