In polar coordinates, the Kampyle has the equation

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

Equivalently, it has a parametric representation as

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

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.

The Kampyle is symmetric about both the x- and y-axes. It crosses the x-axis at (±a,0). It has inflection 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 ${\displaystyle x\to \infty }$ , and in fact can be written as

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

where

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

is the ${\displaystyle n}$ th Catalan 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.