Kampyle of Eudoxus

The Kampyle of Eudoxus (Greek: καμπύλη �ραμμή meaning simply "curved ine 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

:$\left(\pm a\frac,\pm a\frac\right)$

(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

:$y = \frac\sqrt = \frac - \frac \sum_^\infty C_n\left(\frac\right)^,$

where

:$C_n = \frac1 \binom$

is the $n$th Catalan number.

See also

* List of curves

References

*

External links

*
* {{MathWorld, urlname=KampyleofEudoxus, title=Kampyle of Eudoxus

Plane curves