Kampyle of Eudoxus

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

Graph of Kampyle of Eudoxus with a = 1

from which the solution x = y = 0 is excluded.

Alternative parameterizationsEdit

In polar coordinates, the Kampyle has the equation


Equivalently, it has a parametric representation as



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


(four inflections, one in each quadrant). The top half of the curve is asymptotic to   as  , and in fact can be written as




is the  th Catalan number.

See alsoEdit


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

External linksEdit