catenary

File:GaudiCatenaryModel.jpg'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,{{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}}{{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}}

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 equilibrium of arches, by the Abbé Mascheroni. It appears to be a very scientifical 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.{{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] }}}}

It is often said that 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°.{{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 }} 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.Lockwood p. 124

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.{{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 }} Some much older arches approximate catenaries, an example of which is the Arch of Taq-i Kisra in Ctesiphon.{{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 }}

File:Analogy between an arch and a hanging chain and comparison to the dome of St Peter's Cathedral in Rome.png 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 anagramcf. the anagram for Hooke's law, which appeared in the next paragraph. in an appendix to his Description of Helioscopes,{{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 |url-status=dead |archive-url=https://web.archive.org/web/20101113210736/http://www.lindahall.org/events_exhib/exhibit/exhibits/civil/design.shtml |archive-date=2010-11-13 }} 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 anagramThe original anagram was abcccddeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuux: the letters of the Latin phrase, alphabetized. 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; their solutions were published in the Acta Eruditorum for June 1691.{{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=9783764314415}}{{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|page=20140346|doi=10.1098/rsta.2014.0346|pmid=25750153|pmc=4360092|bibcode=2015RSPTA.37340346C}} David Gregory wrote a treatise on the catenary in 1697{{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}} in which he provided an incorrect derivation of the correct differential equation.

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. Nicolas Fuss gave equations describing the equilibrium of a chain under any force in 1796.Routh Art. 455, footnote

==Inverted catenary arch==

Catenary arches are often used in the construction of kilns. 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.{{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 }}

{{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 }}

The Gateway Arch in St. Louis, Missouri, United States is sometimes said to be an (inverted) catenary, but this is incorrect.{{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}} 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.{{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 }}{{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}} {{small|(578 KB)}}

File:LaPedreraParabola.jpg|Catenary{{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 }} 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.{{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 }}

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

{{Clear}}

File:Soderskar-bridge.jpgs are essentially thickened cables, and follow a catenary curve.]]

File:Puentedelabarra(below).jpgs, 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.{{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}} The same is true of a simple suspension bridge or "catenary bridge," where the roadway follows the cable.{{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 }}{{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 }}

A stressed ribbon bridge is a more sophisticated structure with the same catenary shape.{{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}}{{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}}

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.Lockwood p. 122

{{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

}}

File:Comparison catenary parabola.svg (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, $y\; =\; \backslash text\{cosh\; \}\; x$ and $y\; =\; x\; ^\; 2\; [(\backslash text\{cosh\; \}1)\; -\; 1]\; +\; 1$ ]]

File:Catenary.PNG chain forms a catenary, with a low angle of pull on the anchor.]]

The catenary produced by gravity provides an advantage to heavy 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 turbines, 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.{{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}}

Cable ferries and chain boats 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.{{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}}

Image:catenary-pm.svg

The equation of a catenary in Cartesian coordinates has the form

$$y\; =\; a\; \backslash cosh\; \backslash left(\backslash frac\{x\}\{a\}\backslash right)\; =\; \backslash frac\{a\}\{2\}\backslash left(e^\backslash frac\{x\}\{a\}\; +\; e^\{-\backslash frac\{x\}\{a\}\}\backslash right),$$

where {{math|cosh}} is the hyperbolic cosine function, and where {{mvar|a}} is the distance of the lowest point above the x axis.{{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 [...] }} All catenary curves are 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

$$\backslash tan\; \backslash varphi\; =\; \backslash frac\{s\}\{a\},$$

where $\backslash varphi$ is the tangential angle and {{mvar|s}} the arc length.

Differentiating gives

$$\backslash frac\{d\backslash varphi\}\{ds\}\; =\; \backslash frac\{\backslash cos^2\backslash varphi\}\{a\},$$

and eliminating $\backslash varphi$ gives the Cesàro equationMathWorld, eq. 7

$$\backslash kappa=\backslash frac\{a\}\{s^2+a^2\},$$

where $\backslash kappa$ is the curvature.

The radius of curvature is then

$$\backslash rho\; =\; a\; \backslash sec^2\; \backslash varphi,$$

which is the length of the normal between the curve and the {{mvar|x}}-axis.Routh Art. 444

When a parabola is rolled along a straight line, the roulette curve traced by its focus is a catenary. The envelope of the directrix of the parabola is also a catenary.Yates p. 80 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.

