Catenary
In mathematics, the catenary is the shape of a hanging flexible chain or cable when supported at its ends and acted upon by a uniform gravitational force (its own weight). The chain is almost vertical near the points of suspension because this part of the chain has the most weight pulling down on it. Toward the bottom, the slope of the chain decreases because the chain is supporting less and less weight.
The word catenary is derived from the Latin word for "chain." The curve is also called the alysoid, funicular, and chainette. In 1669, Jungius disproved Galileo's claim that the curve of a chain hanging under gravity would be a parabola. The equation was obtained by Leibniz, Christiaan 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.
If you roll a parabola along a straight line, its focus traces out a catenary (see roulette (curve)). 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. Square wheels can roll perfectly smoothly if the road has evenly spaced bumps in the shape of a series of inverted catenary curves.
The intrinsic equation of the shape of the catenary is
<math>y = a \cdot \cosh(x/a).<math>
In railway engineering, a catenary structure consists of overhead lines or cables used to deliver electricity to a railway locomotive, multiple unit, railcar, tram or trolleybus through a pantograph. An alternative system is a third rail.
In structural engineering a catenary shell is a structural form,usually made of concrete, that follows a catenary curve. The profile for the shell is obtained by using flexible material subjected to gravity, converting it into a rigid formwork for pouring the concrete and then using it as required, usually in an inverted manner.
The Catalan architect Antoni Gaudí made extensive use of catenary shapes in his Sagrada Familia.
