A plane curve proposed by Descartes to challenge Fermat's extremum-finding techniques.
In parametric form,
(1) | |||
(2) |
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) |
(7) |
(8) |
(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 Eric W. Weisstein