Kampyle of Eudoxus

Graph of Kampyle of Eudoxus with a = 1

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

$x=a\sec (t),y=a\tan (t)\sec (t).$

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

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

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

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

where

$C_n=\frac{1}{n+1}(\genfrac{}{}{0ex}{}{2n}{n})$

is the $n$th Catalan number.

See also

• List of curves

References

• Template:Cite book

External links

• Template:MacTutor
• Template:MathWorld