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# catenary

/kat"n er'ee/; esp. Brit. /keuh tee"neuh ree/, n., pl. catenaries, adj.
n.
1. Math. the curve assumed approximately by a heavy uniform cord or chain hanging freely from two points not in the same vertical line. Equation: y = k cosh(x/k).
2. (in electric railroads) the cable, running above the track, from which the trolley wire is suspended.
3. of, pertaining to, or resembling a catenary.
4. of or pertaining to a chain or linked series.
[1780-90; < L catenarius relating to a chain, equiv. to caten(a) a chain + -arius -ARY]

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in mathematics, a curve that describes the shape of a flexible hanging chain or cable—the name derives from the Latin catenaria (“chain”). Any freely hanging cable or string assumes this shape, also called a chainette, if the body is of uniform mass per unit of length and is acted upon solely by gravity.

Early in the 17th century, the German astronomer Johannes Kepler (Kepler, Johannes) applied the ellipse to the description of planetary orbits, and the Italian scientist Galileo Galilei (Galileo) employed the parabola to describe projectile motion in the absence of air resistance. Inspired by the great success of conic sections (conic section) in these settings, Galileo incorrectly believed that a hanging chain would take the shape of a parabola. It was later in the 17th century that the Dutch mathematician Christiaan Huygens (Huygens, Christiaan) showed that the chain curve cannot be given by an algebraic equation (one involving only arithmetic operations together with powers and roots (root)); he also coined the term catenary. In addition to Huygens, the Swiss mathematician Jakob Bernoulli (Bernoulli, Jakob) and the German mathematician Gottfried Leibniz (Leibniz, Gottfried Wilhelm) contributed to the complete description of the equation of the catenary.

Precisely, the curve in the xy-plane of such a chain suspended from equal heights at its ends and dropping at x = 0 to its lowest height y = a is given by the equation y = (a/2)(ex/a + ex/a). It can also be expressed in terms of the hyperbolic cosine function (hyperbolic functions) as y = a cosh(x/a). See the figure—>.

Although the catenary curve fails to be described by a parabola, it is of interest to note that it is related to a parabola: the curve traced in the plane by the focus of a parabola as it rolls along a straight line is a catenary. The surface of revolution generated when an upward-opening catenary is revolved around the horizontal axis is called a catenoid. The catenoid was discovered in 1744 by the Swiss mathematician Leonhard Euler (Euler, Leonhard) and it is the only minimal surface, other than the plane, that can be obtained as a surface of revolution.

The catenary and the related hyperbolic functions play roles in other applications. An inverted hanging cable provides the shape for a stable self-standing arch, such as the Gateway Arch located in St. Louis, Missouri. The hyperbolic functions also arise in the description of waveforms, temperature distributions, and the motion of falling bodies subject to air resistance proportional to the square of the speed of the body.

Stephan C. Carlson

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Universalium. 2010.

### Look at other dictionaries:

• Catenary — Cat e*na*ry, Catenarian Cat e*na ri*an, a. [L. catenarius, fr. catena a chain. See {Chain}.] Relating to a chain; like a chain; as, a {catenary} curve. [1913 Webster] …   The Collaborative International Dictionary of English

• Catenary — Cat e*na*ry, n.; pl. {Catenaries}. (Geol.) The curve formed by a rope or chain of uniform density and perfect flexibility, hanging freely between two points of suspension, not in the same vertical line. [1913 Webster] …   The Collaborative International Dictionary of English

• catenary — 1788, from L. catenarius, from catenanus, from catena chain, fetter (see CHAIN (Cf. chain)) …   Etymology dictionary

• catenary — [kat′ənner′ē ənkat′ə ner΄ē; ] chiefly Brit, [kə tē′nər ē] n. pl. catenaries [L catenarius < catena, CHAIN] the curve made by a flexible, uniform chain or cord freely suspended between two fixed points adj. designating or of such a curve: also… …   English World dictionary

• Catenary — This article is about the mathematical curve. For other uses, see Catenary (disambiguation). Chainette redirects here. For the wine grape also known as Chainette, see Cinsaut. A hanging chain forms a catenary …   Wikipedia

• catenary — noun (plural naries) Etymology: New Latin catenaria, from Latin, feminine of catenarius of a chain, from catena Date: 1788 1. the curve assumed by a cord of uniform density and cross section that is perfectly flexible but not ca …   New Collegiate Dictionary

• catenary — grandininė kreivė statusas T sritis fizika atitikmenys: angl. catenary; catenary curve; funicular curve vok. Katenoide, f; Kettenlinie, f; Seilkurve, f rus. веревочная кривая, f; веревочная линия, f; цепная линия, f pranc. courbe funiculaire, f;… …   Fizikos terminų žodynas

• catenary — [kə ti:nəri] noun (plural catenaries) a curve formed by a chain hanging freely from two points on the same horizontal level. ↘a chain forming such a curve. adjective of, involving, or denoting a catenary. Origin C18: from L. catenarius relating… …   English new terms dictionary

• catenary — cat•e•nar•y [[t]ˈkæt nˌɛr i[/t]] esp. brit. [[t]kəˈti nə ri[/t]] n. pl. nar•ies, adj. 1) math. the curve assumed approximately by a heavy uniform cord or chain hanging freely from two points not in the same vertical line. Equation: y = k cosh… …   From formal English to slang

• catenary — /kəˈtinəri/ (say kuh teenuhree) noun (plural catenaries) 1. the curve assumed approximately by a heavy uniform cord or chain hanging freely from two points not in the same vertical line. 2. the slack wire from which the contact wire is suspended… …   Australian English dictionary