Rahul Sharma (Editor)

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

Contents

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 ⁡ θ .

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 ) . Besides these two points, it has no integer points. It has inflection points at

( ± 3 / 2 , ± 3 / 2 )

(four inflections, one in each quadrant). The top half of the curve is asymptotic to x 2 − 1 2 as x → ∞ , and in fact can be written as

y = x 2 1 − x − 2 = x 2 − 1 2 ∑ n ≥ 0 C n ( 2 x ) − 2 n

where

C n = 1 n + 1 ( 2 n n )

is the n th Catalan number.

References

Kampyle of Eudoxus Wikipedia
(Text) CC BY-SA


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