# Quadrifolium

Rotated quadrifolium
Quadrifolium created with gears

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

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

with corresponding algebraic equation

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

Rotated 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]

## Notes

1. ^ C G Gibson, Elementary Geometry of Algebraic Curves, An Undergraduate Introduction, Cambridge University Press, Cambridge, 2001, ISBN 978-0-521-64641-3. Pages 92 and 93
2. ^ Quadrifolium - from Wolfram MathWorld

## References

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. p. 175. ISBN 0-486-60288-5.