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{{Short description|Curve formed by a hanging chain}}
{{About|the mathematical curve}}
{{Redirect|Chainette|the wine grape also known as Chainette|Cinsaut}}

[[File:Kette Kettenkurve Catenary 2008 PD.JPG|thumb|right|This [[chain]], whose ends hang from two points, forms a catenary.]]
[[File:SpiderCatenary.jpg|thumb|right|The silk on this [[spider web]] forms multiple [[elastic deformation|elastic]] catenaries.]]

In [[physics]] and [[geometry]], a '''catenary''' ({{IPAc-en|US|ˈ|k|æ|t|ən|ɛr|i}} {{respell|KAT|ən|err|ee}}, {{IPAc-en|UK|k|ə|ˈ|t|iː|n|ər|i}} {{respell|kə|TEE|nər|ee}}) is the [[curve]] that an idealized hanging [[chain]] or [[wire rope|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 [[Catenary arch|arch]]es 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''',<ref name="MathWorld">[[#MathWorld|MathWorld]]</ref> or, particularly in the materials sciences, an example of a [[funicular curve|funicular]].<ref>''e.g.'':  {{cite book| last = Shodek| first = Daniel L.| title = Structures| edition = 5th| year = 2004| publisher = Prentice Hall| isbn = 978-0-13-048879-4| oclc = 148137330| page = 22 }}</ref> '''Rope statics''' describes catenaries in a classic statics problem involving a hanging rope.<ref>{{cite web|url=https://mae.ufl.edu/~uhk/HANGING-ROPE.pdf |archive-url=https://web.archive.org/web/20180920110549/http://www2.mae.ufl.edu/~uhk/HANGING-ROPE.pdf |archive-date=2018-09-20 |url-status=live |title=Shape of a hanging rope|website=Department of Mechanical & Aerospace Engineering - University of Florida |date=2017-05-02 |access-date=2020-06-04}}</ref>

Mathematically, the catenary curve is the [[Graph of a function|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.<ref>{{cite web|url=http://galileoandeinstein.physics.virginia.edu/7010/CM_02_CalculusVariations.html |title=The Calculus of Variations |date=2015 |access-date=2019-05-03}}</ref> [[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]], [[Christiaan Huygens|Huygens]] and [[Johann Bernoulli]] in 1691.

Catenaries and related curves are used in architecture and engineering (e.g., in the design of bridges and [[Catenary arch|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 line|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.<ref>{{cite book |last1=Luo |first1=Xiangang |title=Catenary optics |date=2019 |publisher=Springer |location=Singapore |isbn=978-981-13-4818-1 |doi=10.1007/978-981-13-4818-1 |s2cid=199492908 }}</ref> The symmetric modes consisting of two [[evanescent waves]] would form a catenary shape.<ref>{{Cite journal|last1=Bourke|first1=Levi|last2=Blaikie|first2=Richard J.|date=2017-12-01|title=Herpin effective media resonant underlayers and resonant overlayer designs for ultra-high NA interference lithography|url=https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-34-12-2243|journal=JOSA A|language=EN|volume=34|issue=12|pages=2243–2249|doi=10.1364/JOSAA.34.002243|pmid=29240100|bibcode=2017JOSAA..34.2243B|issn=1520-8532}}</ref><ref>{{Cite journal|last1=Pu|first1=Mingbo|last2=Guo|first2=Yinghui|last3=Li|first3=Xiong|last4=Ma|first4=Xiaoliang|last5=Luo|first5=Xiangang|date=2018-07-05|title=Revisitation of Extraordinary Young's Interference: from Catenary Optical Fields to Spin–Orbit Interaction in Metasurfaces|journal=ACS Photonics|volume=5|issue=8|pages=3198–3204|language=en|doi=10.1021/acsphotonics.8b00437|bibcode=2018ACSP....5.3198P |s2cid=126267453 |issn=2330-4022}}</ref><ref>{{Cite journal|last1=Pu|first1=Mingbo|last2=Ma|first2=XiaoLiang|last3=Guo|first3=Yinghui|last4=Li|first4=Xiong|last5=Luo|first5=Xiangang|date=2018-07-23|title=Theory of microscopic meta-surface waves based on catenary optical fields and dispersion|url=https://www.osapublishing.org/oe/abstract.cfm?uri=oe-26-15-19555|journal=Optics Express|language=EN|volume=26|issue=15|pages=19555–19562|doi=10.1364/OE.26.019555|pmid=30114126|bibcode=2018OExpr..2619555P|issn=1094-4087|doi-access=free}}</ref>

==History==
[[File:GaudiCatenaryModel.jpg|thumb|250px|[[Antoni Gaudí]]'s catenary model at [[Casa Milà]]]]

The word "catenary" is derived from the Latin word ''catēna'', which means "[[chain]]". The English word "catenary" is usually attributed to [[Thomas Jefferson]],<ref>{{cite web|url=http://www.pballew.net/arithme8.html#catenary |archive-url=https://archive.today/20120906145306/http://www.pballew.net/arithme8.html#catenary |url-status=usurped |archive-date=September 6, 2012 |title="Catenary" at Math Words |publisher=Pballew.net |date=1995-11-21 |access-date=2010-11-17}}</ref><ref>{{cite book| last = Barrow| first = John D.| title = 100 Essential Things You Didn't Know You Didn't Know: Math Explains Your World| year = 2010| publisher = W. W. Norton & Company| isbn = 978-0-393-33867-6| page = [https://archive.org/details/100essentialthin0000barr/page/27 27]| url = https://archive.org/details/100essentialthin0000barr/page/27}}</ref>
who wrote in a letter to [[Thomas Paine]] on the construction of an arch for a bridge:

{{Blockquote|I have lately received from Italy a treatise on the [[Mechanical equilibrium|equilibrium]] of arches, by the Abbé Mascheroni. It appears to be a very scientific work. I have not yet had time to engage in it; but I find that the conclusions of his demonstrations are, that every part of the catenary is in perfect equilibrium.<ref>{{cite book| last = Jefferson| first = Thomas| title = Memoirs, Correspondence and Private Papers of Thomas Jefferson| url = https://archive.org/details/memoirscorrespon02jeffuoft| year = 1829| publisher = Henry Colbura and Richard Bertley| page = [https://archive.org/details/memoirscorrespon02jeffuoft/page/419 419] }}</ref>}}

It is often said<ref name="Lockwood124"/> that [[Galileo Galilei|Galileo]] thought the curve of a hanging chain was parabolic. However, in his ''[[Two New Sciences]]'' (1638), Galileo wrote that a hanging cord is only an approximate parabola, correctly observing that this approximation improves in accuracy as the curvature gets smaller and is almost exact when the elevation is less than 45°.<ref>{{cite book| last = Fahie| first = John Joseph| title = Galileo, His Life and Work| url = https://archive.org/details/galileohislifea01fahigoog| year = 1903| publisher = J. Murray| pages = [https://archive.org/details/galileohislifea01fahigoog/page/n411 359]–360 }}</ref> The fact that the curve followed by a chain is not a parabola was proven by [[Joachim Jungius]] (1587–1657); this result was published posthumously in 1669.<ref name="Lockwood124">[[#Lockwood|Lockwood]] p. 124</ref>

