**Cartesian equation: **

(*x*^{2} + *y*^{2})(*y*^{2} + *x*(*x* + *a*)) = 4*axy*^{2}

**Polar equation: **

*r* = *a* cos*θ* (4sin^{2}*θ* - 1)

**Click below to see one of the Associated curves.**

If your browser can handle JAVA code, click HERE to experiment interactively with this curve and its associated curves.

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= -bcosθ+ 4acosθsin^{2}θ.

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.

JOC/EFR/BS January 1997

The URL of this page is:

http://www-history.mcs.st-andrews.ac.uk/Curves/Trifolium.html