Folium of Descartes

$\begin{figure}\begin{center}\BoxedEPSF{folium_descartes.epsf}\end{center}\end{figure}$

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

. The left wing is generated as

runs from

to 0, the loop as

runs from 0 to $\infty$ , and the right wing as

runs from $-\infty$ to

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

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

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)

$\begin{displaymath} d\theta={dt\over 1+t^2}. \end{displaymath}$

(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 $du=3t^2\,dt$

$\begin{displaymath} A= {\textstyle{3\over 2}} a^2 \int_1^\infty {du\over u^2} = ... ...{\textstyle{3\over 2}} a^2(-0+1) = {\textstyle{3\over 2}} a^2. \end{displaymath}$

(9)

In Cartesian Coordinates,

$\begin{displaymath} x^3+y^3={(3at)^3(1+t^3)\over (1+t^3)^3} = {(3at)^3\over (1+t^3)^2} = 3axy \end{displaymath}$

(10)

(MacTutor Archive). The equation of the Asymptote is

$\begin{displaymath} y=-a-x. \end{displaymath}$

(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.