Catenary

In physics and geometry, the catenary 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 (though mathematically quite different). It also appears in the design of certain types of arches and as a cross section of the shape assumed by a soap film bounded by two parallel circular rings.

Mathematically, the curve is the graph of the hyperbolic cosine function. Its surface of revolution, the catenoid, is a minimal surface and is the only minimal surface of revolution other than the plane.

Contents

History

The word catenary is derived from the Latin word catena, which means "chain". Thomas Jefferson is usually credited with the English word catenary.[1][2] This occurs in a passage 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 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.[3]

The curve is also called the "alysoid", "chainette",[4] or, particularly in the material sciences, "funicular".[5]

It is often stated[6] that Galileo thought the curve followed by a hanging chain was a parabola. A careful reading of his book Two New Sciences[7] shows this to be an oversimplification. Galileo states that a hanging cord is approximated by a parabola, correctly observing that this approximation improves as the curvature gets smaller and is almost exact when the elevation is less than 45°.[8] 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.[9]

The application of the catenary to the construction of arches is due to Robert Hooke, who discovered it in the context of the rebuilding of St Paul's Cathedral,[10] possibly having seen Huygens' work on the catenary. Some much older arches are also approximate catenaries; an example is the Arch of Taq-i Kisra in Ctesiphon. [11]

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[12] in an appendix to his Description of Helioscopes,[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 of this anagram[14] 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.[9] David Gregory wrote a treatise on the catenary in 1697.[9]

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.[4] Nicolas Fuss gave equations describing the equilibrium of a chain under any force in 1796.[15]

The 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.[16][17]

The Gateway Arch in St. Louis, Missouri, United States is sometimes said to be an (inverted) catenary, but this is incorrect.[18] It is close to a more general curve called a flattened catenary, with equation y=Acosh(Bx). (A catenary would have 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.[19][20]

Inverted catenary arches
Catenary[21] arches under the roof of Gaudí's Casa Milà, Barcelona, Spain.  
The Sheffield Winter Garden is enclosed by a series of catenary arches.[22]  
The Gateway Arch (looking East) is a flattened catenary.  
Catenary arch kiln under construction over temporary form  

Catenary bridges

In free-hanging chains the force exerted is uniform with respect to length of the chain and so the chain follows the catenary curve.[23] The same is true of a simple suspension bridge or "catenary bridge," where the roadway follows the cable.[24][25]

A stressed ribbon bridge is a more sophisticated structure with the same catenary shape.[26][27]

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.[28][29]


Anchoring of marine objects

The catenary form given by gravity is taken advantage of in its presence in heavy anchor rodes. An anchor rode (or anchor line) usually consists of chain and/or cable. Anchor rodes are used by ships, oilrigs, docks, floating wind turbines, and other marine assets 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 was nearly straight. This assists 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.[30]

Mathematical description

Equation

The equation of a catenary in Cartesian coordinates has the form[28]

y = a \, \cosh \left ({x \over a} \right ) = {a \over 2} \, \left (e^{x/a} %2B e^{-x/a} \right ),\,

where cosh is the hyperbolic cosine function. All catenary curves are similar to each other. Changing the parameter a is equivalent to a uniform scaling of the curve.[31]

The Whewell equation for the catenary is[28]

\tan \varphi = \frac{s}{a}.\,

Differentiating gives

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

and eliminating \varphi gives the Cesàro equation[32]

\kappa=\frac{a}{s^2%2Ba^2}.\,

The radius of curvature is then

\rho = a \sec^2 \varphi\,

which is the length of the line normal to the curve between it and the x-axis.[33]

Other properties

A parabola rolled along a straight line traces out a catenary (see roulette) with its focus.[4]

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.[34]

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).[35]

The surface of revolution with fixed radii at either end that has minimum surface area is a catenary revolved about the x-axis.[36]

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.[37]

Analysis

Model of chains and arches

In the mathematical model. the chain (or cord, cable, rope, string, etc.) is idealized by assuming that it is so thin that it can be regarded as a curve and that it is so flexible any force of tension exerted by the chain is parallel to the chain.[38] 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.[39] An underlying principle is that the chain may be considered a rigid body once it has attained equilibrium.[40] 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 is 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

\frac{d\mathbf{r}}{ds}=\mathbf{u}\,

where u is a unit tangent vector.