The application of the catenary to the construction of arches is attributed to [[Robert Hooke]], whose "true mathematical and mechanical form" in the context of the rebuilding of [[St Paul's Cathedral]]  alluded to a catenary.<ref>{{Cite journal|jstor=532102 |title=Monuments and Microscopes: Scientific Thinking on a Grand Scale in the Early Royal Society |journal=Notes and Records of the Royal Society of London |volume=55 |issue=2 |pages=289–308 |first=Lisa |last=Jardine|year=2001 |doi=10.1098/rsnr.2001.0145 |s2cid=144311552 }}</ref> Some much older arches approximate catenaries, an example of which is the Arch of [[Taq-i Kisra]] in [[Ctesiphon]].<ref>{{cite book| last = Denny| first = Mark| title = Super Structures: The Science of Bridges, Buildings, Dams, and Other Feats of Engineering| year = 2010| publisher = JHU Press| isbn = 978-0-8018-9437-4| pages = 112–113 }}</ref>

[[File:Analogy between an arch and a hanging chain and comparison to the dome of St Peter's Cathedral in Rome.png|thumb|Analogy between an arch and a hanging chain and comparison to the dome of [[Saint Peter's Basilica]] in Rome ([[Giovanni Poleni]], 1748)]]
In 1671, Hooke announced to the [[Royal Society]] that he had solved the problem of the optimal shape of an arch, and in 1675 published an encrypted solution as a Latin [[anagram]]<ref>[[cf.]] the anagram for [[Hooke's law]], which appeared in the next paragraph.</ref> in an appendix to his ''Description of Helioscopes,''<ref>{{cite web |url=http://www.lindahall.org/events_exhib/exhibit/exhibits/civil/design.shtml |title=Arch Design |publisher=Lindahall.org |date=2002-10-28 |access-date=2010-11-17 |archive-url=https://web.archive.org/web/20101113210736/http://www.lindahall.org/events_exhib/exhibit/exhibits/civil/design.shtml |archive-date=2010-11-13 }}</ref> where he wrote that he had found "a true mathematical and mechanical form of all manner of Arches for Building." He did not publish the solution to this anagram<ref>The original anagram was ''abcccddeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuux'': the letters of the Latin phrase, alphabetized.</ref> in his lifetime, but in 1705 his executor provided it as ''ut pendet continuum flexile, sic stabit contiguum rigidum inversum'', meaning "As hangs a flexible cable so, inverted, stand the touching pieces of an arch."

In 1691, [[Gottfried Leibniz]], [[Christiaan Huygens]], and [[Johann Bernoulli]] derived the [[equation]] in response to a challenge by [[Jakob Bernoulli]];<ref name="Lockwood124"/> their solutions were published in the ''[[Acta Eruditorum]]'' for June 1691.<ref>{{citation|first=C.|last=Truesdell|title=The Rotational Mechanics of Flexible Or Elastic Bodies 1638–1788: Introduction to Leonhardi Euleri Opera Omnia Vol. X et XI Seriei Secundae|location=Zürich|
url=https://books.google.com/books?id=gxrzm6y10EwC&pg=PA66|page=66|publisher=Orell Füssli|date=1960|isbn=978-3-7643-1441-5}}</ref><ref name="calladine" >{{citation|first=C. R.|last=Calladine|title=An amateur's contribution to the design of Telford's Menai Suspension Bridge: a commentary on Gilbert (1826) 'On the mathematical theory of suspension bridges'|journal=Philosophical Transactions of the Royal Society A|date=2015-04-13|volume=373|issue=2039|article-number=20140346|doi=10.1098/rsta.2014.0346|pmid=25750153|pmc=4360092|bibcode=2015RSPTA.37340346C}}</ref> [[David Gregory (mathematician)|David Gregory]] wrote a treatise on the catenary in 1697<ref name="Lockwood124"/><ref>{{citation|first=Davidis|last=Gregorii|title=Catenaria|journal=Philosophical Transactions|volume=19|issue=231|date=August 1697|pages=637–652|doi=10.1098/rstl.1695.0114|doi-access=free}}</ref> in which he provided an incorrect derivation of the correct differential equation.<ref name="calladine" />

[[Leonhard Euler]] proved in 1744 that the catenary is the curve which, when rotated about the {{mvar|x}}-axis, gives the surface of minimum [[surface area]] (the [[catenoid]]) for the given bounding circles.<ref name="MathWorld"/> [[Nicolas Fuss]] gave equations describing the equilibrium of a chain under any [[force]] in 1796.<ref>[[#Routh|Routh]] Art. 455, footnote</ref>

==Inverted catenary arch==<!-- This section is linked from [[Park Güell]] -->

[[Catenary arches]] are often used in the construction of [[kiln]]s. To create the desired curve, the shape of a hanging chain of the desired dimensions is transferred to a form which is then used as a guide for the placement of bricks or other building material.<ref>{{cite book| last1 = Minogue| first1 = Coll| last2 = Sanderson| first2 = Robert| title = Wood-fired Ceramics: Contemporary Practices| year = 2000| publisher = University of Pennsylvania| isbn = 978-0-8122-3514-2| page = 42 }}</ref><ref>
{{cite book| last1 = Peterson| first1 = Susan| last2 = Peterson| first2 = Jan| title = The Craft and Art of Clay: A Complete Potter's Handbook| url = https://books.google.com/books?id=PAZR-A9Ra6EC&pg=PA208| year = 2003| publisher = Laurence King| isbn = 978-1-85669-354-7| page = 224 }}</ref>

The [[Gateway Arch]] in [[St. Louis, Missouri]], United States, is sometimes said to be an (inverted) catenary, but this is incorrect.<ref>{{Citation | last1=Osserman | first1=Robert | title=Mathematics of the Gateway Arch | url=https://www.ams.org/notices/201002/index.html | year=2010 | journal=[[Notices of the American Mathematical Society]] | issn=0002-9920 | volume=57 | issue=2 | pages=220–229}}</ref> It is close to a more general curve called a flattened catenary, with equation {{math|1=''y'' = ''A'' cosh(''Bx'')}}, which is a catenary if {{math|1=''AB'' = 1}}. While a catenary is the ideal shape for a freestanding arch of constant thickness, the Gateway Arch is narrower near the top. According to the U.S. [[National Historic Landmark]] nomination for the arch, it is a "[[weighted catenary]]" instead. Its shape corresponds to the shape that a weighted chain, having lighter links in the middle, would form.<ref>{{cite journal| last = Hicks| first = Clifford B.| title = The Incredible Gateway Arch: America's Mightiest National Monument| url = https://books.google.com/books?id=BuMDAAAAMBAJ&pg=PA89| volume = 120|date=December 1963| page = 89| issn = 0032-4558| issue = 6| journal = [[Popular Mechanics]] }}</ref><ref name="nrhpinv2">{{citation|url={{NHLS url|id=87001423}}|title=National Register of Historic Places Inventory-Nomination: Jefferson National Expansion Memorial Gateway Arch / Gateway Arch; or "The Arch"|year=1985 |first=Laura Soullière |last=Harrison |publisher=National Park Service }} and {{NHLS url|id=87001423|title=''Accompanying one photo, aerial, from 1975''|photos=y}}&nbsp;{{small|(578&nbsp;KB)}}</ref> 

