catenary wikipedia

8/10/2019 Catenary Wikipedia

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Catenary 1

Catenary

This article is about the mathematical curve. For other uses, see Catenary (disambiguation).

"Chainette" redirects here. For the wine grape also known as Chainette, see Cinsaut.

A hanging chain with short links forms a

catenary.

Freely-hanging transmission lines also form

catenaries.

The silk on a spider's web forming multiple

elastic catenaries.

In physics and geometry, a catenary[p] is the curve that an idealized

hanging chain or cable assumes under its own weight when supported

only at its ends. The curve has a U-like shape, superficially similar in

appearance to a parabola, but it is not a parabola: it is a (scaled,

rotated) graph of the hyperbolic cosine. The curve appears in the

design of certain types of arches and as a cross section of the

catenoidthe shape assumed by a soap film bounded by two parallel

circular rings.

The catenary is also called the "alysoid", "chainette",[1] or, particularly

in the material sciences, "funicular".[2]

Mathematically, the catenary curve is the graph of the hyperbolic

cosine function. The surface of revolution of the catenary curve, the

catenoid, is a minimal surface, specifically a minimal surface of

revolution. The mathematical properties of the catenary curve were

first studied by Robert Hooke in the 1670s, and its equation was

derived by Leibniz, Huygens and Johann Bernoulli in 1691.

Catenaries and related curves are used in architecture and engineering,

in the design of bridges and arches, so that forces do not result in

bending moments.

Note also the wider meaning of the word 'catenary' used sincemid-1990s in the offshore oil and gas industry of steel catenary riser.
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Catenary 2

History

Antoni Gaud's catenary model at Casa Mil

The word catenary is derived from the Latin word

catena, which means "chain". The English word

catenary is usually attributed to Thomas Jefferson, who

wrote in a letter to Thomas Paine on the construction of

an arch for a bridge:

I have lately received from Italy a treatise on the

equilibrium of arches, by the Abb 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.

It is often said [3] that Galileo thought the curve of a hanging chain was parabolic. In his Two New Sciences (1638),

Galileo says that a hanging cord is an approximate parabola, and he correctly observes that this approximationimproves as the curvature gets smaller and is almost exact when the elevation is less than 45. That the curve

followed by a chain is not a parabola was proven by Joachim Jungius (15871657); this result was published

posthumously in 1669.[4]

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.[5] Some much

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

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[6] in an appendix to his Description of Helioscopes,

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[7] 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. David Gregory wrote a treatise on the catenary in 1697.

Euler proved in 1744 that the catenary is the curve which, when rotated about the 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.[8]

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.

The Gateway Arch in St. Louis, Missouri, United States is sometimes said to be an (inverted) catenary, but this is

incorrect. It is close to a more general curve called a flattened catenary, with equation y =Acosh(Bx), which is a

catenary ifAB = 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.[9]

Inverted catenary arches
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Catenary 4

The Gateway Arch (looking East) is a

flattened catenary.

Catenary arch kiln under construction overtemporary form

Cross-section of the roof the Keleti Railway Station(Budapest, Hungary).
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Catenary 5

Cross-section of the roof the Keleti Railway Stationforms a catenary.

Catenary bridges

Simple suspension bridges are essentially thickened cables, andfollow a catenary curve.

Stressed ribbon bridges, like this one 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. The same is true of a simplesuspension bridge or "catenary bridge," where the

roadway follows the cable.

A stressed ribbon bridge is a more sophisticated

structure with the same catenary shape.[10]

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.[11]
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Catenary 6

Comparison of a catenary (black dotted curve) and a parabola (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 mass 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

mass 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 arey = cosh(x) andy = (cosh(1) - 1)x2 + 1,

respectively.

Anchoring of marine objects

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 anchor rodes. An anchor rode (or

anchor line) usually consists of chain or cable or both.

Anchor rodes are used by ships, oilrigs, docks, floating

wind turbines, and other marine equipment which must

be anchored to the seabed.

When the rode 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. Thisenhances 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 must rely on the performance of the anchor itself.[12]
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Catenary 7

Mathematical description

Equation

Catenaries for different values of a

Three different catenaries through the same two points, depending horizontal force

being and mass per unit length.

