Kampyle of Eudoxus

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


 * $$x^4 = a^2(x^2+y^2),$$

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

Alternative parameterizations
In polar coordinates, the Kampyle has the equation


 * $$r = a\sec^2\theta.$$

Equivalently, it has a parametric representation as
 * $$x=a\sec(t), \quad y=a\tan(t)\sec(t).$$

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 x- and y-axes. It crosses the x-axis at (±a,0). It has inflection points at


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

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


 * $$y = \frac{x^2}{a}\sqrt{1-\frac{a^2}{x^2}} = \frac{x^2}{a} - \frac{a}{2} \sum_{n=0}^\infty C_n\left(\frac{a}{2x}\right)^{2n},$$

where


 * $$C_n = \frac1{n+1} \binom{2n}{n}$$

is the $$n$$th Catalan number.