## 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 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)

The Area enclosed by the curve is
 (8)

Now let so
 (9)

In Cartesian Coordinates,
 (10)

(MacTutor Archive). The equation of the Asymptote is
 (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.