The equation of a catenary in Cartesian

coordinates has the form

where cosh is the hyperbolic cosine

function. All catenary curves are

similar to each other, having

eccentricity = 2. Changing the

parameter a is equivalent to a uniform

scaling of the curve.

The Whewell equation for the catenary

is

Differentiating gives

and eliminating gives the Cesro

equation[13]

The radius of curvature is then

which is the length of the line normal

to the curve between it and the

x-axis.[14]

Relation to other curves

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.[15] The involute from the vertex, that is the roulette formed 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 if the road has evenly spaced bumps in the shape of a series of inverted catenary curves. 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.
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Catenary 8

Geometrical properties

Over any horizontal interval, the ratio of the area under the catenary to its length equals 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 x-axis.[16]

Science

A charge in a uniform electric field moves along a catenary (which tends to a parabola if the charge velocity is much

less than the speed of light c).

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

the x-axis.

Analysis

Model of chains and archesIn 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.[17] 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. An underlying principle is that the chain may be considered a rigid body

once it has attained equilibrium.[18] 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 r = (x,y) = (x(s),y(s)) where s represents arc length

and r is the position vector. This is the natural parameterization and has the property that

where u is a unit tangent vector.

Diagram of forces acting on a segment of a

catenary from c to r. The forces are the tension T0

at c, the tension T at r, and the weight of the

chain (0, gs). 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.[19] Let

c be the lowest point on the chain, called the vertex of the catenary, [20]

and measure the parameter s from c. Assume r is to the right of c since

the other case is implied by symmetry. The forces acting on the section

of the chain from c to r are the tension of the chain at c, the tension of

the chain at r, and the weight of the chain. The tension at c is tangent

to the curve at c and is therefore horizontal, and it pulls the section to

the left so it may be written (T0, 0) where T

0is the magnitude of the

force. The tension at r is parallel to the curve at r and pulls the section

to the right, so it may be written Tu=(Tcos , Tsin ), where T is the

magnitude of the force and is the angle between the curve at r and

the x-axis (see tangential angle). Finally, the weight of the chain is

represented by (0, gs) where is the mass per unit length, g is the

acceleration of gravity and s is the length of chain between c and r.

The chain is in equilibrium so the sum of three forces is 0, therefore

and
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Catenary 9

and dividing these gives

It is convenient to write

which is the length of chain whose weight is equal in magnitude to the tension at c.[21] Then

is an equation defining the curve.

The horizontal component of the tension, Tcos = T0

is constant and the vertical component of the tension, Tsin =

gs is proportional to the length of chain between the r and the vertex.

Derivation of equations for the curveThe differential equation given above can be solved to produce equations for the curve.[22]

From

the formula for arc length gives

Then

and

The second of these equations can be integrated to give

and by shifting the position of thex-axis, can be taken to be 0. Then

Thex-axis thus chosen is called the directrix of the catenary.

It follows that the magnitude of the tension at a point T = gy which is proportional to the distance between the point

and the directrix.[]

The integral of expression for dx/ds can be found using standard techniques giving[23]

and, again, by shifting the position of they-axis, can be taken to be 0. Then

They-axis thus chosen passes though the vertex and is called the axis of the catenary.

These results can be used to eliminate s giving
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Catenary 10

Alternative derivation

The differential equation can be solved using a different approach.[24]

From

it follows that

and

Integrating gives,

and

As before, thex andy-axes can be shifted so and can be taken to be 0. Then

and taking the reciprocal of both sides

Adding and subtracting the last two equations then gives the solution

and

Determining parameters

In general the parameter a and the position of the axis and directrix are not given but must be determined from other

information. Typically, the information given is that the catenary is suspended at given points P1

and P2

and with

given length s. The equation can be determined in this case as follows:[25] Relabel if necessary so thatP1

is to the left

ofP2

and let h be the horizontal and v be the vertical distance fromP1

toP2. Translate the axes so that the vertex of

the catenary lies on the y-axis and its height a is adjusted so the catenary satisfies the standard equation of the curve

and let the coordinates ofP1

andP2

be (x1,y

1) and (x

2,y

2) respectively. The curve passes through these points, so the

difference of height is

and the length of the curve fromP1

toP2

is

When s2v2 is expanded using these expressions the result is

so
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Catenary 11

This is a transcendental equation in a and must be solved numerically. It can be shown with the methods of

calculus[26] that there is at most one solution with a>0 and so there is at most one position of equilibrium.