Another roulette, formed by rolling a line on a catenary, is another line. This implies that square wheels 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.{{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}}

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.{{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 }}

A moving 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}}).{{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}}

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

$$y\; =\; a\; \backslash cosh^\{-1\}\backslash left(\backslash frac\{x\}\{a\}\backslash right)\; +\; b$$

revolved about the $y$-axis.{{cite book |title=Curves and their Properties |first=Robert C. |last=Yates |publisher=NCTM |year=1952 |page=13}}

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 exerted by the chain is parallel to the chain.Routh Art. 442, p. 316 The analysis of the curve for an optimal arch is similar except that the forces of tension become forces of compression and everything is inverted.{{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]}}

An underlying principle is that the chain may be considered a rigid body once it has attained equilibrium.Whewell p. 65 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 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 natural parameterization and has the property that

$$\backslash frac\{d\backslash mathbf\{r\}\}\{ds\}=\backslash mathbf\{u\}$$

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

File:CatenaryForceDiagram.svg

A differential equation for the curve may be derived as follows.Following Routh Art. 443 p. 316 Let {{math|c}} be the lowest point on the chain, called the vertex of the catenary.Routh Art. 443 p. 317 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|(−T₀, 0)}} where {{math|T₀}} 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=Tu = (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

$$T\; \backslash cos\; \backslash varphi\; =\; T\_0$$

and

$$T\; \backslash sin\; \backslash varphi\; =\; ws\backslash ,,$$

and dividing these gives

$$\backslash frac\{dy\}\{dx\}=\backslash tan\; \backslash varphi\; =\; \backslash frac\{ws\}\{T\_0\}\backslash ,.$$

It is convenient to write

$$a\; =\; \backslash frac\{T\_0\}\{w\}$$

which is the length of chain whose weight is equal in magnitude to the tension at {{math|c}}.Whewell p. 67 Then

$$\backslash frac\{dy\}\{dx\}=\backslash frac\{s\}\{a\}$$

is an equation defining the curve.

The horizontal component of the tension, {{math|1=T cos φ = T₀}} 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.Routh Art 443, p. 318

The differential equation $dy/dx\; =\; s/a$, given above, can be solved

to produce equations for the curve.

A minor variation of the derivation presented here

can be found on page 107 of Maurer.

A different (though ultimately mathematically equivalent) derivation,

which does not make use of hyperbolic function notation,

can be found in Routh

(Article 443, starting in particular at page 317).

We will solve the equation using the boundary condition that

the vertex is positioned at $s\_0=0$ and $(x,y)=(x\_0,y\_0)$.

First, invoke the formula for

arc length

to get

$$\backslash frac\{ds\}\{dx\}$$

= \sqrt{1+\left(\frac{dy}{dx}\right)^2}

= \sqrt{1+\left(\frac{s}{a}\right)^2}\,,

then separate variables

to obtain

$$\backslash frac\{ds\}\{\backslash sqrt\{1+(s/a)^2\}\}$$

= dx\,.

A reasonably straightforward approach to integrate this is to use

hyperbolic substitution,

which gives

$$a\; \backslash sinh^\{-1\}\backslash frac\{s\}\{a\}\; +\; x\_0\; =\; x$$

(where $x\_0$ is a constant of integration),

and hence

$$\backslash frac\{s\}\{a\}\; =\; \backslash sinh\backslash frac\{x-x\_0\}\{a\}\backslash ,.$$

But $s/a\; =\; dy/dx$, so

$$\backslash frac\{dy\}\{dx\}\; =\; \backslash sinh\backslash frac\{x-x\_0\}\{a\}\backslash ,,$$

which integrates as

$$y\; =\; a\; \backslash cosh\backslash frac\{x-x\_0\}\{a\}\; +\; \backslash delta$$

(with $\backslash delta=y\_0-a$ 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 $x\_0=0=\backslash delta$

to simplify the result to

$$y\; =\; a\; \backslash cosh\backslash frac\{x\}\{a\}.$$

For completeness, the $y\; \backslash leftrightarrow\; s$ relation can be derived by

solving each of the $x\; \backslash leftrightarrow\; y$ and $x\; \backslash leftrightarrow\; s$ relations for $x/a$, giving:

$$\backslash cosh^\{-1\}\backslash frac\{y-\backslash delta\}\{a\}\; =\; \backslash frac\{x-x\_0\}\{a\}\; =\; \backslash sinh^\{-1\}\backslash frac\{s\}\{a\}\backslash ,,$$

so

$$y-\backslash delta\; =\; a\backslash cosh\backslash left(\backslash sinh^\{-1\}\backslash frac\{s\}\{a\}\backslash right)\backslash ,,$$

which can be rewritten as

$$y-\backslash delta\; =\; a\backslash sqrt\{1+\backslash left(\backslash frac\{s\}\{a\}\backslash right)^2\}\; =\; \backslash sqrt\{a^2\; +\; s^2\}\backslash ,.$$