<gallery mode="packed" heights="200px"> 
File:LaPedreraParabola.jpg|Catenary<ref>{{cite book| last = Sennott| first = Stephen| title = Encyclopedia of Twentieth Century Architecture| year = 2004| publisher = Taylor & Francis| isbn = 978-1-57958-433-7| page = 224 }}</ref> arches under the roof of [[Gaudí]]'s ''[[Casa Milà]]'', [[Barcelona]], Spain.
File:Sheffield Winter Garden.jpg|The [[Sheffield Winter Garden]] is enclosed by a series of [[catenary arches]].<ref>{{cite book| last = Hymers| first = Paul| title = Planning and Building a Conservatory| year = 2005| publisher = New Holland| isbn = 978-1-84330-910-9| page = 36 }}</ref>
File:Gateway Arch.jpg|The [[Gateway Arch]] ([[St. Louis, Missouri]]) is a flattened catenary.
File:CatenaryKilnConstruction06025.JPG|Catenary arch kiln under construction over temporary form
</gallery>

{{Clear}}

==Catenary bridges==
[[File:Soderskar-bridge.jpg|thumb|right|250px|[[Simple suspension bridge]]s are essentially thickened cables, and follow a catenary curve.]]
[[File:Puentedelabarra(below).jpg|thumb|right|250px|[[Stressed ribbon bridge]]s, like the [[Leonel Viera Bridge]] in [[Maldonado, Uruguay]], also follow a catenary curve, with cables embedded in a rigid deck.]]

In free-hanging chains, the force exerted is uniform with respect to length of the chain, and so the chain follows the catenary curve.<ref>{{cite book| first1=Owen |last1=Byer |first2=Felix |last2=Lazebnik |first3=Deirdre L. |last3=Smeltzer |author3-link=Deirdre Smeltzer |title=Methods for Euclidean Geometry |url=https://books.google.com/books?id=QkuVb672dWgC&pg=PA210 |date=2010-09-02 | publisher=MAA | isbn=978-0-88385-763-2 |page=210}}</ref> The same is true of a [[simple suspension bridge]] or "catenary bridge," where the roadway follows the cable.<ref>{{cite book| first=Leonardo| last=Fernández Troyano| title=Bridge Engineering: A Global Perspective| url=https://books.google.com/books?id=0u5G8E3uPUAC&pg=PA514| year=2003| publisher=Thomas Telford| isbn = 978-0-7277-3215-6| page = 514 }}</ref><ref>{{cite book| first1= W. |last1=Trinks|first2=M. H. |last2=Mawhinney|first3=R. A. |last3=Shannon|first4=R. J. |last4=Reed| first5=J. R.| last5=Garvey| title=Industrial Furnaces| url=https://books.google.com/books?id=EqRTAAAAMAAJ&pg=PA132| date=2003-12-05| publisher=Wiley| isbn=978-0-471-38706-0| page=132 }}</ref>

A [[stressed ribbon bridge]] is a more sophisticated structure with the same catenary shape.<ref>{{cite book| first= John S. |last=Scott| title=Dictionary Of Civil Engineering| date=1992-10-31| publisher=Springer| isbn=978-0-412-98421-1| page=433}}</ref><ref>{{cite journal| title=Cranked stress ribbon design to span Medway| url=https://www.architectsjournal.co.uk/archive/cranked-stress-ribbon-design-to-span-medway| first=Paul| last=Finch| journal=[[Architects' Journal]] |volume=207 |date=19 March 1998 |page=51}}</ref>

However, in a [[suspension bridge]] with a suspended roadway, the chains or cables support the weight of the bridge, and so do not hang freely. In most cases the roadway is flat, so when the weight of the cable is negligible compared with the weight being supported, the force exerted is uniform with respect to horizontal distance, and the result is a [[parabola]], as discussed below (although the term "catenary" is often still used, in an informal sense). If the cable is heavy then the resulting curve is between a catenary and a parabola.<ref name="Lockwood122">[[#Lockwood|Lockwood]] p. 122</ref><ref>
{{cite web
|title=Hanging With Galileo
|date=June 30, 2006
|first=Paul
|last=Kunkel
|publisher=Whistler Alley Mathematics
|url=http://whistleralley.com/hanging/hanging.htm
|access-date=March 27, 2009
}}</ref>
[[File:Comparison catenary parabola.svg|thumb|none|400px|Comparison of a [[catenary arch]] (black dotted curve) and a [[parabolic arch]] (red solid curve) with the same span and sag. The catenary represents the profile of a simple suspension bridge, or the cable of a suspended-deck suspension bridge on which its deck and hangers have negligible weight compared to its cable. The parabola represents the profile of the cable of a suspended-deck suspension bridge on which its cable and hangers have negligible weight compared to its deck. The profile of the cable of a real suspension bridge with the same span and sag lies between the two curves. The catenary and parabola equations are respectively, <math>y = \text{cosh } x </math> and <math>y = x ^ 2 [(\text{cosh }1) - 1] + 1</math> ]]

==Anchoring of marine objects==
[[File:Catenary.PNG|thumb|right|250px|A heavy [[anchor]] chain forms a catenary, with a low angle of pull on the anchor.]]
The catenary produced by gravity provides an advantage to heavy [[wikt:rode#Noun|anchor rodes]]. An anchor rode (or anchor line) usually consists of chain or cable or both. Anchor rodes are used by ships, oil rigs, docks, [[floating wind turbine]]s, and other marine equipment which must be anchored to the seabed.

When the rope is slack, the catenary curve presents a lower angle of pull on the [[anchor]] or mooring device than would be the case if it were nearly straight. This enhances the performance of the anchor and raises the level of force it will resist before dragging. To maintain the catenary shape in the presence of wind, a heavy chain is needed, so that only larger ships in deeper water can rely on this effect. Smaller boats also rely on catenary to maintain maximum holding power.<ref>{{cite web |url=http://www.petersmith.net.nz/boat-anchors/catenary.php |title=Chain, Rope, and Catenary – Anchor Systems For Small Boats |website=Petersmith.net.nz |access-date=2010-11-17}}</ref>

[[Cable ferries]] and [[chain boat]]s present a special case of marine vehicles moving although moored by the two catenaries each of one or more cables (wire ropes or chains) passing through the vehicle and moved along by motorized sheaves. The catenaries can be evaluated graphically.<ref>{{cite web |url=http://hupi.org/HPeJ/0033/0033.html |title=Efficiency of Cable Ferries - Part 2 |website=Human Power eJournal |access-date=2023-12-08}}</ref>

==Mathematical description==

===Equation===
[[Image:catenary-pm.svg|thumb|350px|right|Catenaries for different values of {{mvar|a}}]]

The equation of a catenary in [[Cartesian coordinate system|Cartesian coordinates]] has the form<ref name="Lockwood122"/>

<math display=block>y = a \cosh \left(\frac{x}{a}\right) = \frac{a}{2}\left(e^\frac{x}{a} + e^{-\frac{x}{a}}\right),</math>
where {{math|cosh}} is the [[hyperbolic function|hyperbolic cosine function]], and where {{mvar|a}} is the distance of the lowest point above the x axis.<ref>{{cite web |url=http://mathworld.wolfram.com/Catenary.html |title=Catenary |last=Weisstein |first=Eric W. |author-link=Eric W. Weisstein |work=MathWorld--A Wolfram Web Resource |access-date=2019-09-21 |quote=The parametric equations for the catenary are given by x(t) = t, y(t) = [...] a cosh(t/a), where t=0 corresponds to the vertex [...] }}</ref> All catenary curves are [[Similarity (geometry)|similar]] to each other, since changing the parameter {{mvar|a}} is equivalent to a [[uniform scaling]] of the curve.