Generalizations with vertical force

Nonuniform chainsIf 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.[27]

Let w denote the weight per unit length of the chain, then the weight of the chain has magnitude

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

and

and therefore

Differentiation then gives

In terms of and the radius of curvature this becomes

Suspension bridge curve

Golden Gate Bridge. Most suspension bridge cables follow a parabolic, not a catenary

curve, due to the weight of the roadway being much greater than that of the cable.

A similar analysis can be done to find

the curve followed by the cable

supporting a suspension bridge with a

horizontal roadway.[28] If the weight of

the roadway per unit length is w and

the weight of the cable and the wiresupporting the bridge is negligible in

comparison, then the weight on the

cable from c to r is wx where x is the

horizontal distance between c to r. Proceeding as before gives the differential equation

This is solved by simple integration to get
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Catenary 13

where p is the natural length of the segment from c to r and 0

is the mass per unit length of the spring with no

tension and g is the acceleration of gravity. Write

so

Then

and

from which

and

Integrating gives the parametric equations

Again, the x and y-axes can be shifted so and can be taken to be 0. So

are parametric equations for the curve. At the rigid limit where 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 have been made regarding the force G acting on the chain, the following analysis can be

made.[34]

First, let T=T(s) be the force of tension as a function of 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, T must be parallel to the chain. In other

words,

where T is the magnitude of T and u is the unit tangent vector.

Second, let G=G(s) be the external force per unit length acting on a small segment of a chain as a function of s. Theforces acting on the segment of the chain between s and s+s are the force of tension T(s+s) at one end of the
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Catenary 14

segment, the nearly opposite force T(s) at the other end, and the external force acting on the segment which is

approximately Gs. These forces must balance so

Divide by s and take the limit as s 0 to obtain

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, G = (0, g) where the chain has mass per unit length and g is the acceleration

of gravity.

Notes

[p] ^ Word "catenary" is said as either /kt..nr.i//'Kat-a-nr-ee/, or British /ktinri//ka'Teen'Ree/.

[1][1] MathWorld

[2] e.g.:

[3] For example Lockwood,A Book of Curves, p. 124.

[4][4] Lockwood p. 124

[5] "Monuments and Microscopes: Scientific Thinking on a Grand Scale in the Early Royal Society" by Lisa Jardine (http://www.jstor.org/

stable/532102)

[6] cf. the anagram for Hooke's law, which appeared in the next paragraph.

[7][7] The original anagram was "abcccddeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuux": the letters of the Latin phrase, alphabetized.

[8][8] Routh Art. 455, footnote

[9][9] and

[10] The Architects' Journal, Volume 207, The Architectural Press Ltd., 1998, p. 51.

[11][11] Lockwood p. 122

[12][12] (for section)

[13][13] MathWorld, eq. 7

[14][14] Routh Art. 444

[15][15] Yates p. 80[16] Parker, Edward (2010), "A Property Characterizing the Catenary",Mathematics Magazine83: 6364

[17][17] Routh Art. 442, p. 316

[18][18] Whewell p. 65

[19][19] Following Routh Art. 443 p. 316

[20][20] Routh Art. 443 p. 317

[21][21] Whewell p. 67

[22][22] Following Routh Art. 443 p/ 317

[23][23] Use of hyperbolic functions follows Maurer p. 107

[24][24] Following Lamb p. 342

[25][25] Following Todhunter Art. 186

[26][26] See Routh art. 447

[27][27] Following Routh Art. 450


[29][29] Ira Freeman investigated the case where the 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.


[31][31] Routh Art. 489

[32][32] Routh Art. 494


[34][34] Follows Routh Art. 455
http://en.wikipedia.org/w/index.php?title=Hooke%27s_lawhttp://en.wikipedia.org/w/index.php?title=Cf.http://www.jstor.org/stable/532102http://www.jstor.org/stable/532102http://en.wikipedia.org/w/index.php?title=Wikipedia:RESPELLhttp://en.wikipedia.org/w/index.php?title=Help:IPA_for_Englishhttp://en.wikipedia.org/w/index.php?title=Help:IPA_for_English%23Keyhttp://en.wikipedia.org/w/index.php?title=Help:IPA_for_Englishhttp://en.wikipedia.org/w/index.php?title=Wikipedia:RESPELLhttp://en.wikipedia.org/w/index.php?title=Help:IPA_for_Englishhttp://en.wikipedia.org/w/index.php?title=Help:IPA_for_English%23Keyhttp://en.wikipedia.org/w/index.php?title=Help:IPA_for_English


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Catenary 15

Bibliography

Lockwood, E.H. (1961). "Chapter 13: The Tractrix and Catenary" (http://www.archive.org/details/

bookofcurves006299mbp).A Book of Curves. Cambridge.

