Quadrifolium

Rotated quadrifolium

Quadrifolium created with gears

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

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

Dual quadrifolium

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

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

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

References

• Template:Cite book

External links

• Interactive example with JSXGraph