The path traced out by a point on the Edge of a Circle of Radius rolling on the outside of a Circle of Radius .

It is given by the equations

(1) | |||

(2) |

(3) | |||

(4) |

(5) |

(6) |

(7) |

Note that is the parameter here,

(8) |

To get Cusps in the epicycloid, , because then rotations of bring the point on
the edge back to its starting position.

(9) |

so

(10) |

An epicycloid with one cusp is called a Cardioid, one with two cusps is called a Nephroid, and one with five cusps is called a Ranunculoid.

-epicycloids can also be constructed by beginning with the Diameter of a Circle, offsetting one end by a series of steps while at the same time offsetting the other end by steps times as large. After traveling around the Circle once, an -cusped epicycloid is produced, as illustrated above (Madachy 1979).

Epicycloids have Torsion

(11) |

(12) |

**References**

Bogomolny, A. ``Cycloids.'' http://www.cut-the-knot.com/pythagoras/cycloids.html.

Lawrence, J. D. *A Catalog of Special Plane Curves.* New York: Dover, pp. 160-164 and 169, 1972.

Lee, X. ``Epicycloid and Hypocycloid.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/EpiHypocycloid_dir/epiHypocycloid.html.

MacTutor History of Mathematics Archive. ``Epicycloid.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Epicycloid.html.

Madachy, J. S. *Madachy's Mathematical Recreations.* New York: Dover, pp. 219-225, 1979.

Wagon, S. *Mathematica in Action.* New York: W. H. Freeman, pp. 50-52, 1991.

Yates, R. C. ``Epi- and Hypo-Cycloids.'' *A Handbook on Curves and Their Properties.* Ann Arbor, MI: J. W. Edwards, pp. 81-85, 1952.

© 1996-9

1999-05-25