Salmon, George (1879).Higher Plane Curves. Hodges, Foster and Figgis. pp. 287289.

Routh, Edward John (1891). "Chapter X: On Strings" (http://books.google.com/?id=3N5JAAAAMAAJ&

pg=PA315).A Treatise on Analytical Statics. University Press. Maurer, Edward Rose (1914). "Art. 26 Catenary Cable" (http://books.google.com/?id=L98uAQAAIAAJ&

pg=PA107). Technical Mechanics. J. Wiley & Sons.

Lamb, Sir Horace (1897). "Art. 134 Transcendental Curves; Catenary, Tractrix" (http://books.google.com/

?id=eDM6AAAAMAAJ&pg=PA342).An Elementary Course of Infinitesimal Calculus. University Press.

Todhunter, Isaac (1858). "XI Flexible Strings. Inextensible, XII Flexible Strings. Extensible" (http://books.

google.com/?id=-iEuAAAAYAAJ&pg=PA199).A Treatise on Analytical Statics. Macmillan.

Whewell, William (1833). "Chapter V: The Eqilibruim of a Flexible Body" (http://books.google.com/

?id=BF8JAAAAIAAJ&pg=PA65).Analytical Statics. J. & J.J. Deighton. p. 65.

Weisstein, Eric W., "Catenary" (http://mathworld.wolfram.com/Catenary.html),MathWorld.

Further reading

Swetz, Frank (1995).Learn from the Masters(http://books.google.com/?id=gqGLoh-WYrEC&pg=PA128).

MAA. pp. 1289. ISBN 0-88385-703-0.

Venturoli, Giuseppe (1822). "Chapter XXIII: On the Catenary" (http://books.google.com/

?id=kHhBAAAAYAAJ&pg=PA67).Elements of the Theory of Mechanics. Trans. Daniel Cresswell. J.

Nicholson & Son.

External links

O'Connor, John J.; Robertson, Edmund F., "Catenary" (http://www-history.mcs.st-andrews.ac.uk/Curves/Catenary.html),MacTutor History of Mathematics archive, University of St Andrews.

Catenary (http://planetmath.org/encyclopedia/Catenary.html) at PlanetMath.org.

Catenary (http://www.geom.uiuc.edu/zoo/diffgeom/surfspace/catenoid/catenary.html) at The Geometry

Center

Encyclopdie des Formes Mathmatiques Remarquables

"Chanette" (http://www.mathcurve.com/courbes2d/chainette/chainette.shtml)

"Chanette lastique" (http://www.mathcurve.com/courbes2d/chainette/chainetteelastique.shtml)

"Chanette d'gale Rsistance" (http://www.mathcurve.com/courbes2d/chainettedegaleresistance/

chainettedegaleresistance.shtml)

"Courbe de la corde sauter" (http://www.mathcurve.com/courbes2d/cordeasauter/cordeasauter.shtml)

"Catenary" at Visual Dictionary of Special Plane Curves (http://xahlee.org/SpecialPlaneCurves_dir/

Catenary_dir/catenary.html)

The Catenary - Chains, Arches, and Soap Films. (http://www.maththoughts.com/blog/2013/catenary)

Hanging With Galileo (http://whistleralley.com/hanging/hanging.htm) mathematical derivation of formula

for suspended and free-hanging chains; interactive graphical demo of parabolic versus hyperbolic suspensions.

