Quadrifolium

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Rotated Quadrifolium

This article is about a geometric shape. For the article about the plant, please see Four-leaf clover

The quadrifolium (also known as four-leaved clover^[1]) 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

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

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 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.^[2]

Notes

↑ 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
↑ 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

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