Quadrifolium

The quadrifolium (also known as four-leaved clover[1]) is a type of rose curve with an angular frequency of 2. It has the polar equation:

${\displaystyle r=\cos(2\theta ),\,}$

Rotated quadrifolium

Quadrifolium created with gears

with corresponding algebraic equation

${\displaystyle (x^{2}+y^{2})^{3}=(x^{2}-y^{2})^{2}.\,}$

Rotated counter-clockwise by 45°, this becomes

${\displaystyle r=\sin(2\theta )\,}$

with corresponding algebraic equation

${\displaystyle (x^{2}+y^{2})^{3}=4x^{2}y^{2}.\,}$

In either form, it is a plane algebraic curve of genus zero.

The dual curve to the quadrifolium is

${\displaystyle (x^{2}-y^{2})^{4}+837(x^{2}+y^{2})^{2}+108x^{2}y^{2}=16(x^{2}+7y^{2})(y^{2}+7x^{2})(x^{2}+y^{2})+729(x^{2}+y^{2}).\,}$

Dual quadrifolium

The area inside the curve is ${\displaystyle {\tfrac {1}{2}}\pi }$, which is exactly half of the area of the circumcircle of the quadrifolium. The length of the curve is about 9.6884.[2]