Quadrifolium

File:Quadrifolium.svg

Rotated Quadrifolium

This article is about the geometric shape. For the plant, see Four-leaf clover. For the symmetrical shape framework, see Quatrefoil.

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

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

with corresponding algebraic equation

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

Rotated by 45°, this becomes

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

with corresponding algebraic equation

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

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

The dual curve to the quadrifolium is

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

Dual Quadrifolium

The area inside the curve is $\tfrac 12 \pi$, which is exactly half of the area of the circumcircle of the quadrifolium. The length of the curve is ca. 9.6884.

External links

• Interactive example with JSXGraph