Quadrifolium Explained

The quadrifolium (also known as four-leaved clover^[1]) 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} = 4 a ^{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} + 108 x ^{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 is8 a\operatorname{E}\left( (\sqrt{ 3)/() }\right)=4\pi a\left(\frac+\frac\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.^[2]

Notes

1. C G Gibson, Elementary Geometry of Algebraic Curves, An Undergraduate Introduction, Cambridge University Press, Cambridge, 2001, . Pages 92 and 93
2. http://mathworld.wolfram.com/Quadrifolium.html Quadrifolium - from Wolfram MathWorld

References

• Book: J. Dennis Lawrence . A catalog of special plane curves . Dover Publications . 1972 . 0-486-60288-5 . 175 . registration .

External links

• Interactive example with JSXGraph