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.

Associated Curves

Definitions of the Associated curves