The [[Whewell equation]] for the catenary is<ref name="Lockwood122"/>
<math display=block>\tan \varphi = \frac{s}{a},</math>
where <math>\varphi</math> is the [[tangential angle]] and {{mvar|s}} the [[arc length]].

Differentiating gives
<math display=block>\frac{d\varphi}{ds} = \frac{\cos^2\varphi}{a},</math>
and eliminating <math>\varphi</math> gives the [[Cesàro equation]]<ref>[[#MathWorld|MathWorld]], eq. 7</ref>
<math display=block>\kappa=\frac{a}{s^2+a^2},</math>
where <math>\kappa</math> is the [[curvature]].

The [[radius of curvature]] is then
<math display=block>\rho = a \sec^2 \varphi,</math>
which is the length of the [[normal line|normal]] between the curve and the {{mvar|x}}-axis.<ref>[[#Routh|Routh]] Art. 444</ref>

===Relation to other curves===
When a [[parabola]] is rolled along a straight line, the [[roulette (curve)|roulette]] curve traced by its [[Conic section#Eccentricity, focus and directrix|focus]] is a catenary.<ref name="Yates 13"/> The [[Envelope (mathematics)|envelope]] of the [[Conic section#Eccentricity, focus and directrix|directrix]] of the parabola is also a catenary.<ref>Yates p. 80</ref> The [[involute]] from the vertex, that is the roulette traced by a point starting at the vertex when a line is rolled on a catenary, is the [[tractrix]].<ref name="Yates 13"/>

Another roulette, formed by rolling a line on a catenary, is another line. This implies that [[square wheel]]s can roll perfectly smoothly on a road made of a series of bumps in the shape of an inverted catenary curve. The wheels can be any [[regular polygon]] except a triangle, but the catenary must have parameters corresponding to the shape and dimensions of the wheels.<ref>{{cite journal |last1=Hall |first1=Leon |last2=Wagon |first2=Stan |author2-link=Stan Wagon|year=1992 |title=Roads and Wheels |journal=Mathematics Magazine |volume=65 |issue=5 |pages=283–301 |jstor=2691240 |doi=10.2307/2691240}}
</ref>

===Geometrical properties===
Over any horizontal interval, the ratio of the area under the catenary to its length equals {{mvar|a}}, independent of the interval selected. The catenary is the only plane curve other than a horizontal line with this property. Also, the geometric centroid of the area under a stretch of catenary is the midpoint of the perpendicular segment connecting the centroid of the curve itself and the {{mvar|x}}-axis.<ref>{{cite journal |last=Parker |first=Edward |year=2010 |title=A Property Characterizing the Catenary |journal=Mathematics Magazine |volume=83 |pages=63–64 |doi=10.4169/002557010X485120 |s2cid=122116662 }}</ref>

===Science===
A moving [[electric charge|charge]] in a uniform [[electric field]] travels along a catenary (which tends to a [[parabola]] if the charge velocity is much less than the [[speed of light]] {{mvar|c}}).<ref>{{cite book| last=Landau| first=Lev Davidovich| url=https://books.google.com/books?id=X18PF4oKyrUC&pg=PA56| title=The Classical Theory of Fields| year=1975| publisher=Butterworth-Heinemann| isbn=978-0-7506-2768-9| page=56}}</ref>

The [[surface of revolution]] with fixed radii at either end that has minimum surface area is a catenary

<math display="block">y = a \cosh^{-1}\left(\frac{x}{a}\right) + b</math>

revolved about the <math>y</math>-axis.<ref name="Yates 13">{{cite book |title=Curves and their Properties |first=Robert C. |last=Yates |publisher=NCTM |year=1952 |page=13}}</ref>

==Analysis==

===Model of chains and arches===
In the [[mathematical model]] the chain (or cord, cable, rope, string, etc.) is idealized by assuming that it is so thin that it can be regarded as a [[curve]] and that it is so flexible any force of [[Tension (physics)|tension]] exerted by the chain is parallel to the chain.<ref>[[#Routh|Routh]] Art. 442, p. 316</ref> The analysis of the curve for an optimal arch is similar except that the forces of tension become forces of [[Compression (physics)|compression]] and everything is inverted.<ref>{{cite book| last=Church| first=Irving Porter| title=Mechanics of Engineering| url=https://archive.org/details/mechanicsengine06churgoog| year=1890| publisher=Wiley| page=[https://archive.org/details/mechanicsengine06churgoog/page/n410 387]}}</ref>
An underlying principle is that the chain may be considered a rigid body once it has attained equilibrium.<ref>[[#Whewell|Whewell]] p. 65</ref> Equations which define the shape of the curve and the tension of the chain at each point may be derived by a careful inspection of the various forces acting on a segment using the fact that these forces must be in balance if the chain is in [[static equilibrium]].

Let the path followed by the chain be given [[parametric equations|parametrically]] by {{math|1='''r''' = (''x'', ''y'') = (''x''(''s''), ''y''(''s''))}} where {{mvar|s}} represents [[arc length]] and {{math|'''r'''}} is the [[position vector]]. This is the [[Unit speed parametrization|natural parameterization]] and has the property that

<math display=block>\frac{d\mathbf{r}}{ds}=\mathbf{u}</math>

where {{math|'''u'''}} is a [[unit tangent vector]].

[[File:CatenaryForceDiagram.svg|thumb|Diagram of forces acting on a segment of a catenary from {{math|'''c'''}} to {{math|'''r'''}}. The forces are the tension {{math|'''T'''<sub>0</sub>}} at {{math|'''c'''}}, the tension {{math|'''T'''}} at {{math|'''r'''}}, and the weight of the chain {{math|(0, −''ws'')}}. Since the chain is at rest the sum of these forces must be zero.]]
A [[differential equation]] for the curve may be derived as follows.<ref>Following [[#Routh|Routh]] Art. 443 p. 316</ref> Let {{math|'''c'''}} be the lowest point on the chain, called the vertex of the catenary.<ref>[[#Routh|Routh]] Art. 443 p. 317</ref> The slope {{math|{{sfrac|''dy''|''dx''}}}} of the curve is zero at {{math|'''c'''}} since it is a minimum point. Assume {{math|'''r'''}} is to the right of {{math|'''c'''}} since the other case is implied by symmetry. The forces acting on the section of the chain from {{math|'''c'''}} to {{math|'''r'''}} are the tension of the chain at {{math|'''c'''}}, the tension of the chain at {{math|'''r'''}}, and the weight of the chain. The tension at {{math|'''c'''}} is tangent to the curve at {{math|'''c'''}} and is therefore horizontal without any vertical component and it pulls the section to the left so it may be written {{math|(−''T''<sub>0</sub>, 0)}} where {{math|''T''<sub>0</sub>}} is the magnitude of the force. The tension at {{math|'''r'''}} is parallel to the curve at {{math|'''r'''}} and pulls the section to the right. The tension at {{math|'''r'''}} can be split into two components so it may be written {{math|1=''T'''''u''' = (''T'' cos ''φ'', ''T'' sin ''φ'')}}, where {{mvar|T}} is the magnitude of the force and {{mvar|φ}} is the angle between the curve at {{math|'''r'''}} and the {{mvar|x}}-axis (see [[tangential angle]]). Finally, the weight of the chain is represented by {{math|(0, −''ws'')}} where {{mvar|w}} is the weight per unit length and {{mvar|s}} is the length of the segment of chain between {{math|'''c'''}} and {{math|'''r'''}}.

