 Folium of Descartes

For nonzero real $a$, the folium of Descartes is the algebraic curve $x^3+y^3=3axy$, with rational parametrization of an algebraic curveIt is symmetric about $y=x$ and has a node at the origin, with the coordinate axes as tangents. For $a>0$, $0\le t<\infty$ traces a loop in the first quadrant, returning to the origin as $t\to \infty$. Its enclosed area, by Green's theorem, isThe other branches approach the asymptote $x+y+a=0$ as $t\to -1$. Negative $a$ rotates the positive-$a$ picture through $\pi$.

The Folium of Descartes is a plane algebraic curve defined by the equation: \[ x^3 + y^3 - 3axy = 0 \] where \( a \) is a parameter that determines the shape and position of the curve. The term "folium" comes from the Latin word for "leaf," referring to the leaf-like shape of the curve.