# Kampyle of Eudoxus

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Graph of Kampyle of Eudoxus

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

${\displaystyle x^{4}=x^{2}+y^{2}}$

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

## Alternative parameterizations

In polar coordinates, the Kampyle has the equation

${\displaystyle r=\sec ^{2}\theta \,.}$

Equivalently, it has a parametric representation as,

${\displaystyle x=a\sec(t),y=a\tan(t)\sec(t)}$.

## 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 ${\displaystyle x}$- and ${\displaystyle y}$-axes. It crosses the ${\displaystyle x}$-axis at ${\displaystyle (-1,0)}$ and ${\displaystyle (1,0)}$. It has inflection points at

${\displaystyle (\pm {\sqrt {3/2}},\pm {\sqrt {3}}/2)}$

(four inflections, one in each quadrant). The top half of the curve is asymptotic to ${\displaystyle x^{2}-{\frac {1}{2}}}$ as ${\displaystyle x\to \infty }$, and in fact can be written as

${\displaystyle y=x^{2}{\sqrt {1-x^{-2}}}=x^{2}-{\frac {1}{2}}\sum _{n\geq 0}C_{n}(2x)^{-2n}}$

where

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

## References

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