Folium of Descartes
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 runsfrom 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.
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