Kampyle of Eudoxus

For the company, see Kampyle (software).

Graph of 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=x^2+y^2

from which the solution x = y = 0 should be excluded.

Alternative parameterizations

In polar coordinates, the Kampyle has the equation

r= \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 $(-1,0)$ and $(1,0)$ . It has inflection points at

(\pm\sqrt{3/2},\pm\sqrt{3}/2)

(four inflections, one in each quadrant). The top half of the curve is asymptotic to $x^2-\frac12$ as $x \to \infty$ , and in fact can be written as

y = x^2\sqrt{1-x^{-2}} = x^2 - \frac12 \sum_{n \ge 0} C_n(2x)^{-2n}

where

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

is the $n$ th Catalan number.

References

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

External links