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| [[File:Quadrifolium.svg|thumb|Rotated Quadrifolium]]
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| {{Dablink|This article is about a geometric shape. For the article about the plant, please see [[Four-leaf clover]]}}
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| The '''quadrifolium'''
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| (also known as '''four-leaved clover'''<ref>
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| 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
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| </ref>)
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| is a type of [[Rose (mathematics)|rose curve]] with n=2. It has [[polar equation]]:
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| :<math>r = \cos(2\theta), \,</math>
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| with corresponding algebraic equation
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| :<math>(x^2+y^2)^3 = (x^2-y^2)^2. \,</math>
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| Rotated by 45°, this becomes
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| :<math>r = \sin(2\theta) \,</math>
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| with corresponding algebraic equation
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| :<math>(x^2+y^2)^3 = 4x^2y^2. \,</math>
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| In either form, it is a [[algebraic curve|plane algebraic curve]] of [[geometric genus|genus]] zero.
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| The [[dual curve]] to the quadrifolium is
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| :<math>(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). \,</math>
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| [[File:Dualrose.png|thumb|Dual Quadrifolium]]
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| The area inside the curve is <math>\tfrac 12 \pi</math>, which is exactly half of the area of the circumcircle of the quadrifolium. The length of the curve is ca. 9.6884.<ref>[http://mathworld.wolfram.com/Quadrifolium.html Quadrifolium - from Wolfram MathWorld]</ref>
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| ==Notes== | |
| <references />
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| ==References==
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| * {{cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | page=175 }}
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| ==External links==
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| * [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Rose Interactive example with JSXGraph]
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| [[Category:Sextic curves]]
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