The chain is in equilibrium so the sum of three forces is {{math|'''0'''}}, therefore

<math display=block>T \cos \varphi = T_0</math>
and
<math display=block>T \sin \varphi = ws\,,</math>

and dividing these gives

<math display=block>\frac{dy}{dx}=\tan \varphi = \frac{ws}{T_0}\,.</math>

It is convenient to write

<math display=block>a = \frac{T_0}{w}</math>

which is the length of chain whose weight is equal in magnitude to the tension at {{math|'''c'''}}.<ref>[[#Whewell|Whewell]] p. 67</ref>  Then

<math display=block>\frac{dy}{dx}=\frac{s}{a}</math>

is an equation defining the curve.

The horizontal component of the tension, {{math|1=''T'' cos ''φ'' = ''T''<sub>0</sub>}} is constant and the vertical component of the tension, {{math|1=''T'' sin ''φ'' = ''ws''}} is proportional to the length of chain between {{math|'''r'''}} and the vertex.<ref>[[#Routh|Routh]] Art 443, p. 318</ref>

===Derivation of equations for the curve===
The differential equation <math>dy/dx = s/a</math>, given above, can be solved
to produce equations for the curve.
<ref>
  A minor variation of the derivation presented here
  can be found on page 107 of [[#Maurer|Maurer]].
  A different (though ultimately mathematically equivalent) derivation,
  which does not make use of hyperbolic function notation,
  can be found in [[#Routh|Routh]]
  (Article 443, starting in particular at page 317).
</ref>
We will solve the equation using the boundary condition that
the vertex is positioned at <math>s_0=0</math> and <math>(x,y)=(x_0,y_0)</math>.

First, invoke the formula for
[[Arc length#Finding arc lengths by integration|arc length]]
to get
<math display=block>\frac{ds}{dx}
  = \sqrt{1+\left(\frac{dy}{dx}\right)^2}
  = \sqrt{1+\left(\frac{s}{a}\right)^2}\,,</math>
then [[Separation of variables#Ordinary differential equations (ODE)|separate variables]]
to obtain
<math display=block>\frac{ds}{\sqrt{1+(s/a)^2}}
    = dx\,.</math>

A reasonably straightforward approach to integrate this is to use
[[Trigonometric substitution#Hyperbolic substitution|hyperbolic substitution]],
which gives
<math display=block>a \sinh^{-1}\frac{s}{a} + x_0 = x</math>
(where <math>x_0</math> is a [[constant of integration]]),
and hence
<math display=block>\frac{s}{a} = \sinh\frac{x-x_0}{a}\,.</math>

But <math display=inline>s/a = dy/dx</math>, so
<math display=block>\frac{dy}{dx} = \sinh\frac{x-x_0}{a}\,,</math>
which [[Hyperbolic functions#Standard integrals|integrates as]]
<math display=block>y = a \cosh\frac{x-x_0}{a} + \delta</math>
(with <math>\delta=y_0-a</math> being the constant of integration satisfying the boundary condition).

Since the primary interest here is simply the shape of the curve,
the placement of the coordinate axes are arbitrary;
so make the convenient choice of <math display=inline>x_0=0=\delta</math>
to simplify the result to
<math display=block>y = a \cosh\frac{x}{a}.</math>

For completeness, the <math>y \leftrightarrow s</math> relation can be derived by
solving each of the <math>x \leftrightarrow y</math> and <math>x \leftrightarrow s</math> relations for <math>x/a</math>, giving:
<math display=block>\cosh^{-1}\frac{y-\delta}{a} = \frac{x-x_0}{a} = \sinh^{-1}\frac{s}{a}\,,</math>
so
<math display=block>y-\delta = a\cosh\left(\sinh^{-1}\frac{s}{a}\right)\,,</math>
which [[Inverse hyperbolic function#Composition of hyperbolic and inverse hyperbolic functions|can be rewritten]] as
<math display=block>y-\delta = a\sqrt{1+\left(\frac{s}{a}\right)^2} = \sqrt{a^2 + s^2}\,.</math>

===Alternative derivation===
The differential equation can be solved using a different approach.<ref>Following Lamb p. 342</ref> From

<math display=block>s = a \tan \varphi</math>

it follows that

<math display=block>\frac{dx}{d\varphi} = \frac{dx}{ds}\frac{ds}{d\varphi}=\cos \varphi \cdot a \sec^2 \varphi= a \sec \varphi</math>
and
<math display=block>\frac{dy}{d\varphi} = \frac{dy}{ds}\frac{ds}{d\varphi}=\sin \varphi \cdot a \sec^2 \varphi= a \tan \varphi \sec \varphi\,.</math>

Integrating gives,

<math display=block>x = a \ln(\sec \varphi + \tan \varphi) + \alpha</math>
and
<math display=block>y = a \sec \varphi + \beta\,.</math>

As before, the {{mvar|x}} and {{mvar|y}}-axes can be shifted so {{mvar|α}} and {{mvar|β}} can be taken to be 0. Then

<math display=block>\sec \varphi + \tan \varphi = e^\frac{x}{a}\,,</math>
and taking the reciprocal of both sides
<math display=block>\sec \varphi - \tan \varphi = e^{-\frac{x}{a}}\,.</math>

Adding and subtracting the last two equations then gives the solution
<math display=block>y = a \sec \varphi = a \cosh\left(\frac{x}{a}\right)\,,</math>
and
<math display=block>s = a \tan \varphi = a \sinh\left(\frac{x}{a}\right)\,.</math>

===Determining parameters===
[[Image:Catenary-tension.svg|350px|thumb|Three catenaries through the same two points, for different lengths of the catenary.]]
The parameter {{mvar|a}} is the minimal {{mvar|y}} coordinate of the points of the catenary. In practical applications, neither this parameter, nor the axes where the catenary has its standard form are known ''a priori''. 

