# Curves

### Trifolium

- Cartesian equation:
- $(x^{2} + y^{2})(y^{2} + x(x + a)) = 4axy^{2}$

- Polar equation:
- $r = a \cos \theta (4\sin^{2} \theta - 1)$

### Description

The general form of the folium is given by the formula$(x^{2} + y^{2})(y^{2} + x(x + b)) = 4axy^{2}$

or, in polar coordinates
$r = -b \cos\theta + 4a \cos\theta \sin^{2}\theta$.

The word *folium*means 'leaf-shaped'.

There are three special forms of the folium, the simple folium, the double folium and the trifolium. These correspond to the cases

$b = 4a, b = 0, b = a$

respectively in the formula for the general form.
The graph plotted above is the trifolium. There are separate entries for the simple folium and the double folium.