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Folium of Descartes

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FoliumofDescartes

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

x=(3at)/(1+t^3)
(1)
y=(3at^2)/(1+t^3).
(2)

The curve has a discontinuity at t=-1. The left wing is generated as t runs from -1 to 0, the loop as t runs from 0 to infty, and the right wing as t runs from -infty to -1.

In Cartesian coordinates,

x^3+y^3=3axy
(3)

(MacTutor Archive). The equation of the asymptote is

y=-a-x.
(4)

The curvature and tangential angle of the folium of Descartes are

kappa(t)=(2(1+t^3)^4)/(3a(1+4t^2-4t^3-4t^5+4t^6+t^8)^(3/2))
(5)
phi(t)=tan^(-1)[(t(t^3-2))/((2t^3-1))]+H(t-2^(-1/3)),
(6)

where H(t) is the Heaviside step function.

Converting the parametric equations to polar coordinates gives

r^2=(9a^2t^2(1+t^2))/((1+t^3)^2)
(7)
theta=tan^(-1)t,
(8)

so the polar equation is

r=(3asecthetatantheta)/(1+tan^3theta).
(9)
FoliumofDescartesArea

The area enclosed by the curve is

A=1/2intr^2dtheta
(10)
=int_0^(pi/2)(3asecthetatantheta)/(1+tan^3theta)dtheta
(11)
=3/2a^2.
(12)

The arc length of the loop is given by

s=3aint_0^infty(sqrt(1+t^2(4-4t-4t^3+4t^4+t^6)))/((1+t^3)^2)dt
(13)
=4.917488...a.
(14)

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. 77-82, 1997.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.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, 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.

Cite this as:

Weisstein, Eric W. "Folium of Descartes." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FoliumofDescartes.html

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