# Kampyle of Eudoxus

The Kampyle of Eudoxus (Greek: καμπύλη [γραμμή], meaning simply "curved [line], curve") is a curve with a Cartesian equation of

$x^{4}=a^{2}(x^{2}+y^{2}),$ from which the solution x = y = 0 is excluded.

## Alternative parameterizations

In polar coordinates, the Kampyle has the equation

$r=a\sec ^{2}\theta .$

Equivalently, it has a parametric representation as

$x=a\sec(t),\quad y=a\tan(t)\sec(t).$

## 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 x- and y-axes. It crosses the x-axis at (±a,0). It has inflection points at

$\left(\pm a{\frac {\sqrt {6}}{2}},\pm a{\frac {\sqrt {3}}{2}}\right)$

(four inflections, one in each quadrant). The top half of the curve is asymptotic to $x^{2}/a-a/2$  as $x\to \infty$ , and in fact can be written as

$y={\frac {x^{2}}{a}}{\sqrt {1-{\frac {a^{2}}{x^{2}}}}}={\frac {x^{2}}{a}}-{\frac {a}{2}}\sum _{n=0}^{\infty }C_{n}\left({\frac {a}{2x}}\right)^{2n},$

where

$C_{n}={\frac {1}{n+1}}{\binom {2n}{n}}$

is the $n$ th Catalan number.