The Kampyle of Eudoxus ( Greek: καμπύλη �ραμμή meaning simply "curved ine curve") is a

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...

with a

Cartesian equation A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

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), \quad y=a\tan(t)\sec(t).

History

This

quartic curve In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation: :Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0, with at least one o ...

was studied by the Greek astronomer and mathematician

Eudoxus of Cnidus Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his original works are lost, though some fragments are ...

(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 In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...

at :

\left(\pm a\frac,\pm a\frac\right)

(four inflections, one in each quadrant). The top half of the curve is asymptotic to

x^2/a-a/2

x \to \infty

, and in fact can be written as :

y = \frac\sqrt = \frac - \frac \sum_^\infty C_n\left(\frac\right)^,

where :

C_n = \frac1 \binom

is the

n

th Catalan number.

References

External links

* * {{MathWorld, urlname=KampyleofEudoxus, title=Kampyle of Eudoxus Plane curves

Alternative parameterizations

History

Properties

See also

References

External links