# folium

The folium of Descartes (green) with asymptote (blue) when *a* = 1.

A folium is a curve, first described by Johannes Kepler in 1609, that corresponds to the general equation

(*x *^{2} + *y *^{2})(*y *^{2} + *x*(*x* + *b*)) = 4*axy *^{2}, in
Cartesian form, or

*r* = –*b* cos *θ* + 4a cos *θ* sin^{2}*θ*, in polar
coordinates.

The Latin *folium* means "leap-shaped." Three types, known as the **simple folium**, the **bifolium** (or **double
folium**), and the **trifolium**, correspond to the
cases when *b* = 4*a*, *b* = 0, and *b* = *a*,
respectively.

The **folium of Descartes** is given by the Cartesian equation *x*^{ 3} + *y *^{3} = 3*axy* and was
first discussed by René Descartes in 1638.
Although he found the correct shape of the curve in the positive quadrant,
he wrongly thought that this leaf shape was repeated in each quadrant like
the four petals of a flower. The problem to determine the tangent to the curve was proposed to Gilles de Roberval who, having made the same
incorrect assumption, called the curve **fleur de jasmin** after the four-petal jasmine bloom – a name that was later dropped.
The folium of Descartes has an asymptote *x* + *y* + *a* = 0.