Quadrifolium

Rotated quadrifolium

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

External links

• Interactive example with JSXGraph