Folium of Descartes

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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 $x^{2}-{\frac {1}{2}}$ as $x\to \infty$, and in fact can be written as

$y=x^{2}{\sqrt {1-x^{-2}}}=x^{2}-{\frac {1}{2}}\sum _{n\geq 0}C_{n}(2x)^{-2n}$

where

$C_{n}={\frac {1}{n+1}}{\binom {2n}{n}}$

is the $n$th Catalan number.

See also

• List of curves

References

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