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^2x^2y^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^2y^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 12 \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.