Catenary

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The curve a hanging flexible wire or chain assumes when supported at its ends and acted upon by a uniform gravitational force. The word catenary is derived from the Latin word for ``chain.'' In 1669, Jungius disproved Galileo's claim that the curve of a chain hanging under gravity would be a Parabola (MacTutor Archive). The curve is also called the Alysoid and Chainette. The equation was obtained by Leibniz, Huygens, and Johann Bernoulli in 1691 in response to a challenge by Jakob Bernoulli.

Huygens was the first to use the term catenary in a letter to Leibniz in 1690, and David Gregory wrote a treatise on the catenary in 1690 (MacTutor Archive). If you roll a Parabola along a straight line, its Focus traces out a catenary. As proved by Euler in 1744, the catenary is also the curve which, when rotated, gives the surface of minimum Surface Area (the Catenoid) for the given bounding Circle.

The parametric equations for the catenary are given by

$\displaystyle x$	$\textstyle =$	$\displaystyle t$	(1)
$\displaystyle y$	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}a(e^{t/a}+e^{-t/a}) = a\cosh\left({t\over a}\right),$	(2)

and the Cesàro Equation is

$\begin{displaymath} (s^2+a^2)\kappa=-a. \end{displaymath}$

(3)

The catenary gives the shape of the road over which a regular polygonal ``wheel'' can travel smoothly. For a regular

-gon, the Cartesian equation of the corresponding catenary is $y=-A\cosh(x/A)$ , where $A\equiv R\cos(\pi/n)$ .

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The Arc Length, Curvature, and Tangential Angle are

$\displaystyle s$	$\textstyle =$	$\displaystyle a\sinh\left({t\over a}\right),$	(4)
$\displaystyle \kappa$	$\textstyle =$	$\displaystyle -{1\over a}\mathop{\rm sech}\nolimits ^2\left({t\over a}\right),$	(5)
$\displaystyle \phi$	$\textstyle =$	$\displaystyle -2\tan^{-1}\left[{\tanh\left({t\over 2a}\right)}\right].$	(6)

The slope is proportional to the Arc Length as measured from the center of symmetry.

References

Geometry Center. ``The Catenary.'' http://www.geom.umn.edu/zoo/diffgeom/surfspace/catenoid/catenary.html.

Gray, A. ``The Evolute of a Tractrix is a Catenary.'' §5.3 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 80-81, 1993.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 195 and 199-200, 1972.

Lockwood, E. H. ``The Tractrix and Catenary.'' Ch. 13 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 118-124, 1967.

MacTutor History of Mathematics Archive. ``Catenary.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Catenary.html.

Pappas, T. ``The Catenary & the Parabolic Curves.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 34, 1989.

Yates, R. C. ``Catenary.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 12-14, 1952.