The catenary curve, often called chainette and sometimes alysoid, has the following parametric equation:

y = (a/2)(exp(x/a)+exp(-x/a))

or, equivalently,

y = a cosh(x/a)

where "cosh" denotes the hyperbolic cosine. As mentioned in the Webster, this is the curve taken by a homogenous, flexible and inextensible chain hanged by its two extremities. Galileo thought this curve would be a parabola; in 1690, Jakob Bernouilli challenged his colleagues to find the exact mathematical expression, and in 1691, Leibniz, Huygens and Johann Bernouilli all found independently the answer to this question. The catenary curve is the solution to the following differential equation:

y'' = (m/c) sqrt(1+y'2)

which describes the condition for static equilibrium, m being the mass density of the chain and c the tension along the horizontal direction. Follow "A hanging string forms a hyperbolic cosine curve" for a complete demonstration... It is interesting to note that when a parabola is rolling on a line, the path followed by the focus is a catenary.

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