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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
[28][28] Following Routh Art. 452
[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.
[30][30] Following Routh Art. 453
[31][31] Routh Art. 489
[32][32] Routh Art. 494
[33][33] Following Routh Art. 500
[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 -
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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 -
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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
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