Folium of Descartes

The Folium of Descartes ("folium" means "leaf") is a curve defined implicitly by the equation $x^3+y^3=3axy$, first proposed by René Descartes in 1638 as a challenge to Pierre de Fermat. Descartes originally believed the curve repeated a flower-petal pattern into every quadrant, but the true shape — a single leaf-shaped loop in the first quadrant, with two branches trailing off toward a diagonal asymptote in the third quadrant — was not correctly worked out until later that century. The loop is symmetric about the line $y=x$ and crosses itself at the origin. Substituting $t=y/x$ gives a clean parametric form:

$x(t) = \dfrac{3at}{1+t^3}$

$y(t) = \dfrac{3at^2}{1+t^3}$