These data can be determined from the position of two given points {{tmath|P_1}} and {{tmath|P_2}} and the length {{mvar|L}} of the catenary between these points as follows:<ref>Following Todhunter Art. 186</ref>

Relabel if necessary so that {{math|''P''<sub>1</sub>}} is to the left of {{math|''P''<sub>2</sub>}} and let {{mvar|H}} be the horizontal and {{mvar|v}} be the vertical distance from {{math|''P''<sub>1</sub>}} to {{math|''P''<sub>2</sub>}}. [[Translation (geometry)|Translate]] the axes so that the vertex of the catenary lies on the {{mvar|y}}-axis and its height {{mvar|a}} is adjusted so the catenary satisfies the standard equation of the curve
 
<math display=block>y = a \cosh\left(\frac{x}{a}\right)</math>

and let the coordinates of {{math|''P''<sub>1</sub>}} and {{math|''P''<sub>2</sub>}} be {{math|(''x''<sub>1</sub>, ''y''<sub>1</sub>)}} and {{math|(''x''<sub>2</sub>, ''y''<sub>2</sub>)}} respectively. The curve passes through these points, so the difference of height is

<math display=block>v = a \cosh\left(\frac{x_2}{a}\right) - a \cosh\left(\frac{x_1}{a}\right)\,.</math>

and the length of the curve from {{math|''P''<sub>1</sub>}} to {{math|''P''<sub>2</sub>}} is

<math display="block">L = a \sinh\left(\frac{x_2}{a}\right) - a \sinh\left(\frac{x_1}{a}\right)\,.</math>

When {{math|''L''<sup>2</sup> − ''v''<sup>2</sup>}} is expanded using these expressions the result is

<math display="block">L^2-v^2=2a^2\left(\cosh\left(\frac{x_2-x_1}{a}\right)-1\right)=4a^2\sinh^2\left(\frac{H}{2a}\right)\,,</math>
so
<math display="block">\frac 1H \sqrt{L^2-v^2}=\frac{2a}H \sinh\left(\frac{H}{2a}\right)\,.</math>

This is a transcendental equation in {{mvar|a}} and must be solved [[Numerical analysis|numerically]]. Since <math>\sinh(x)/x</math> is strictly monotonic on <math>x > 0</math>,<ref>See [[#Routh|Routh]] art. 447</ref> there is at most one solution with {{math|''a'' > 0}} and so there is at most one position of equilibrium.

However, if both ends of the curve ({{math|''P''<sub>1</sub>}} and {{math|''P''<sub>2</sub>}}) are at the same level ({{math|''y''<sub>1</sub> {{=}} ''y''<sub>2</sub>}}), it can be shown that<ref>Archived at [https://ghostarchive.org/varchive/youtube/20211205/T-gUVEs51-c Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20201028222841/https://www.youtube.com/watch?v=T-gUVEs51-c Wayback Machine]{{cbignore}}: {{cite web| url = https://www.youtube.com/watch?v=T-gUVEs51-c| title = Chaînette - partie 3: longueur | website=[[YouTube]]| date = 6 January 2015 }}{{cbignore}}</ref>
<math display=block>a = \frac {\frac14 L^2-h^2} {2h}\,</math>
where L is the total length of the curve between {{math|''P''<sub>1</sub>}} and {{math|''P''<sub>2</sub>}} and {{mvar|h}} is the sag (vertical distance between {{math|''P''<sub>1</sub>}}, {{math|''P''<sub>2</sub>}} and the vertex of the curve).

It can also be shown that
<math display=block>L = 2a \sinh \frac {H} {2a}\,</math>
and
<math display=block>H = 2a \operatorname {arcosh} \frac {h+a} {a}\,</math>
where H is the horizontal distance between {{math|''P''<sub>1</sub>}} and {{math|''P''<sub>2</sub>}} which are located at the same level  ({{math|''H'' {{=}} ''x''<sub>2</sub> − ''x''<sub>1</sub>}}).

The horizontal traction force at {{math|''P''<sub>1</sub>}} and {{math|''P''<sub>2</sub>}} is {{math|''T<sub>0</sub>'' {{=}} ''wa''}}, where {{mvar|w}} is the weight per unit length of the chain or cable.

===Tension relations===
There is a simple relationship between the tension in the cable at a point and its {{mvar|x}}- and/or {{mvar|y}}- coordinate. Begin by combining the squares of the vector components of the tension:
<math display=block>(T\cos\varphi)^2 + (T\sin\varphi)^2 = T_0^2 + (ws)^2</math>
which (recalling that <math>T_0=wa</math>) can be rewritten as
<math display=block>\begin{align}
T^2(\cos^2\varphi + \sin^2\varphi) &= (wa)^2 + (ws)^2 \\[6pt]
T^2 &= w^2 (a^2 + s^2) \\[6pt]
T &= w\sqrt{a^2+s^2} \,.
\end{align}</math>
But, [[#Derivation of equations for the curve|as shown above]],
<math>y = \sqrt{a^2 + s^2}</math> (assuming that <math>y_0=a</math>), so we get the simple relations<ref>[[#Routh|Routh]] Art 443, p. 318</ref>
<math display=block>T = wy = wa \cosh\frac{x}{a}\,.</math>

=== Variational formulation ===
Consider a chain of length <math>L</math> suspended from two points of equal height and at distance <math>D</math>. The curve has to minimize its potential energy
<math display=block> U = \int_0^D w y\sqrt{1+y'^2} dx </math>
(where {{mvar|w}} is the weight per unit length) and is subject to the constraint 
<math display=block> \int_0^D \sqrt{1+y'^2} dx = L\,.</math>

The modified [[Calculus of variations|Lagrangian]] is therefore 
<math display=block> \mathcal{L} = (w y - \lambda )\sqrt{1+y'^2}</math>
where <math>\lambda </math> is the [[Lagrange multiplier]] to be determined. As the independent variable <math>x</math> does not appear in the Lagrangian, we can use the [[Beltrami identity]] 
<math display=block> \mathcal{L}-y' \frac{\partial \mathcal{L} }{\partial y'} = C </math>
where <math>C</math> is an integration constant, in order to obtain a first integral
<math display=block>\frac{(w y - \lambda )}{\sqrt{1+y'^2}} = -C</math>

This is an ordinary first order differential equation that can be solved by the method of [[separation of variables]]. Its solution is the usual hyperbolic cosine where the parameters are obtained from the constraints.

==Generalizations with vertical force==

===Nonuniform chains===
If the density of the chain is variable then the analysis above can be adapted to produce equations for the curve given the density, or given the curve to find the density.<ref>Following [[#Routh|Routh]] Art. 450</ref>

Let {{mvar|w}} denote the weight per unit length of the chain, then the weight of the chain has magnitude

<math display=block>\int_\mathbf{c}^\mathbf{r} w\, ds\,,</math>

where the limits of integration are {{math|'''c'''}} and {{math|'''r'''}}. Balancing forces as in the uniform chain produces

<math display=block>T \cos \varphi = T_0</math>
and
<math display=block>T \sin \varphi = \int_\mathbf{c}^\mathbf{r} w\, ds\,,</math>
and therefore
<math display=block>\frac{dy}{dx}=\tan \varphi = \frac{1}{T_0} \int_\mathbf{c}^\mathbf{r} w\, ds\,.</math>

Differentiation then gives

<math display=block>w=T_0 \frac{d}{ds}\frac{dy}{dx} = \frac{T_0 \dfrac{d^2y}{dx^2}}{\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}}\,.</math>

In terms of {{mvar|φ}} and the radius of curvature {{mvar|ρ}} this becomes

<math display=block>w= \frac{T_0}{\rho \cos^2 \varphi}\,.</math>

===Suspension bridge curve===
[[File:Golden Gate Bridge, SF.jpg|thumb|right|480px|[[Golden Gate Bridge]]. Most [[suspension bridge]] cables follow a parabolic, not a catenary curve, because the roadway is much heavier than the cable.]]