Catenary Demonstration Experiment (http://jonathan.lansey.net/pastimes/catenary/index.html) An easy

way to demonstrate the Mathematical properties of a cosh using the hanging cable effect. Devised by Jonathan

Lansey

Catenary curve derived (http:/

/

members.

chello.

nl/

j.

beentjes3/

Ruud/

catfiles/

catenary.

pdf)

The shape of acatenary is derived, plus examples of a chain hanging between 2 points of unequal height, including C program to

calculate the curve.
http://members.chello.nl/j.beentjes3/Ruud/catfiles/catenary.pdfhttp://jonathan.lansey.net/pastimes/catenary/index.htmlhttp://whistleralley.com/hanging/hanging.htmhttp://www.maththoughts.com/blog/2013/catenaryhttp://xahlee.org/SpecialPlaneCurves_dir/Catenary_dir/catenary.htmlhttp://xahlee.org/SpecialPlaneCurves_dir/Catenary_dir/catenary.htmlhttp://www.mathcurve.com/courbes2d/cordeasauter/cordeasauter.shtmlhttp://www.mathcurve.com/courbes2d/chainettedegaleresistance/chainettedegaleresistance.shtmlhttp://www.mathcurve.com/courbes2d/chainettedegaleresistance/chainettedegaleresistance.shtmlhttp://www.mathcurve.com/courbes2d/chainette/chainetteelastique.shtmlhttp://www.mathcurve.com/courbes2d/chainette/chainette.shtmlhttp://en.wikipedia.org/w/index.php?title=The_Geometry_Centerhttp://en.wikipedia.org/w/index.php?title=The_Geometry_Centerhttp://www.geom.uiuc.edu/zoo/diffgeom/surfspace/catenoid/catenary.htmlhttp://en.wikipedia.org/w/index.php?title=PlanetMathhttp://planetmath.org/encyclopedia/Catenary.htmlhttp://en.wikipedia.org/w/index.php?title=University_of_St_Andrewshttp://en.wikipedia.org/w/index.php?title=MacTutor_History_of_Mathematics_archivehttp://www-history.mcs.st-andrews.ac.uk/Curves/Catenary.htmlhttp://www-history.mcs.st-andrews.ac.uk/Curves/Catenary.htmlhttp://en.wikipedia.org/w/index.php?title=Edmund_F._Robertsonhttp://en.wikipedia.org/w/index.php?title=John_J._O%27Connor_%28mathematician%29http://books.google.com/?id=kHhBAAAAYAAJ&pg=PA67http://books.google.com/?id=kHhBAAAAYAAJ&pg=PA67http://en.wikipedia.org/w/index.php?title=Special:BookSources/0-88385-703-0http://en.wikipedia.org/w/index.php?title=International_Standard_Book_Numberhttp://books.google.com/?id=gqGLoh-WYrEC&pg=PA128http://en.wikipedia.org/w/index.php?title=MathWorldhttp://mathworld.wolfram.com/Catenary.htmlhttp://en.wikipedia.org/w/index.php?title=Eric_W._Weissteinhttp://books.google.com/?id=BF8JAAAAIAAJ&pg=PA65http://books.google.com/?id=BF8JAAAAIAAJ&pg=PA65http://en.wikipedia.org/w/index.php?title=William_Whewellhttp://books.google.com/?id=-iEuAAAAYAAJ&pg=PA199http://books.google.com/?id=-iEuAAAAYAAJ&pg=PA199http://en.wikipedia.org/w/index.php?title=Isaac_Todhunterhttp://books.google.com/?id=eDM6AAAAMAAJ&pg=PA342http://books.google.com/?id=eDM6AAAAMAAJ&pg=PA342http://books.google.com/?id=L98uAQAAIAAJ&pg=PA107http://books.google.com/?id=L98uAQAAIAAJ&pg=PA107http://books.google.com/?id=3N5JAAAAMAAJ&pg=PA315http://books.google.com/?id=3N5JAAAAMAAJ&pg=PA315http://en.wikipedia.org/w/index.php?title=Edward_Routhhttp://www.archive.org/details/bookofcurves006299mbphttp://www.archive.org/details/bookofcurves006299mbp


16/17

Catenary 16

Cable Sag Error Calculator (http://www.spaceagecontrol.com/calccabl.htm) Calculates the deviation from a

straight line of a catenary curve and provides derivation of the calculator and references.

Hexagonal Geodesic Domes Catenary Domes (http://hexdome.com/essays/catenary_domes/index.php), an

article about creating catenary domes

Dynamic as well as static cetenary curve equations derived (http://www.subhrajit.net/files/Projects-Work/

OilBoom_Catenary_2010/catenary.pdf) The equations governing the shape (static case) as well as dynamics

(dynamic case) of a centenary is derived. Solution to the equations discussed.

