A plane curve proposed by Descartes to challenge Fermat's extremum-finding techniques.
In parametric form,

(1) | |||

(2) |

The curve has a discontinuity at . The left wing is generated as runs from to 0, the loop as runs from 0 to , and the right wing as runs from to .

The Curvature and Tangential Angle of the folium of Descartes, illustrated above, are

(3) | |||

(4) |

Converting the parametric equations to
Polar Coordinates gives

(5) | |||

(6) |

so

(7) |

(8) |

Now let so

(9) |

(10) |

(11) |

**References**

Gray, A. *Modern Differential Geometry of Curves and Surfaces.* Boca Raton, FL: CRC Press, pp. 59-62, 1993.

Lawrence, J. D. *A Catalog of Special Plane Curves.* New York: Dover, pp. 106-109, 1972.

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

Stroeker, R. J. ``Brocard Points, Circulant Matrices, and Descartes' Folium.'' *Math. Mag.* **61**, 172-187, 1988.

Yates, R. C. ``Folium of Descartes.'' In *A Handbook on Curves and Their Properties.* Ann Arbor, MI: J. W. Edwards, pp. 98-99, 1952.

© 1996-9

1999-05-26