DPLL algorithm: Difference between revisions

Revision as of 00:06, 2 September 2013

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}).\,

The area inside the curve is ${\tfrac {1}{2}}\pi$ , 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

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External links

Interactive example with JSXGraph

[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

[1]

[2]

@@ Line 1: / Line 1: @@
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+[[File:Quadrifolium.svg|thumb|Rotated Quadrifolium]]
+{{Dablink|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'''<ref>
+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
+</ref>)
+is a type of [[Rose (mathematics)|rose curve]] with n=2. It has [[polar equation]]:
+:<math>r = \cos(2\theta), \,</math>
+with corresponding algebraic equation
+:<math>(x^2+y^2)^3 = (x^2-y^2)^2. \,</math>
+Rotated by 45°, this becomes
+:<math>r = \sin(2\theta) \,</math>
+with corresponding algebraic equation
+:<math>(x^2+y^2)^3 = 4x^2y^2. \,</math>
+In either form, it is a [[algebraic curve|plane algebraic curve]] of [[geometric genus|genus]] zero.
+The [[dual curve]] to the quadrifolium is
+:<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>
+[[File:Dualrose.png|thumb|Dual Quadrifolium]]
+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>
+==Notes==
+<references />
+==References==
+* {{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 }}
+==External links==
+* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Rose Interactive example with JSXGraph]
+[[Category:Sextic curves]]

DPLL algorithm: Difference between revisions

Revision as of 00:06, 2 September 2013

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