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Folium of Descartes


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

$\displaystyle x$ $\textstyle =$ $\displaystyle {3at\over 1+t^3}$ (1)
$\displaystyle y$ $\textstyle =$ $\displaystyle {3at^2\over 1+t^3}.$ (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 $-1$.

\begin{figure}\begin{center}\BoxedEPSF{FoliumOfDescartesInfo.epsf scaled 1000}\end{center}\end{figure}

The Curvature and Tangential Angle of the folium of Descartes, illustrated above, are

$\displaystyle \kappa(t)$ $\textstyle =$ $\displaystyle {2(1+t^3)^4\over 3(1+4t^2-4t^3-4t^5+4t^6+t^8)^{3/2}}$ (3)
$\displaystyle \phi(t)$ $\textstyle =$ $\displaystyle {1\over 2}\left[{\pi+\tan^{-1}\left({1-2t^3\over t^4-2t}\right)-\tan^{-1}\left({2t^3-1\over t^4-2t}\right)}\right].$  

Converting the parametric equations to Polar Coordinates gives

$\displaystyle r^2$ $\textstyle =$ $\displaystyle {(3at)^2(1+t^2)\over (1+t^3)^2}$ (5)
$\displaystyle \theta$ $\textstyle =$ $\displaystyle \tan^{-1}\left({y\over x}\right)= \tan^{-1} t,$ (6)

d\theta={dt\over 1+t^2}.
\end{displaymath} (7)

The Area enclosed by the curve is
$\displaystyle A$ $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\int r^2\,d\theta ={\textstyle{1\over 2}}\int_0^\infty {(3at)^2(1+t^2)\over (1+t^3)^2} {dt\over 1+t^2}$  
  $\textstyle =$ $\displaystyle {\textstyle{3\over 2}} a^2 \int_0^\infty {3t^2\,dt\over (1+t^3)^2}.$ (8)

Now let $u\equiv 1+t^3$ so $du=3t^2\,dt$
A= {\textstyle{3\over 2}} a^2 \int_1^\infty {du\over u^2} = ...
...{\textstyle{3\over 2}} a^2(-0+1) = {\textstyle{3\over 2}} a^2.
\end{displaymath} (9)

In Cartesian Coordinates,
x^3+y^3={(3at)^3(1+t^3)\over (1+t^3)^3} = {(3at)^3\over (1+t^3)^2} = 3axy
\end{displaymath} (10)

(MacTutor Archive). The equation of the Asymptote is
\end{displaymath} (11)


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.''

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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© 1996-9 Eric W. Weisstein