Catenary

In physics and geometry, a catenary (US: , UK: ) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends.
The catenary curve has a U-like shape, superficially similar in appearance to a parabolic arch, but it is not a parabola.
The curve appears in the design of certain types of arches and as a cross section of the catenoid—the shape assumed by a soap film bounded by two parallel circular rings.
The catenary is also called the alysoid, chainette, or, particularly in the materials sciences, funicular. Rope statics describes catenaries in a classic statics problem involving a hanging rope.Mathematically, the catenary curve is the graph of the hyperbolic cosine function. The surface of revolution of the catenary curve, the catenoid, is a minimal surface, specifically a minimal surface of revolution. A hanging chain will assume a shape of least potential energy which is a catenary. Galileo Galilei in 1638 discussed the catenary in the book Two New Sciences recognizing that it was different from a parabola. The mathematical properties of the catenary curve were studied by Robert Hooke in the 1670s, and its equation was derived by Leibniz, Huygens and Johann Bernoulli in 1691.
Catenaries and related curves are used in architecture and engineering (e.g., in the design of bridges and arches so that forces do not result in bending moments). In the offshore oil and gas industry, "catenary" refers to a steel catenary riser, a pipeline suspended between a production platform and the seabed that adopts an approximate catenary shape. In the rail industry it refers to the overhead wiring that transfers power to trains. (This often supports a lighter contact wire, in which case it does not follow a true catenary curve.)
In optics and electromagnetics, the hyperbolic cosine and sine functions are basic solutions to Maxwell's equations. The symmetric modes consisting of two evanescent waves would form a catenary shape.

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