A differential equation for the curve may be derived as follows.[41] Let c be the lowest point on the chain, called the vertex of the catenary, [42] 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 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 T0 is 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 φ, Tsin φ), where T is the magnitude of the force and φ is the is the angle between the curve at r and the x-axis (see tangential angle). Finally, weight of the chain is represented by (0, −λgs) where λ is the mass per unit length and g is the acceleration of gravity.

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

T \cos \varphi = T_0\,

and

T \sin \varphi = \lambda gs,\,

and dividing these gives

\frac{dy}{dx}=\tan \varphi = \frac{\lambda gs}{T_0}.\,

It is convenient to write

a = \frac{T_0}{\lambda g}\,

which is the length of chain whose weight is equal in magnitude to the tension at c.[43] Then

\frac{dy}{dx}=\frac{s}{a}\,

is an equation defining the curve.

It is immediate that the horizontal component of the tension, Tcos φ = λga is constant and the vertical component of the tension, Tsin φ = λgs is proportional to the length of chain between the r and the vertex.[44]

Derivation of equations for the curve

The differential equation given above can be solved to produce equations for the curve.[45]

From

\frac{dy}{dx} = \frac{s}{a},\,

the formula for arc length gives

\frac{ds}{dx} = \sqrt{1%2B\left(\dfrac{dy}{dx}\right)^2} = \frac{\sqrt{a^2%2Bs^2}}{a}.\,

Then

\frac{dx}{ds} = \frac{1}{\frac{ds}{dx}} = \frac{a}{\sqrt{a^2%2Bs^2}}\,

and

\frac{dy}{ds} = \frac{\frac{dy}{dx}}{\frac{ds}{dx}} = \frac{s}{\sqrt{a^2%2Bs^2}}.\,

The second of these equations can be integrated to give

y = \sqrt{a^2%2Bs^2} %2B \beta\,

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

y = \sqrt{a^2%2Bs^2},\ y^2=a^2%2Bs^2.\,

The x-axis thus chosen is called the directrix of the catenary.

It follows that the magnitude of the tension at a point T = λgy which is proportional to the distance between the point and the directrix.[44]

The integral of expression for dx/ds can be found using standard techniques giving[46]

x = a\ \operatorname{arcsinh}(s/a) %2B \alpha.\,

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

x = a\ \operatorname{arcsinh}(s/a),\ s=a \sinh{x \over a}.\,

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

These results can be used to eliminate s giving

y = a \cosh \frac{x}{a}.\,

Alternative derivation

The differential equation can be solved using a different approach.[47]

From

s = a \tan \varphi\,

it follows that

\frac{dx}{d\varphi} = \frac{dx}{ds}\frac{ds}{d\varphi}=\cos \varphi \cdot a \sec^2 \varphi= a \sec \varphi\,

and

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

Integrating gives,

x = a \ln(\sec \varphi %2B \tan \varphi) %2B \alpha,\,

and

y = a \sec \varphi %2B \beta.\,

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

\sec \varphi %2B \tan \varphi = e^{x/a},\,

and taking the reciprocal of both sides

\sec \varphi - \tan \varphi = e^{-x/a}.\,

Adding and subtracting the last two equations then gives the solution

y = a \sec \varphi = a \cosh \tfrac{x}{a},\,

and

s = a \tan \varphi = a \sinh \tfrac{x}{a}.\,

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 A and A′ and with given length l. The equation can be determined in this case as follows:[48] Relabel if necessary so that A is to the left of A′ and let h and k be the vertical and horizontal distances from A to A′. Translate the axes so that the origin is at the vertex of the catenary so the equation of the curve is

y = a \cosh \tfrac{x}{a}\,

and let the coordinates of A and A′ be (b, c) and (b′, c′) respectively. The curve passes through these points, so

c = a \cosh \tfrac{b}{a},\,c' = a \cosh \tfrac{b'}{a},\,

from which

k = a (\cosh \tfrac{b}{a} - \cosh \tfrac{b'}{a}).\,

The lengths of the curve from the vertex to A and from the vertex to A′ are

a \sinh \tfrac{b}{a},\, a \sinh \tfrac{b'}{a}\,

respectively, so the length from A to A′ is

l = a (\sinh \tfrac{b}{a} - \sinh \tfrac{b'}{a}).\,

When l2−k2 is expanded using these expressions the result is

l^2-k^2=a^2(-2%2B2\cosh \tfrac{b%2Bb'}{a})=4a^2\sinh^2 \tfrac{h}{2a},\,

so

\sqrt{l^2-k^2}=2a\sinh \tfrac{h}{2a}.\,

This is a transcendental equation in a and must be solved numerically. However, it can be shown with the methods of calculus[49] that there is at most one solution with a>0 and so there is at most one position of equilibrium. Furthermore, a solution exists only when

\sqrt{l^2-k^2}>h,\,

in other words l is greater than the distance from A to A′. That is, a solution exists only when the length of the chain is longer than the distance between the two points.

