TheKampyle of Eudoxus (Greek: καμπύλη [γραμμή], meaning simply "curved [line], curve") is acurve with aCartesian equation of

${\displaystyle x^4=a^2(x^2+y^2),}$

from which the solutionx =y = 0 is excluded.

Inpolar coordinates, the Kampyle has the equation

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

Equivalently, it has a parametric representation as

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

Thisquartic curve was studied by the Greek astronomer and mathematicianEudoxus of Cnidus (c. 408 BC – c.347 BC) in relation to the classical problem ofdoubling the cube.

The Kampyle is symmetric about both thex- andy-axes. It crosses thex-axis at (±a,0). It hasinflection points at

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

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

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

where

${\displaystyle C_n=\frac{1}{n+1}(\genfrac{}{}{0ex}{}{2n}{n})}$

is the${\displaystyle n}$thCatalan number.

• List of curves

• J. Dennis Lawrence (1972).A catalog of special plane curves. Dover Publications. pp. 141–142.ISBN 0-486-60288-5.

• O'Connor, John J.;Robertson, Edmund F.,"Kampyle of Eudoxus",MacTutor History of Mathematics Archive,University of St Andrews
• Weisstein, Eric W."Kampyle of Eudoxus".MathWorld.

• Quartic curves
• Greek mathematics