Kampyle of Eudoxus
The Kampyle of Eudoxus (Greek: καμπύλη [γραμμή], meaning simply "curved [line], curve") is a curve with a Cartesian equation of
- [math]\displaystyle{ x^4 = a^2(x^2+y^2), }[/math]
from which the solution x = y = 0 is excluded.
Alternative parameterizations
In polar coordinates, the Kampyle has the equation
- [math]\displaystyle{ r = a\sec^2\theta. }[/math]
Equivalently, it has a parametric representation as
- [math]\displaystyle{ x=a\sec(t), \quad y=a\tan(t)\sec(t). }[/math]
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
- [math]\displaystyle{ \left(\pm a\frac{\sqrt{6}}{2},\pm a\frac{\sqrt{3}}{2}\right) }[/math]
(four inflections, one in each quadrant). The top half of the curve is asymptotic to [math]\displaystyle{ x^2/a-a/2 }[/math] as [math]\displaystyle{ x \to \infty }[/math], and in fact can be written as
- [math]\displaystyle{ y = \frac{x^2}{a}\sqrt{1-\frac{a^2}{x^2}} = \frac{x^2}{a} - \frac{a}{2} \sum_{n=0}^\infty C_n\left(\frac{a}{2x}\right)^{2n}, }[/math]
where
- [math]\displaystyle{ C_n = \frac1{n+1} \binom{2n}{n} }[/math]
is the [math]\displaystyle{ n }[/math]th Catalan number.
See also
References
- J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 141–142. ISBN 0-486-60288-5. https://archive.org/details/catalogofspecial00lawr/page/141.
External links
- O'Connor, John J.; Robertson, Edmund F., "Kampyle of Eudoxus", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Curves/Kampyle.html.
- Weisstein, Eric W.. "Kampyle of Eudoxus". http://mathworld.wolfram.com/KampyleofEudoxus.html.
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