Generalizations with vertical force

Nonuniform chains

If the density of the chain is variable then the analysis above can be adapted to produce equations for the curve given the density, or given the curve to find the density.[50]

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

\int w\ ds,\,

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

T \cos \varphi = T_0\,

and

T \sin \varphi = \int w\ ds,\,

and therefore

\frac{dy}{dx}=\tan \varphi = \frac{1}{T_0} \int w\ ds.\,

Differentiation then gives

w=T_0 \frac{d}{ds}\frac{dy}{dx} = \frac{T_0 \frac{d^2y}{dx^2}}{\sqrt{1%2B\left(\frac{dy}{dx}\right)^2}}.\,

In terms of φ and the radius of curvature ρ this becomes

w= \frac{T_0}{\rho \cos^2 \varphi}.\,

Suspension bridge curve

A similar analysis can be done to find the curve followed by the cable supporting a suspension bridge with a horizontal roadway.[51] If the weight of the roadway per unit length is w and the weight of the cable and the wire supporting 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 a before gives the differential equation

\frac{dy}{dx}=\tan \varphi = \frac{w}{T_0}x.\,

This is solved by simple integration to get

y=\frac{w}{2T_0}x^2 %2B \beta\,

and so the cable follows a parabola. If weight of the cable and supporting wires are not negligible then the analysis is more difficult.[52]

Catenary of equal strength

In a catenary of equal strength, 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, w, per unit length of the chain can be written T/c, where c is constant, and the analysis for nonuniform chains can be applied.[53]

In this case the equations for tension are

T \cos \varphi = T_0,\,
T \sin \varphi = \frac{1}{c}\int T\ ds.\,

Combining gives

c \tan \varphi = \int \sec \varphi\ ds\,

and by differentiation

c = \rho \cos \varphi\,

where ρ is the radius of curvature.

The solution to this is

y = c \ln \sec \frac{x}{c}.\,

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

x = c\varphi,\ s = \ln \tan \tfrac{1}{4} (\pi%2B2\varphi).\,

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

Elastic catenary

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 p is the natural length of a section of spring, then the length of the spring with tension T applied has length

s=(1%2B\frac{T}{E})p,\,

where E is a constant.[54] In the catenary the value of T is variable, but ratio remains valid at a local level, so[55]

\frac{ds}{dp}=1%2B\frac{T}{E}.

The curve followed by an elastic spring can now be derived following a similar method as for the inelastic spring.[56]

The equations for tension of the spring are

T \cos \varphi = T_0,\,

and

T \sin \varphi = \lambda_0 gp,\,

from which

\frac{dy}{dx}=\tan \varphi = \frac{\lambda_0 gp}{T_0},\ T=\sqrt{T_0^2%2B\lambda_0^2 g^2p^2},\,

where p is the natural length of the segment from c to r and λ0 is the mass per unit length of the spring with no tension and g is the acceleration of gravity. Write

a = \frac{T_0}{\lambda_0 g}\,

so

\frac{dy}{dx}=\tan \varphi = \frac{p}{a},\ T=\frac{T_0}{a}\sqrt{a^2%2Bp^2}.

Then

\frac{dx}{ds} = \cos \varphi = \frac{T_0}{T}

and

\frac{dy}{ds} = \sin \varphi = \frac{\lambda_0 gp}{T},

from which

\frac{dx}{dp} = \frac{T_0}{T}\frac{ds}{dp} = T_0(\frac{1}{T}%2B\frac{1}{E})=\frac{a}{\sqrt{a^2%2Bp^2}}%2B\frac{T_0}{E}

and

\frac{dy}{dp} = \frac{\lambda_0 gp}{T}\frac{ds}{dp} = \frac{T_0p}{a}(\frac{1}{T}%2B\frac{1}{E})=\frac{p}{\sqrt{a^2%2Bp^2}}%2B\frac{T_0p}{Ea}.

