A plane curve proposed by Descartes to challenge Fermat's extremumfinding techniques. In parametric form,
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 .
In Cartesian coordinates,

(3)

(MacTutor Archive). The equation of the asymptote is

(4)

The curvature and tangential
angle of the folium of Descartes are
where
is the Heaviside step function.
Converting the parametric equations to polar coordinates gives
so the polar equation is

(9)

The area enclosed by the curve is
The arc length of the loop is given by
See also
Folium
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References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 218,
1987.Gray, A. Modern
Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca
Raton, FL: CRC Press, pp. 7782, 1997.Lawrence, J. D. A
Catalog of Special Plane Curves. New York: Dover, pp. 106109, 1972.MacTutor
History of Mathematics Archive. "Folium of Descartes." http://wwwgroups.dcs.stand.ac.uk/~history/Curves/Foliumd.html.Smith,
D. E. History
of Mathematics, Vol. 2: Special Topics of Elementary Mathematics. New
York: Dover, p. 328, 1958.Stroeker, R. J. "Brocard Points,
Circulant Matrices, and Descartes' Folium." Math. Mag. 61, 172187,
1988.Yates, R. C. "Folium of Descartes." In A
Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards,
pp. 9899, 1952.
Cite this as:
Weisstein, Eric W. "Folium of Descartes."
From MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/FoliumofDescartes.html
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