Ira Freeman "A General Form of the Suspension Bridge Catenary"Bulletin of the AMS(http://www.ams.org/

journals/bull/1925-31-08/S0002-9904-1925-04083-5/S0002-9904-1925-04083-5.pdf)
http://www.ams.org/journals/bull/1925-31-08/S0002-9904-1925-04083-5/S0002-9904-1925-04083-5.pdfhttp://www.ams.org/journals/bull/1925-31-08/S0002-9904-1925-04083-5/S0002-9904-1925-04083-5.pdfhttp://www.subhrajit.net/files/Projects-Work/OilBoom_Catenary_2010/catenary.pdfhttp://www.subhrajit.net/files/Projects-Work/OilBoom_Catenary_2010/catenary.pdfhttp://en.wikipedia.org/w/index.php?title=Catenary_domehttp://hexdome.com/essays/catenary_domes/index.phphttp://www.spaceagecontrol.com/calccabl.htm


17/17

Article Sources and Contributors 17

Article Sources and ContributorsCatenary Source: http://en.wikipedia.org/w/index.php?oldid=613419840 Contributors: -Ozone-, 11kravitzn, Abolen, Aetheling, Agne27, Alexey Muranov, Altmany, Anagogist, Andre Engels,Anthony Appleyard, Arthur Rubin, Atchius, AugPi, Axl, Baccyak4H, Badmonkey, Bjf, BoH, Br77rino, Bruguiea, Bse3, CAkira, Cctoide, Chabuk, Charles Matthews, CiaPan, Cjw54321,ClementSeveillac, Cmglee, CommonsDelinker, Conversion script, Crowsnest, Curb Chain, DRTllbrg, David Eppstein, DavidCary, Davius, Dfuss, Discospinster, Djhnsn, Dmcq, Dmmaus,DocendoDiscimus, Dolovis, Dolphin51, Doncram, Dtvjho, Dweir, EEMIV, EagleFan, Electricmic, Eramesan, Error, Etan J. Tal, Eudaemonic3, Ewlyahoocom, Fizped, Furrykef, Gandalf61,Geremia, Gerfriedc, Gesslein, Giftlite, Gloucks, Gr0ff, Graham87, Hairy Dude, Hellbus, Huw Powell, IanOfNorwich, Iful, Incnis Mrsi, Ingolfson, InternetMeme, JCSantos, JackofOz, Jbergquist,Jefferson1957, Jfclemay, John of Reading, JohnBlackburne, JohnOwens, Jonpin, KRS, Khurlbutt, Kipb9, Kri, Kurykh, Laurens-af, Laurentius, Lensovet, Leonard G., Lfstevens, LiDaobing,

LilHelpa, Loodog, Lowellian, Lwphillips, MBisanz, MEJ119, MSGJ, Magister Mathematicae, Martin S Taylor, MegaPedant, Megaboz, Membender, Michael Devore, Michael Hardy, Mik 0,Mikez, Mormegil, Mossig, Mr Stephen, Mtpaley, Mwarren us, N2e, NT27, NawlinWiki, Nbarth, Ninly, NipponBill, Nopetro, Nothingofwater, Notthe9, Oleg Alexandrov, Oliphaunt, Onore BakaSama, Parveson, Pascal Steger, Patrick, PerezTerron, PerryTachett, Peterpj77, Phancy Physicist, Phes11129, Phuzion, Phydend, Piero Montesacro, PierreAbbat, Pjacobi, Pleasantville, Pol098,Poor Yorick, Quadrescence, Quibik, R. S. Shaw, R.e.b., RDBury, RL0919, RadiantRay, Rausch, Raven in Orbit, Rich Farmbrough, Richard Stephens, Rjwilmsi, Robert Brook, RockMagnetist,Salix alba, SandraShklyaeva, SarekOfVulcan, Sc147, Scxnwa, Seahorseruler, Segv11, Septegram, Shadowjams, SimonTrew, Smack, Spiffulent, Stamcose, Stemonitis, Stikonas, StradivariusTV,Subh83, Svick, Syd1435, Tai89ch, Tamfang, Tarquin, Telso, Tennisstud1234, The Anome, The Thing That Should Not Be, Theodore Kloba, Thinking of England, Timrollpickering, Tman12321,Trovatore, Una Smith, VanBuren, Vanish2, Vstarsky, Werieth, Whatever1111, Whispering, Whoop whoop pull up, Wikfr, Wikid77, Wikipelli, William Avery, Woohookitty, Wordsmith, Xtv,Zkamran53, Zoicon5, , 219 anonymous edits