The differential equation can be solved using a different approach.Following Lamb p. 342 From

$$s\; =\; a\; \backslash tan\; \backslash varphi$$

it follows that

$$\backslash frac\{dx\}\{d\backslash varphi\}\; =\; \backslash frac\{dx\}\{ds\}\backslash frac\{ds\}\{d\backslash varphi\}=\backslash cos\; \backslash varphi\; \backslash cdot\; a\; \backslash sec^2\; \backslash varphi=\; a\; \backslash sec\; \backslash varphi$$

and

$$\backslash frac\{dy\}\{d\backslash varphi\}\; =\; \backslash frac\{dy\}\{ds\}\backslash frac\{ds\}\{d\backslash varphi\}=\backslash sin\; \backslash varphi\; \backslash cdot\; a\; \backslash sec^2\; \backslash varphi=\; a\; \backslash tan\; \backslash varphi\; \backslash sec\; \backslash varphi\backslash ,.$$

Integrating gives,

$$x\; =\; a\; \backslash ln(\backslash sec\; \backslash varphi\; +\; \backslash tan\; \backslash varphi)\; +\; \backslash alpha$$

and

$$y\; =\; a\; \backslash sec\; \backslash varphi\; +\; \backslash beta\backslash ,.$$

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

$$\backslash sec\; \backslash varphi\; +\; \backslash tan\; \backslash varphi\; =\; e^\backslash frac\{x\}\{a\}\backslash ,,$$

and taking the reciprocal of both sides

$$\backslash sec\; \backslash varphi\; -\; \backslash tan\; \backslash varphi\; =\; e^\{-\backslash frac\{x\}\{a\}\}\backslash ,.$$

Adding and subtracting the last two equations then gives the solution

$$y\; =\; a\; \backslash sec\; \backslash varphi\; =\; a\; \backslash cosh\backslash left(\backslash frac\{x\}\{a\}\backslash right)\backslash ,,$$

and

$$s\; =\; a\; \backslash tan\; \backslash varphi\; =\; a\; \backslash sinh\backslash left(\backslash frac\{x\}\{a\}\backslash right)\backslash ,.$$

Image:Catenary-tension.svg

In general the parameter {{mvar|a}} is the position of the axis. The equation can be determined in this case as follows:Following Todhunter Art. 186

Relabel if necessary so that {{math|P₁}} is to the left of {{math|P₂}} and let {{mvar|H}} be the horizontal and {{mvar|v}} be the vertical distance from {{math|P₁}} to {{math|P₂}}. 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

$$y\; =\; a\; \backslash cosh\backslash left(\backslash frac\{x\}\{a\}\backslash right)$$

and let the coordinates of {{math|P₁}} and {{math|P₂}} be {{math|(x₁, y₁)}} and {{math|(x₂, y₂)}} respectively. The curve passes through these points, so the difference of height is

$$v\; =\; a\; \backslash cosh\backslash left(\backslash frac\{x\_2\}\{a\}\backslash right)\; -\; a\; \backslash cosh\backslash left(\backslash frac\{x\_1\}\{a\}\backslash right)\backslash ,.$$

and the length of the curve from {{math|P₁}} to {{math|P₂}} is

$$L\; =\; a\; \backslash sinh\backslash left(\backslash frac\{x\_2\}\{a\}\backslash right)\; -\; a\; \backslash sinh\backslash left(\backslash frac\{x\_1\}\{a\}\backslash right)\backslash ,.$$

When {{math|L² − v²}} is expanded using these expressions the result is

$$L^2-v^2=2a^2\backslash left(\backslash cosh\backslash left(\backslash frac\{x\_2-x\_1\}\{a\}\backslash right)-1\backslash right)=4a^2\backslash sinh^2\backslash left(\backslash frac\{H\}\{2a\}\backslash right)\backslash ,,$$

so

$$\backslash frac\; 1H\; \backslash sqrt\{L^2-v^2\}=\backslash frac\{2a\}H\; \backslash sinh\backslash left(\backslash frac\{H\}\{2a\}\backslash right)\backslash ,.$$

This is a transcendental equation in {{mvar|a}} and must be solved numerically. Since $\backslash sinh(x)/x$ is strictly monotonic on $x\; >\; 0$,See Routh art. 447 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|P₁}} and {{math|P₂}}) are at the same level ({{math|y₁ {{=}} y₂}}), it can be shown thatArchived 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}}

$$a\; =\; \backslash frac\; \{\backslash frac14\; L^2-h^2\}\; \{2h\}\backslash ,$$

where L is the total length of the curve between {{math|P₁}} and {{math|P₂}} and {{mvar|h}} is the sag (vertical distance between {{math|P₁}}, {{math|P₂}} and the vertex of the curve).

