Curves

Trifolium

Main
Cartesian equation:
(x2+y2)(y2+x(x+a))=4axy2(x^{2} + y^{2})(y^{2} + x(x + a)) = 4axy^{2}
Polar equation:
r=acosθ(4sin2θ1)r = a \cos \theta (4\sin^{2} \theta - 1)

Description

The general form of the folium is given by the formula
(x2+y2)(y2+x(x+b))=4axy2(x^{2} + y^{2})(y^{2} + x(x + b)) = 4axy^{2}
or, in polar coordinates
r=bcosθ+4acosθsin2θr = -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=ab = 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.