A similar analysis can be done to find the curve followed by the cable supporting a [[suspension bridge]] with a horizontal roadway.<ref>Following [[#Routh|Routh]] Art. 452</ref> If the weight of the roadway per unit length is {{mvar|w}} and the weight of the cable and the wire supporting the bridge is negligible in comparison, then the weight on the cable (see the figure in [[Catenary#Model of chains and arches]]) from {{math|'''c'''}} to {{math|'''r'''}} is {{mvar|wx}} where {{mvar|x}} is the horizontal distance between {{math|'''c'''}} and {{math|'''r'''}}. Proceeding as before gives the differential equation

<math display=block>\frac{dy}{dx}=\tan \varphi = \frac{w}{T_0}x\,. </math>

This is solved by simple integration to get

<math display=block>y=\frac{w}{2T_0}x^2 + \beta</math>

and so the cable follows a parabola. If the weight of the cable and supporting wires is not negligible then the analysis is more complex.<ref>Ira Freeman investigated the case where only the cable and roadway are significant, see the External links section. [[#Routh|Routh]] gives the case where only the supporting wires have significant weight as an exercise.</ref>

===Catenary of equal strength===
In a catenary of equal strength, the cable is strengthened according to the magnitude of the tension at each point, so its resistance to breaking is constant along its length. Assuming that the strength of the cable is proportional to its density per unit length, the weight, {{mvar|w}}, per unit length of the chain can be written {{mvar|{{sfrac|T|c}}}}, where {{mvar|c}} is constant, and the analysis for nonuniform chains can be applied.<ref>Following [[#Routh|Routh]] Art. 453</ref>

In this case the equations for tension are

<math display=block>\begin{align}
T \cos \varphi &= T_0\,,\\
T \sin \varphi &= \frac{1}{c}\int T\, ds\,.
\end{align}</math>

Combining gives

<math display=block>c \tan \varphi = \int \sec \varphi\, ds</math>

and by differentiation

<math display=block>c = \rho \cos \varphi</math>

where {{mvar|ρ}} is the radius of curvature.

The solution to this is

<math display=block>y = c \ln\left(\sec\left(\frac{x}{c}\right)\right)\,.</math>

In this case, the curve has vertical asymptotes and this limits the span to {{math|π''c''}}. Other relations are

<math display=block>x = c\varphi\,,\quad s = \ln\left(\tan\left(\frac{\pi+2\varphi}{4}\right)\right)\,.</math>

The curve was studied 1826 by [[Davies Gilbert]] and, apparently independently, by [[Gaspard-Gustave Coriolis]] in 1836.

Recently, it was shown that this type of catenary could act as a building block of [[electromagnetic metasurface]] and was known as "catenary of equal phase gradient".<ref>{{cite journal 
|last1=Pu|first1=Mingbo|last2=Li|first2=Xiong|last3=Ma|first3=Xiaoliang|last4=Luo|first4=Xiangang|year=2015|title=Catenary Optics for Achromatic Generation of Perfect Optical Angular Momentum
|journal=Science Advances|volume=1|issue= 9|article-number=e1500396 |url= |doi=10.1126/sciadv.1500396|pmid=26601283|pmc=4646797|bibcode=2015SciA....1E0396P}}
</ref>

===Elastic catenary===
In an [[Elasticity (physics)|elastic]] catenary, the chain is replaced by a [[Spring (device)|spring]] which can stretch in response to tension. The spring is assumed to stretch in accordance with [[Hooke's law]]. Specifically, if {{math|p}} is the natural length of a section of spring, then the length of the spring with tension {{mvar|T}} applied has length

<math display=block>s=\left(1+\frac{T}{E}\right)p\,,</math>

where {{mvar|E}} is a constant equal to {{mvar|kp}}, where {{mvar|k}} is the [[stiffness]] of the spring.<ref>[[#Routh|Routh]] Art. 489</ref> In the catenary the value of {{mvar|T}} is variable, but ratio remains valid at a local level, so<ref>[[#Routh|Routh]] Art. 494</ref>
<math display=block>\frac{ds}{dp}=1+\frac{T}{E}\,.</math>
The curve followed by an elastic spring can now be derived following a similar method as for the inelastic spring.<ref>Following [[#Routh|Routh]] Art. 500</ref>

The equations for tension of the spring are

<math display=block>T \cos \varphi = T_0\,,</math>
and
<math display=block>T \sin \varphi = w_0 p\,,</math>

from which

<math display=block>\frac{dy}{dx}=\tan \varphi = \frac{w_0 p}{T_0}\,,\quad T=\sqrt{T_0^2+w_0^2 p^2}\,,</math>

where {{mvar|p}} is the natural length of the segment from {{math|'''c'''}} to {{math|'''r'''}} and {{math|''w''<sub>0</sub>}} is the weight per unit length of the spring with no tension. Write
<math display=block>a = \frac{T_0}{w_0}</math>
so
<math display=block>\frac{dy}{dx}=\tan \varphi = \frac{p}{a} \quad\text{and}\quad T=\frac{T_0}{a}\sqrt{a^2+p^2}\,.</math>

Then 
<math display=block>\begin{align}
\frac{dx}{ds} &= \cos \varphi = \frac{T_0}{T} \\[6pt]
\frac{dy}{ds} &= \sin \varphi = \frac{w_0 p}{T}\,,
\end{align}</math>
from which
<math display=block>\begin{alignat}{3}
\frac{dx}{dp} &= \frac{T_0}{T}\frac{ds}{dp} &&= T_0\left(\frac{1}{T}+\frac{1}{E}\right) &&= \frac{a}{\sqrt{a^2+p^2}}+\frac{T_0}{E} \\[6pt]
\frac{dy}{dp} &= \frac{w_0 p}{T}\frac{ds}{dp} &&= \frac{T_0p}{a}\left(\frac{1}{T}+\frac{1}{E}\right) &&= \frac{p}{\sqrt{a^2+p^2}}+\frac{T_0p}{Ea}\,.
\end{alignat}</math>

Integrating gives the parametric equations

<math display=block>\begin{align}
x&=a\operatorname{arsinh}\left(\frac{p}{a}\right)+\frac{T_0}{E}p + \alpha\,, \\[6pt]
y&=\sqrt{a^2+p^2}+\frac{T_0}{2Ea}p^2+\beta\,.
\end{align}</math>

Again, the {{mvar|x}} and {{mvar|y}}-axes can be shifted so {{mvar|α}} and {{mvar|β}} can be taken to be 0. So

<math display=block>\begin{align}
x&=a\operatorname{arsinh}\left(\frac{p}{a}\right)+\frac{T_0}{E}p\,, \\[6pt]
y&=\sqrt{a^2+p^2}+\frac{T_0}{2Ea}p^2
\end{align}</math>

are parametric equations for the curve. At the rigid [[Limit (mathematics)|limit]] where {{mvar|E}} is large, the shape of the curve reduces to that of a non-elastic chain.

