It forms a loop in the first quadrant with a double point at the origin and asymptote

It is symmetrical about y = x.

The name comes from the Latin word folium which means "leaf".

The curve was featured, along with a portrait of Descartes, on an Albanian stamp in 1966.

The curve was first proposed by Descartes in 1638. Its claim to fame lies in an incident in the development of calculus. Descartes challenged Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines. Fermat solved the problem easily, something Descartes was unable to do. Since the invention of calculus, the slope of the tangent line can be found easily using implicit differentiation.

Since the equation is degree 3 in both x and y, and does not factor, it is difficult to solve for one of the variables. However, the equation in polar coordinates is:

which can be plotted easily. Another technique is to write y = px and solve for x and y in terms of p. This yields the parametric equations:

.

We can see that the parameter is related to the position on the curve as follows:

The folium of Descartes is related to the trisectrix of Maclaurin by affine transformation. To see this, start with the equation