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Kampyle of Eudoxus

From Wikipedia, the free encyclopedia

Kampyle of Eudoxus
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The Kampyle of Eudoxus (Greek: καμπύλη [γραμμή], meaning simply "curved [line], curve") is a curve with a Cartesian equation of

Thumb
Graph of Kampyle of Eudoxus with a = 1

from which the solution x = y = 0 is excluded.

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Alternative parameterizations

In polar coordinates, the Kampyle has the equation

Equivalently, it has a parametric representation as

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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

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The Kampyle is symmetric about both the x- and y-axes. It crosses the x-axis at (±a,0). It has inflection points at

(four inflections, one in each quadrant). The top half of the curve is asymptotic to as , and in fact can be written as

where

is the th Catalan number.

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See also

References

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