Catenary

Catenary (from Lat. catena, a chain), in mathematics, the curve assumed by a uniform chain or string hanging freely between two supports. It was investigated by Galileo, who erroneously determined it to be a parabola; Jungius detected Galileo’s error, but the true form was not discovered until 1691, when James Bernoulli published it as a problem in the Acta Eruditorum. Bernoulli also considered the cases when (1) the chain was of variable density, (2) extensible, (3) acted upon at each point by a force directed to a fixed centre. These curves attracted much attention and were discussed by John Bernoulli, Leibnitz, Huygens, David Gregory and others.  
  
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The mechanical properties of the curves are treated in the article Mechanics, where various forms are illustrated. The simple catenary is shown in the figure. The cartesian equation referred to the axis and directrix is y = c cosh (x/c) or y = ½c(ex/c \+ e−x/c); other forms are s = c sinh (x/c) and y² = c² + s², s being the arc measured from the vertex; the intrinsic equation is s = c tan ψ. The radius of curvature and normal are each equal to c sec² ψ.

The surface formed by revolving the catenary about its directrix is named the alysseide. It is a minimal surface, i.e. the catenary solves the problem: to find a curve joining two given points, which when revolved about a line co-planar with the points traces a surface of minimum area (see Variations, Calculus of).

The involute of the catenary is called the tractory, tractrix or antifriction curve; it has a cusp at the vertex of the catenary, and is asymptotic to the directrix. The cartesian equation is

x = √ (c² − y²) + ½c log [ {c − √ (c² − y²)} / {c + √ (c² + y²)} ],

and the curve has the geometrical property that the length of its tangent is constant. It is named the tractory, since a weight placed on the ground and drawn along by means of a flexible string by a person travelling in a straight line, the weight not being in this line, describes the curve in question. It is named the antifriction curve, since a pivot and step having the form of the surface generated by revolving the curve about its vertical axis wear away equally (see Mechanics: Applied).

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