Folium of Descartes
= Folium of Descartes
{title2=$x^3+y^3=3axy$}
For nonzero real $a$, the <folium of Descartes> is the <algebraic curve> $x^3+y^3=3axy$, with <rational parametrization of an algebraic curve>
$$
x=\frac{3at}{1+t^3},\qquad y=\frac{3at^2}{1+t^3},\qquad t\ne-1.
$$
It is symmetric about $y=x$ and has a node at the origin, with the coordinate axes as tangents. For $a>0$, $0\leq t<\infty$ traces a loop in the first quadrant, returning to the origin as $t\to\infty$. Its enclosed <area>, by <Green's theorem>, is
$$
\frac12\int_0^\infty(xy'-yx')\,dt
=\frac{9a^2}{2}\int_0^\infty\frac{t^2\,dt}{(1+t^3)^2}=\frac{3a^2}{2}.
$$
The other branches approach the <asymptote> $x+y+a=0$ as $t\to-1$. Negative $a$ rotates the positive-$a$ picture through $\pi$.