Catenary
Curve that an idealized hanging chain or cable assumes
In physics and geometry, a catenary (UK: kə-TEE-nər-ee, US: KAT-ən-err-ee) is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field.
The catenary curve has a U-like shape, superficially similar in appearance to 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, an example of a 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.
Begin with the source’s own compact description: “Catenary” is curve that an idealized hanging chain or cable assumes. The dossier treats that line as a proposition to test through Catenary, Curve and idealized, not as a finished interpretation.
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The phrase “curve that an idealized hanging chain or cable assumes” supplies a clear boundary for inquiry. It also exposes the unanswered questions: who defined that boundary, when it became stable and which sources sit outside it.
Vocabulary and entity names are the principal evidence signals here, because they determine the precision of every later search. The source revision retrieved here is dated Jul 21, 2026. The linked authority identifier is Q214561. None of the 0 selected statements returned an explicit reference. The first chronological checks are 1638 and 1691.
The absence of detail may reflect summary conventions rather than a lack of surviving documentation. The lead is largely declarative, so disagreement and counter-evidence require a deliberate search beyond the opening account. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
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This entry incorporates text from “Catenary” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.