Kampyle of Eudoxus

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

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

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 ${\displaystyle x^{2}/a-a/2}$ as ${\displaystyle x\to \infty }$, and in fact can be written as

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