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| {{For|the company|Kampyle (software)}}
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| [[Image:Eudoxus.png|thumb|Graph of Kampyle of Eudoxus]]The '''Kampyle of Eudoxus''' ([[Ancient Greek|Greek]]: καμπύλη [γραμμή], meaning simply "curved [line], curve") is a [[curve]], with a [[Cartesian equation]] of
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| :<math>x^4=x^2+y^2</math>
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| from which the solution ''x'' = ''y'' = 0 should be excluded.
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| ==Alternative parameterizations==
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| In [[polar coordinates]], the Kampyle has the equation
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| :<math>r= \sec^2\theta\,.</math>
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| Equivalently, it has a parametric representation as,
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| :<math>x=a\sec(t), y=a\tan(t)\sec(t)</math>.
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| ==History==
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| This [[quartic curve]] was studied by the Greek astronomer and mathematician [[Eudoxus of Cnidus]] (c. 408 BC – c.347 BC) in relation to the classical problem of [[doubling the cube]].
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| ==Properties==
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| The Kampyle is symmetric about both the <math>x</math>- and <math>y</math>-axes. It crosses the <math>x</math>-axis at <math>(-1,0)</math> and <math>(1,0)</math>. It has [[inflection points]] at
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| :<math>(\pm\sqrt{3/2},\pm\sqrt{3}/2)</math> | |
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| (four inflections, one in each quadrant). The top half of the curve is asymptotic to <math>x^2-\frac12</math> as <math>x \to \infty</math>, and in fact can be written as
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| :<math>y = x^2\sqrt{1-x^{-2}} = x^2 - \frac12 \sum_{n \ge 0} C_n(2x)^{-2n}</math>
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| where
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| :<math>C_n = \frac1{n+1} \binom{2n}{n}</math>
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| is the <math>n</math>th [[Catalan number]].
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| ==See also==
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| * [[List of curves]]
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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 | pages=141–142 }}
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| ==External links==
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| * {{MacTutor|class=Curves|id=Kampyle|title=Kampyle of Eudoxus}}
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| * {{MathWorld|urlname=KampyleofEudoxus|title=Kampyle of Eudoxus}}
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| [[Category:Curves]]
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