It can also be shown that

$$L\; =\; 2a\; \backslash sinh\; \backslash frac\; \{H\}\; \{2a\}\backslash ,$$

and

$$H\; =\; 2a\; \backslash operatorname\; \{arcosh\}\; \backslash frac\; \{h+a\}\; \{a\}\backslash ,$$

where H is the horizontal distance between {{math|P₁}} and {{math|P₂}} which are located at the same level ({{math|H {{=}} x₂ − x₁}}).

The horizontal traction force at {{math|P₁}} and {{math|P₂}} is {{math|T₀ {{=}} wa}}, where {{mvar|w}} is the weight per unit length of the chain or cable.

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:

$$(T\backslash cos\backslash varphi)^2\; +\; (T\backslash sin\backslash varphi)^2\; =\; T\_0^2\; +\; (ws)^2$$

which (recalling that $T\_0=wa$) can be rewritten as

$$\backslash begin\{align\}$$

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}

But, as shown above,

$y\; =\; \backslash sqrt\{a^2\; +\; s^2\}$ (assuming that $y\_0=a$), so we get the simple relationsRouth Art 443, p. 318

$$T\; =\; wy\; =\; wa\; \backslash cosh\backslash frac\{x\}\{a\}\backslash ,.$$

Consider a chain of length $L$ suspended from two points of equal height and at distance $D$. The curve has to minimize its potential energy

$$U\; =\; \backslash int\_0^D\; w\; y\backslash sqrt\{1+y\text{'}^2\}\; dx$$

(where {{mvar|w}} is the weight per unit length) and is subject to the constraint

$$\backslash int\_0^D\; \backslash sqrt\{1+y\text{'}^2\}\; dx\; =\; L\backslash ,.$$

The modified Lagrangian is therefore

$$\backslash mathcal\{L\}\; =\; (w\; y\; -\; \backslash lambda\; )\backslash sqrt\{1+y\text{'}^2\}$$

where $\backslash lambda$ is the Lagrange multiplier to be determined. As the independent variable $x$ does not appear in the Lagrangian, we can use the Beltrami identity

$$\backslash mathcal\{L\}-y\text{'}\; \backslash frac\{\backslash partial\; \backslash mathcal\{L\}\; \}\{\backslash partial\; y\text{'}\}\; =\; C$$

where $C$ is an integration constant, in order to obtain a first integral

$$\backslash frac\{(w\; y\; -\; \backslash lambda\; )\}\{\backslash sqrt\{1+y\text{'}^2\}\}\; =\; -C$$

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.

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.Following Routh Art. 450

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

$$\backslash int\_\backslash mathbf\{c\}^\backslash mathbf\{r\}\; w\backslash ,\; ds\backslash ,,$$

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

$$T\; \backslash cos\; \backslash varphi\; =\; T\_0$$

and

$$T\; \backslash sin\; \backslash varphi\; =\; \backslash int\_\backslash mathbf\{c\}^\backslash mathbf\{r\}\; w\backslash ,\; ds\backslash ,,$$

and therefore

$$\backslash frac\{dy\}\{dx\}=\backslash tan\; \backslash varphi\; =\; \backslash frac\{1\}\{T\_0\}\; \backslash int\_\backslash mathbf\{c\}^\backslash mathbf\{r\}\; w\backslash ,\; ds\backslash ,.$$

Differentiation then gives

$$w=T\_0\; \backslash frac\{d\}\{ds\}\backslash frac\{dy\}\{dx\}\; =\; \backslash frac\{T\_0\; \backslash dfrac\{d^2y\}\{dx^2\}\}\{\backslash sqrt\{1+\backslash left(\backslash dfrac\{dy\}\{dx\}\backslash right)^2\}\}\backslash ,.$$

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

$$w=\; \backslash frac\{T\_0\}\{\backslash rho\; \backslash cos^2\; \backslash varphi\}\backslash ,.$$

File:Golden Gate Bridge, SF.jpg. 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.Following Routh Art. 452 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

$$\backslash frac\{dy\}\{dx\}=\backslash tan\; \backslash varphi\; =\; \backslash frac\{w\}\{T\_0\}x\backslash ,.$$

This is solved by simple integration to get

$$y=\backslash frac\{w\}\{2T\_0\}x^2\; +\; \backslash beta$$

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.Ira Freeman investigated the case where only the cable and roadway are significant, see the External links section. Routh gives the case where only the supporting wires have significant weight as an exercise.

