Catenary

Cartesian equation:
y = a cosh(x/a)


Click below to see one of the Associated curves.

Definitions of the Associated curves Evolute
Involute 1 Involute 2
Inverse curve wrt origin Inverse wrt another circle
Pedal curve wrt origin Pedal wrt another point
Negative pedal curve wrt origin Negative pedal wrt another point
Caustic wrt horizontal rays Caustic curve wrt another point


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The catenary is the shape of a perfectly flexible chain suspended by its ends and acted on by gravity. Its equation was obtained by Leibniz, Huygens and Johann Bernoulli in 1691. They were responding to a challenge put out by Jacob Bernoulli to find the equation of the 'chain-curve'.

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. Jungius (1669) disproved Galileo's claim that the curve of a chain hanging under gravity would be a parabola.

The catenary is the locus of the focus of a parabola rolling along a straight line.

The catenary is the evolute of the tractrix. It is the locus of the mid-point of the vertical line segment between the curves ex and e-x.

Euler showed in 1744 that a catenary revolved about its asymptote generates the only minimal surface of revolution.


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JOC/EFR/BS January 1997

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/Curves/Catenary.html