Integrating gives the parametric equations

x=a \operatorname{arcsinh}(p/a)%2B\frac{T_0}{E}p %2B \alpha,
y=\sqrt{a^2%2Bp^2}%2B\frac{T_0}{2Ea}p^2%2B\beta.

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

x=a\ \operatorname{arcsinh}(p/a)%2B\frac{T_0}{E}p,\,
y=\sqrt{a^2%2Bp^2}%2B\frac{T_0}{2Ea}p^2\,

are parametric equations for the curve.

Other generalizations

A 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.[57]

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,

\mathbf{T} = T \mathbf{u},\,

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. The forces acting on the segment of the chain between s and s+Δs are the force of tension T(s+Δs) at one end of the segment, the nearly opposite force −T(s) at the other end, and the external force acting on the segment which is approximately GΔs. These forces must balance so

\mathbf{T}(s%2B\Delta s)-\mathbf{T}(s)%2B\mathbf{G}\Delta s \approx \mathbf{0}.\,

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

\frac{d\mathbf{T}}{ds} %2B \mathbf{G} = \mathbf{0}.\,

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, −λg) where the chain has mass λ per unit length and g is the acceleration of gravity.

See also

Notes

  1. ^ ""Catenary" at Math Words". Pballew.net. 1995-11-21. http://www.pballew.net/arithme8.html#catenary. Retrieved 2010-11-17. 
  2. ^ Barrow, John D. (2010). 100 Essential Things You Didn't Know You Didn't Know: Math Explains Your World. W. W. Norton & Company. p. 27. ISBN 0-393-33867-3. 
  3. ^ Jefferson, Thomas (1829). Memoirs, Correspondence and Private Papers of Thomas Jefferson. Henry Colbura and Richard Bertley. p. 419. http://books.google.com/books?id=wFlq_7_IAEUC&pg=PA419#v=onepage&f=false. 
  4. ^ a b c MathWorld
  5. ^ e.g.: Shodek, Daniel L. (2004). Structures (5th ed.). Prentice Hall. p. 22. ISBN 978-0-13-048879-4. OCLC 148137330. 
  6. ^ For example Lockwood p. 124
  7. ^ Galileo Galilei (1914). Dialogues concerning two new sciences. Trans. Henry Crew & Alfonso de Salvio. Macmillan. pp. 149, 290. http://books.google.com/books?id=SPhnaiERbWcC. 
  8. ^ Fahie, John Joseph (1903). Galileo, His Life and Work. J. Murray. pp. 359–360. http://books.google.com/books?id=iX0RAAAAYAAJ&pg=PA359#v=onepage&q&f=false. 
  9. ^ a b c Lockwood p. 124
  10. ^ "Monuments and Microscopes: Scientific Thinking on a Grand Scale in the Early Royal Society" by Lisa Jardine
  11. ^ Denny, Mark (2010). Super Structures: The Science of Bridges, Buildings, Dams, and Other Feats of Engineering Super Structures. JHU Press. pp. 112–113. ISBN 0-8018-9437-9. 
  12. ^ cf. the anagram for Hooke's law, which appeared in the next paragraph.
  13. ^ "Arch Design". Lindahall.org. 2002-10-28. http://www.lindahall.org/events_exhib/exhibit/exhibits/civil/design.shtml. Retrieved 2010-11-17. 
  14. ^ The original anagram was "abcccddeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuux": the letters of the Latin phrase, alphabetized.
  15. ^ Routh Art. 455, footnote
  16. ^ Minogue, Coll; Sanderson, Robert (2000). Wood-fired Ceramics: Contemporary Practices. University of Pennsylvania. p. 42. ISBN 0-8122-3514-2. 
  17. ^ Peterson, Susan; Peterson, Jan (2003). The Craft and Art of Clay: A Complete Potter's Handbook. Laurence King. p. 224. ISBN 1-85669-354-6. http://books.google.com/books?id=PAZR-A9Ra6EC&lpg=PP1&pg=PA208#v=onepage&f=false. 