==Other generalizations==

===Chain under a general force===
With no assumptions being made regarding the force {{math|'''G'''}} acting on the chain, the following analysis can be made.<ref>Follows [[#Routh|Routh]] Art. 455</ref>

First, let {{math|1='''T''' = '''T'''(''s'')}} be the force of tension as a function of {{mvar|s}}. The chain is flexible so it can only exert a force parallel to itself. Since tension is defined as the force that the chain exerts on itself, {{math|'''T'''}} must be parallel to the chain. In other words,

<math display=block>\mathbf{T} = T \mathbf{u}\,,</math>

where {{mvar|T}} is the magnitude of {{math|'''T'''}} and {{math|'''u'''}} is the unit tangent vector.

Second, let {{math|1='''G''' = '''G'''(''s'')}} be the external force per unit length acting on a small segment of a chain as a function of {{mvar|s}}. The forces acting on the segment of the chain between {{mvar|s}} and {{math|''s'' + Δ''s''}} are the force of tension {{math|'''T'''(''s'' + Δ''s'')}} at one end of the segment, the nearly opposite force {{math|−'''T'''(''s'')}} at the other end, and the external force acting on the segment which is approximately {{math|'''G'''Δ''s''}}. These forces must balance so

<math display=block>\mathbf{T}(s+\Delta s)-\mathbf{T}(s)+\mathbf{G}\Delta s \approx \mathbf{0}\,.</math>

Divide by {{math|Δ''s''}} and take the limit as {{math|Δ''s'' → 0}} to obtain

<math display=block>\frac{d\mathbf{T}}{ds} + \mathbf{G} = \mathbf{0}\,.</math>

These equations can be used as the starting point in the analysis of a flexible chain acting under any external force. In the case of the standard catenary, {{math|1='''G''' = (0, −''w'')}} where the chain has weight {{mvar|w}} per unit length.

==See also==
* [[Catenary arch]]
* [[Chain fountain]] or self-siphoning beads
* [[Funicular curve]]
* [[Overhead catenary]] – power lines suspended over rail or tram vehicles
* [[Roulette (curve)]] – an elliptic/hyperbolic catenary
* [[Troposkein]] – the shape of a spun rope
* [[Weighted catenary]]

==Notes==
{{Reflist}}

==Bibliography==
* {{anchor|Lockwood}}{{cite book |title=A Book of Curves|first=E.H.|last=Lockwood|publisher=Cambridge|year=1961
|chapter=Chapter 13: The Tractrix and Catenary|chapter-url=https://archive.org/details/bookofcurves006299mbp}}
* {{cite book|last=Salmon|first=George | title=Higher Plane Curves
|url=https://archive.org/details/3edtreatiseonhighesalmuoft|publisher=Hodges, Foster and Figgis|year=1879|pages=[https://archive.org/details/3edtreatiseonhighesalmuoft/page/287 287]–289
}}
* {{anchor|Routh}}{{cite book| last = Routh| first = Edward John| author-link = Edward Routh| title = A Treatise on Analytical Statics| chapter-url = https://books.google.com/books?id=3N5JAAAAMAAJ&pg=PA315| year = 1891| publisher = University Press| chapter = Chapter X: On Strings }}
* {{anchor|Maurer}}{{cite book| last = Maurer| first = Edward Rose| title = Technical Mechanics| chapter-url = https://books.google.com/books?id=L98uAQAAIAAJ&pg=PA107| year = 1914| publisher = J. Wiley & Sons| chapter = Art. 26 Catenary Cable }}
* {{cite book| last = Lamb| first = Sir Horace| title = An Elementary Course of Infinitesimal Calculus| chapter-url = https://books.google.com/books?id=eDM6AAAAMAAJ&pg=PA342| year = 1897| publisher = University Press| chapter = Art. 134 Transcendental Curves; Catenary, Tractrix }}
* {{cite book| last = Todhunter| first = Isaac| author-link = Isaac Todhunter| title = A Treatise on Analytical Statics| chapter-url = https://books.google.com/books?id=-iEuAAAAYAAJ&pg=PA199| year = 1858| publisher = Macmillan| chapter = XI Flexible Strings. Inextensible, XII Flexible Strings. Extensible }}
* {{anchor|Whewell}}{{cite book| last = Whewell| first = William| author-link = William Whewell| title = Analytical Statics| chapter-url = https://books.google.com/books?id=BF8JAAAAIAAJ&pg=PA65| year = 1833| publisher = J. & J.J. Deighton| page = 65| chapter = Chapter V: The Equilibrium of a Flexible Body }}
* {{anchor|MathWorld}}{{mathworld|Catenary|Catenary}}

==Further reading==
* {{cite book| last = Swetz| first = Frank| title = Learn from the Masters| url = https://books.google.com/books?id=gqGLoh-WYrEC&pg=PA128| year = 1995| publisher = MAA| isbn = 978-0-88385-703-8| pages = 128–9 }}
* {{cite book| last = Venturoli| first = Giuseppe| others = Trans. Daniel Cresswell| title = Elements of the Theory of Mechanics| chapter-url = https://books.google.com/books?id=kHhBAAAAYAAJ&pg=PA67| year = 1822| publisher = J. Nicholson & Son| chapter = Chapter XXIII: On the Catenary }}

==External links==
{{Commons category}}
{{wikiquote}}
{{Wikisource1911Enc|Catenary}}
* {{MacTutor|class=Curves|id=Catenary|title=Catenary}}
* {{PlanetMath|urlname=Catenary|title=Catenary}}
* [http://www.fxsolver.com/browse/formulas/Catenary+curve Catenary curve calculator]
* [http://www.geom.uiuc.edu/zoo/diffgeom/surfspace/catenoid/catenary.html Catenary] at [[The Geometry Center]]
* [http://xahlee.org/SpecialPlaneCurves_dir/Catenary_dir/catenary.html "Catenary" at Visual Dictionary of Special Plane Curves]
* [http://www.maththoughts.com/blog/2013/catenary The Catenary - Chains, Arches, and Soap Films.]
* [http://www.spaceagecontrol.com/calccabl.htm Cable Sag Error Calculator] – Calculates the deviation from a straight line of a catenary curve and provides derivation of the calculator and references.
* [http://www.subhrajit.net/files/Projects-Work/OilBoom_Catenary_2010/catenary.pdf Dynamic as well as static cetenary curve equations derived] – The equations governing the shape (static case) as well as dynamics (dynamic case) of a centenary is derived. Solution to the equations discussed.
* [https://arxiv.org/abs/1401.2660 The straight line, the catenary, the brachistochrone, the circle, and Fermat] Unified approach to some geodesics.
* [https://www.ams.org/journals/bull/1925-31-08/S0002-9904-1925-04083-5/S0002-9904-1925-04083-5.pdf Ira Freeman "A General Form of the Suspension Bridge Catenary" ''Bulletin of the AMS'']

{{Mathematics and art}}

[[Category:Roulettes (curve)]]
[[Category:Exponentials]]
[[Category:Analytic geometry]]