Folium of Descartes
A plane curve proposed by Descartes to challenge Fermat's extremum-finding techniques. In parametric form,
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.
| 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
![画像:tan^(-1)[(t(t^3-2))/((2t^3-1))]+H(t-2^(-1/3)),](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/FoliumofDescartes/Inline20.svg){kind=link}
where H(t) is the Heaviside step function.
Converting the parametric equations to polar coordinates gives
so the polar equation is
The area enclosed by the curve is
The arc length of the loop is given by
See also
FoliumExplore with Wolfram|Alpha
More things to try:
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." https://mathshistory.st-andrews.ac.uk/Curves/Foliumd/.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.Referenced on Wolfram|Alpha
Folium of DescartesCite this as:
Weisstein, Eric W. "Folium of Descartes." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FoliumofDescartes.html