Kampyle of Eudoxus explained

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

\left(\pm a (\sqrt{ 6)/() },\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 inf ty , and in fact can be written asy = (x ^{2})/(\sqrt{ 1- a) (a ^{2})/(x ^{2}) } = \frac - \frac \sum_^\infty C_n\left(\frac\right)^,

where

C _{n} = (1 {n +1 })/(\binom{ 2 n}{n})

is the

n th Catalan number.

See also

• List of curves

References

• Book: J. Dennis Lawrence . A catalog of special plane curves . Dover Publications . 1972 . 0-486-60288-5 . 141–142 . registration .