A polar curve also called Lemniscate of Bernoulli which is the Locus of points the product of whose distances
from two points (called the Foci) is a constant. Letting the Foci be located at , the Cartesian equation is

(1) |

(2) |

(3) |

(4) |

(5) | |||

(6) |

The bipolar equation of the lemniscate is

(7) |

(8) |

(9) |

Jakob Bernoulli published an article in *Acta Eruditorum* in 1694 in which he called
this curve the lemniscus (``a pendant ribbon''). Jakob Bernoulli was not aware that the curve he was describing was a
special case of Cassini Ovals which had been described by Cassini in 1680. The general properties of
the lemniscate were discovered by G. Fagnano in 1750 (MacTutor Archive). Gauss's and
Euler's investigations of the Arc Length of the curve led to later work on Elliptic
Functions.

The Curvature of the lemniscate is

(10) |

(11) |

(12) |

(13) |

(14) |

(15) |

(16) |

and

(17) |

(18) |

(19) |

(20) |

(21) |

The Area of one loop of the lemniscate is

(22) |

**References**

Ayoub, R. ``The Lemniscate and Fagnano's Contributions to Elliptic Integrals.'' *Arch. Hist. Exact Sci.* **29**, 131-149, 1984.

Borwein, J. M. and Borwein, P. B. *Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.*
New York: Wiley, 1987.

Gray, A. ``Lemniscates of Bernoulli.'' §3.2 in *Modern Differential Geometry of Curves and Surfaces.*
Boca Raton, FL: CRC Press, pp. 39-41, 1993.

Lawrence, J. D. *A Catalog of Special Plane Curves.* New York: Dover, pp. 120-124, 1972.

Le Lionnais, F. *Les nombres remarquables.* Paris: Hermann, p. 37, 1983.

Lee, X. ``Lemniscate of Bernoulli.'' http://www.best.com/~xah/SpecialPlaneCurves_dir/LemniscateOfBernoulli_dir/lemniscateOfBernoulli.html.

Lockwood, E. H. *A Book of Curves.* Cambridge, England: Cambridge University Press, 1967.

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

Yates, R. C. ``Lemniscate.'' *A Handbook on Curves and Their Properties.* Ann Arbor, MI: J. W. Edwards, pp. 143-147, 1952.

© 1996-9

1999-05-26