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.Following Routh Art. 453

In this case the equations for tension are

$$\backslash begin\{align\}$$

T \cos \varphi &= T_0\,,\\

T \sin \varphi &= \frac{1}{c}\int T\, ds\,.

\end{align}

Combining gives

$$c\; \backslash tan\; \backslash varphi\; =\; \backslash int\; \backslash sec\; \backslash varphi\backslash ,\; ds$$

and by differentiation

$$c\; =\; \backslash rho\; \backslash cos\; \backslash varphi$$

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

The solution to this is

$$y\; =\; c\; \backslash ln\backslash left(\backslash sec\backslash left(\backslash frac\{x\}\{c\}\backslash right)\backslash right)\backslash ,.$$

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

$$x\; =\; c\backslash varphi\backslash ,,\backslash quad\; s\; =\; \backslash ln\backslash left(\backslash tan\backslash left(\backslash frac\{\backslash pi+2\backslash varphi\}\{4\}\backslash right)\backslash right)\backslash ,.$$

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".{{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|pages=e1500396 |url= |doi=10.1126/sciadv.1500396|pmid=26601283|pmc=4646797|bibcode=2015SciA....1E0396P}}

In an elastic catenary, the chain is replaced by a 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

$$s=\backslash left(1+\backslash frac\{T\}\{E\}\backslash right)p\backslash ,,$$

where {{mvar|E}} is a constant equal to {{mvar|kp}}, where {{mvar|k}} is the stiffness of the spring.Routh Art. 489 In the catenary the value of {{mvar|T}} is variable, but ratio remains valid at a local level, soRouth Art. 494

$$\backslash frac\{ds\}\{dp\}=1+\backslash frac\{T\}\{E\}\backslash ,.$$

The curve followed by an elastic spring can now be derived following a similar method as for the inelastic spring.Following Routh Art. 500

The equations for tension of the spring are

$$T\; \backslash cos\; \backslash varphi\; =\; T\_0\backslash ,,$$

and

$$T\; \backslash sin\; \backslash varphi\; =\; w\_0\; p\backslash ,,$$

from which

$$\backslash frac\{dy\}\{dx\}=\backslash tan\; \backslash varphi\; =\; \backslash frac\{w\_0\; p\}\{T\_0\}\backslash ,,\backslash quad\; T=\backslash sqrt\{T\_0^2+w\_0^2\; p^2\}\backslash ,,$$

where {{mvar|p}} is the natural length of the segment from {{math|c}} to {{math|r}} and {{math|w₀}} is the weight per unit length of the spring with no tension. Write

$$a\; =\; \backslash frac\{T\_0\}\{w\_0\}$$

so

$$\backslash frac\{dy\}\{dx\}=\backslash tan\; \backslash varphi\; =\; \backslash frac\{p\}\{a\}\; \backslash quad\backslash text\{and\}\backslash quad\; T=\backslash frac\{T\_0\}\{a\}\backslash sqrt\{a^2+p^2\}\backslash ,.$$

Then

$$\backslash begin\{align\}$$

\frac{dx}{ds} &= \cos \varphi = \frac{T_0}{T} \\[6pt]

\frac{dy}{ds} &= \sin \varphi = \frac{w_0 p}{T}\,,

\end{align}

from which

$$\backslash begin\{alignat\}\{3\}$$

\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}

Integrating gives the parametric equations

$$\backslash begin\{align\}$$

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}

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

$$\backslash begin\{align\}$$

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}

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

With no assumptions being made regarding the force {{math|G}} acting on the chain, the following analysis can be made.Follows Routh Art. 455

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,

$$\backslash mathbf\{T\}\; =\; T\; \backslash mathbf\{u\}\backslash ,,$$

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 + Δs}} are the force of tension {{math|T(s + Δs)}} 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|GΔs}}. These forces must balance so

$$\backslash mathbf\{T\}(s+\backslash Delta\; s)-\backslash mathbf\{T\}(s)+\backslash mathbf\{G\}\backslash Delta\; s\; \backslash approx\; \backslash mathbf\{0\}\backslash ,.$$

Divide by {{math|Δs}} and take the limit as {{math|Δs → 0}} to obtain

$$\backslash frac\{d\backslash mathbf\{T\}\}\{ds\}\; +\; \backslash mathbf\{G\}\; =\; \backslash mathbf\{0\}\backslash ,.$$

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.

• 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

{{Reflist}}

• {{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}}

• {{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 }}

{{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}}

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