Image Sources, Licenses and ContributorsFile:Kette Kettenkurve Catenary 2008 PD.JPG Source: http://en.wikipedia.org/w/index.php?title=File:Kette_Kettenkurve_Catenary_2008_PD.JPG License: Creative CommonsAttribution-Sharealike 3.0 Contributors: Kamel15

File:PylonsSunset-5982.jpg Source: http://en.wikipedia.org/w/index.php?title=File:PylonsSunset-5982.jpg License: Creative Commons Attribution-Sharealike 3.0 Contributors: Loadmaster(David R. Tribble)

File:SpiderCatenary.jpg Source: http://en.wikipedia.org/w/index.php?title=File:SpiderCatenary.jpg License: Creative Commons Attribution 3.0 Contributors: Mtpaley (talk)File:GaudiCatenaryModel.jpg Source: http://en.wikipedia.org/w/index.php?title=File:GaudiCatenaryModel.jpg License: Creative Commons Attribution 3.0 Contributors: Etan J. Tal

Image:LaPedreraParabola.jpg Source: http://en.wikipedia.org/w/index.php?title=File:LaPedreraParabola.jpg License: Creative Commons Attribution-Sharealike 2.5 Contributors: Diseo deAntonio Gaud. Foto de Error.

Image:Sheffield Winter Garden.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Sheffield_Winter_Garden.jpg License: Public Domain Contributors: Photos taken by JanWedekind using a Voigtlnder Virtus D800, stitched using Hugin. Original uploader was Wedesoft at en.wikipedia

Image:Gateway Arch.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Gateway_Arch.jpg License: Creative Commons Attribution-Sharealike 2.5 Contributors: David K. Staub aten.wikipedia

Image:CatenaryKilnConstruction06025.JPG Source: http://en.wikipedia.org/w/index.php?title=File:CatenaryKilnConstruction06025.JPG License: Public Domain Contributors:User:Leonard G.

Image:Budapest_Keleti_teto 1.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Budapest_Keleti_teto_1.jpg License: Creative Commons Attribution-Sharealike 3.0 Contributors:Ztonyi Sndor (ifj.), Fizped

Image:Budapest_Keleti_teto_2.svg Source: http://en.wikipedia.org/w/index.php?title=File:Budapest_Keleti_teto_2.svg License: Creative Commons Attribution-Sharealike 3.0 Contributors:Ztonyi Sndor (ifj.), Fizped

File:Soderskar-bridge.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Soderskar-bridge.jpg License: Creative Commons Attribution-Sharealike 2.0 Contributors: A333, Dodo,Ejdzej, Ronaldino

File:Puentedelabarra(below).jpg Source: http://en.wikipedia.org/w/index.php?title=File:Puentedelabarra(below).jpg License: Public Domain Contributors: M.schwedFile:Comparison catenary parabola.svg Source: http://en.wikipedia.org/w/index.php?title=File:Comparison_catenary_parabola.svg License: Creative Commons Attribution-Sharealike 3.0Contributors: Cmglee

File:Catenary.PNG Source: http://en.wikipedia.org/w/index.php?title=File:Catenary.PNG License: Public Domain Contributors: BoH, Darapti, Kilom691

Image:catenary-pm.svg Source: http://en.wikipedia.org/w/index.php?title=File:Catenary-pm.svg License: Creative Commons Attribution-Sharealike 3.0 Contributors: Geek3

Image:Catenary-tension.png Source: http://en.wikipedia.org/w/index.php?title=File:Catenary-tension.png License: Public Domain Contributors: Davius

File:CatenaryForceDiagram.svg Source: http://en.wikipedia.org/w/index.php?title=File:CatenaryForceDiagram.svg License: Creative Commons Attribution-Sharealike 3.0 Contributors:User:RDBury

File:Golden Gate Bridge, SF.jpg Source: http://en.wikipedia.org/w/index.php?title=File:Golden_Gate_Bridge,_SF.jpg License: Creative Commons Attribution-Sharealike 3.0,2.5,2.0,1.0Contributors: Bernard Gagnon

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