  18. ^ Osserman, Robert (2010), "Mathematics of the Gateway Arch", Notices of the American Mathematical Society 57 (2): 220–229, ISSN 0002-9920, http://www.ams.org/notices/201002/index.html 
  19. ^ Hicks, Clifford B. (December 1963). "The Incredible Gateway Arch: America's Mightiest National Monument". Popular Mechanics 120 (6): 89. ISSN 0032-4558. http://books.google.com/books?id=BuMDAAAAMBAJ&pg=PA89&dq=weighted+catenary#v=onepage. 
  20. ^ Laura Soullière Harrison (1985) (PDF), National Register of Historic Places Inventory-Nomination: Jefferson National Expansion Memorial Gateway Arch / Gateway Arch; or "The Arch", National Park Service, http://pdfhost.focus.nps.gov/docs/NHLS/Text/87001423.pdf, retrieved 2009-06-21  and Accompanying one photo, aerial, from 1975PDF (578 KB)
  21. ^ Sennott, Stephen (2004). Encyclopedia of Twentieth Century Architecture. Taylor & Francis. p. 224. ISBN 1-57958-433-0. 
  22. ^ Hymers, Paul (2005). Planning and Building a Conservatory. New Holland. p. 36. ISBN 1-84330-910-6. 
  23. ^ Owen Byer, Felix Lazebnik, and Deirdre L. Smeltzer, Methods for Euclidean Geometry, MAA, 2010, ISBN 0-88385-763-4, p. 210.
  24. ^ Leonardo Fernández Troyano, Bridge Engineering: A global perspective, Thomas Telford, 2003, ISBN 0-7277-3215-3, p. 514.
  25. ^ Willibald Trinks, Industrial Furnaces, 6th ed., Wiley-IEEE, 2003, ISBN 0-471-38706-1, p. 132.
  26. ^ John S. Scott, Dictionary of Civil Engineering, 4th ed., Springer, 1992, ISBN 0-412-98421-0, p. 433.
  27. ^ The Architects' Journal, Volume 207, The Architectural Press Ltd., 1998, p. 51.
  28. ^ a b c Lockwood p. 122
  29. ^ Paul Kunkel (June 30, 2006). "Hanging With Galileo". Whistler Alley Mathematics. http://whistleralley.com/hanging/hanging.htm. Retrieved March 27, 2009. 
  30. ^ "Chain, Rope, and Catenary – Anchor Systems For Small Boats". Petersmith.net.nz. http://www.petersmith.net.nz/boat-anchors/catenary.php. Retrieved 2010-11-17.  (for section)
  31. ^ "Catenary". Xahlee.org. 2003-05-28. http://xahlee.org/SpecialPlaneCurves_dir/Catenary_dir/catenary.html. Retrieved 2010-11-17. 
  32. ^ MathWorld, eq. 7
  33. ^ Routh Art. 444
  34. ^ "Roulette: A Comfortable Ride on an n-gon Bicycle" by Borut Levart, Wolfram Demonstrations Project, 2007.
  35. ^ Landau, Lev Davidovich (1975). The Classical Theory of Fields. Butterworth-Heinemann. p. 56. ISBN 0-7506-2768-9. 
  36. ^ Yates p. 13
  37. ^ Parker, Edward (2010), "A Property Characterizing the Catenary", Mathematics Magazine 83: 63–64
  38. ^ Routh Art. 442, p. 316
  39. ^ Church, Irving Porter (1890). Mechanics of Engineering. Wiley. p. 387. http://books.google.com/books?id=-iAPAAAAYAAJ&pg=PA387#v=onepage&f=false. 
  40. ^ Whewell p. 65
  41. ^ Following Routh Art. 443 p. 316
  42. ^ Routh Art. 443 p. 317
  43. ^ Whewell p. 67
  44. ^ a b Routh Art. 443 p. 318
  45. ^ Following Routh Art. 443 p/ 317
  46. ^ Use of hyperbolic functions follows Maurer p. 107
  47. ^ Following Lamb p. 342
  48. ^ Following Todhunter Art. 186
  49. ^ See Routh art. 447
  50. ^ Following Routh Art. 450
  51. ^ Following Routh Art. 452
  52. ^ 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.
  53. ^ Following Routh Art. 453
  54. ^ Routh Art. 489
  55. ^ Routh Art. 494
  56. ^ Following Routh Art. 500
  57. ^ Follows Routh Art. 455

Bibliography

Further reading

External links