Catenary

The silk on a spider's web forming multiple elastic catenaries.

In physics and geometry, a catenary (US: /ˈkætənɛri/ KAT-ən-err-ee, UK: /kəˈtiːnəri/ kə-TEE-nər-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, which it is not.

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,^[1] or, particularly in the materials sciences, an example of a funicular.^[2] Rope statics describes catenaries in a classic statics problem involving a hanging rope.^[3]

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.^[4] 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 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.^[5] The symmetric modes consisting of two evanescent waves would form a catenary shape.^[6]^[7]^[8]

^ MathWorld
^ e.g.: Shodek, Daniel L. (2004). Structures (5th ed.). Prentice Hall. p. 22. ISBN 978-0-13-048879-4. OCLC 148137330.
^ "Shape of a hanging rope" (PDF). Department of Mechanical & Aerospace Engineering - University of Florida. 2017-05-02. Archived (PDF) from the original on 2018-09-20. Retrieved 2020-06-04.
^ "The Calculus of Variations". 2015. Retrieved 2019-05-03.
^ Luo, Xiangang (2019). Catenary optics. Singapore: Springer. doi:10.1007/978-981-13-4818-1. ISBN 978-981-13-4818-1. S2CID 199492908.
^ Bourke, Levi; Blaikie, Richard J. (2017-12-01). "Herpin effective media resonant underlayers and resonant overlayer designs for ultra-high NA interference lithography". JOSA A. 34 (12): 2243–2249. Bibcode:2017JOSAA..34.2243B. doi:10.1364/JOSAA.34.002243. ISSN 1520-8532. PMID 29240100.
^ Pu, Mingbo; Guo, Yinghui; Li, Xiong; Ma, Xiaoliang; Luo, Xiangang (2018-07-05). "Revisitation of Extraordinary Young's Interference: from Catenary Optical Fields to Spin–Orbit Interaction in Metasurfaces". ACS Photonics. 5 (8): 3198–3204. doi:10.1021/acsphotonics.8b00437. ISSN 2330-4022. S2CID 126267453.
^ Pu, Mingbo; Ma, XiaoLiang; Guo, Yinghui; Li, Xiong; Luo, Xiangang (2018-07-23). "Theory of microscopic meta-surface waves based on catenary optical fields and dispersion". Optics Express. 26 (15): 19555–19562. Bibcode:2018OExpr..2619555P. doi:10.1364/OE.26.019555. ISSN 1094-4087. PMID 30114126.

[MathWorld-1] MathWorld

[2] e.g.: Shodek, Daniel L. (2004). Structures (5th ed.). Prentice Hall. p. 22. ISBN 978-0-13-048879-4. OCLC 148137330.

[3] "Shape of a hanging rope" (PDF). Department of Mechanical & Aerospace Engineering - University of Florida. 2017-05-02. Archived (PDF) from the original on 2018-09-20. Retrieved 2020-06-04.

[4] "The Calculus of Variations". 2015. Retrieved 2019-05-03.

[5] Luo, Xiangang (2019). Catenary optics. Singapore: Springer. doi:10.1007/978-981-13-4818-1. ISBN 978-981-13-4818-1. S2CID 199492908.

[6] Bourke, Levi; Blaikie, Richard J. (2017-12-01). "Herpin effective media resonant underlayers and resonant overlayer designs for ultra-high NA interference lithography". JOSA A. 34 (12): 2243–2249. Bibcode:2017JOSAA..34.2243B. doi:10.1364/JOSAA.34.002243. ISSN 1520-8532. PMID 29240100.

[7] Pu, Mingbo; Guo, Yinghui; Li, Xiong; Ma, Xiaoliang; Luo, Xiangang (2018-07-05). "Revisitation of Extraordinary Young's Interference: from Catenary Optical Fields to Spin–Orbit Interaction in Metasurfaces". ACS Photonics. 5 (8): 3198–3204. doi:10.1021/acsphotonics.8b00437. ISSN 2330-4022. S2CID 126267453.

[8] Pu, Mingbo; Ma, XiaoLiang; Guo, Yinghui; Li, Xiong; Luo, Xiangang (2018-07-23). "Theory of microscopic meta-surface waves based on catenary optical fields and dispersion". Optics Express. 26 (15): 19555–19562. Bibcode:2018OExpr..2619555P. doi:10.1364/OE.26.019555. ISSN 1094-4087. PMID 30114126.

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