Quadrifolium
# Quadrifolium

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

    $r=a\cos(2\theta ),\,$

with corresponding algebraic equation

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

Rotated counter-clockwise by 45°, this becomes

    $r=a\sin(2\theta )\,$

with corresponding algebraic equation

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

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

The dual curve to the quadrifolium is

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

The area inside the quadrifolium is ${\tfrac {1}{2}}\pi a^{2}$, which is exactly half of the area of the circumcircle of the quadrifolium. The perimeter of the quadrifolium is

    $8a\operatorname {E} \left({\frac {\sqrt {3}}{2}}\right)=4\pi a\left({\frac {(52{\sqrt {3}}-90)\operatorname {M} '(1,7-4{\sqrt {3}})}{\operatorname {M} ^{2}(1,7-4{\sqrt {3}})}}+{\frac {7-4{\sqrt {3}}}{\operatorname {M} (1,7-4{\sqrt {3}})}}\right)$

where $\operatorname {E} (k)$ is the complete elliptic integral of the second kind with modulus $k$, $\operatorname {M}$ is the arithmetic–geometric mean and $'$ denotes the derivative with respect to the second variable.

