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{{For|the company|Kampyle (software)}}
[[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

:<math>x^4=x^2+y^2</math>

from which the solution ''x'' = ''y'' = 0 should be excluded.

==Alternative parameterizations==
In [[polar coordinates]], the Kampyle has the equation

:<math>r= \sec^2\theta\,.</math>

Equivalently, it has a parametric representation as,
:<math>x=a\sec(t), y=a\tan(t)\sec(t)</math>.

==History==
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]].

==Properties==
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

:<math>(\pm\sqrt{3/2},\pm\sqrt{3}/2)</math>

(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

:<math>y = x^2\sqrt{1-x^{-2}} = x^2 - \frac12 \sum_{n \ge 0} C_n(2x)^{-2n}</math>

where

:<math>C_n = \frac1{n+1} \binom{2n}{n}</math>

is the <math>n</math>th [[Catalan number]].

==See also==
* [[List of curves]]

==References==
* {{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 }}

==External links==
* {{MacTutor|class=Curves|id=Kampyle|title=Kampyle of Eudoxus}}
* {{MathWorld|urlname=KampyleofEudoxus|title=Kampyle of Eudoxus}}

[[Category:Curves]]