Catenary Explained

In physics and geometry, a catenary is the curve that an idealized hanging chain or cable assumes under its own weight when supported only at its ends in a uniform gravitational field.

The catenary curve has a U-like shape, superficially similar in appearance to a parabola, which it is not.

The curve appears in the design of certain types of arches and as a cross section of the catenoid—the shape assumed by a soap film bounded by two parallel circular rings.

The catenary is also called the alysoid, chainette,^[1] or, particularly in the materials sciences, an example of a funicular.^[2] Rope statics describes catenaries in a classic statics problem involving a hanging rope.^[3]

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. A hanging chain will assume a shape of least potential energy which is a catenary.^[4] Galileo Galilei in 1638 discussed the catenary in the book Two New Sciences recognizing that it was different from a parabola. The mathematical properties of the catenary curve were studied by Robert Hooke in the 1670s, and its equation was derived by Leibniz, Huygens and Johann Bernoulli in 1691.

Catenaries and related curves are used in architecture and engineering (e.g., in the design of bridges and arches so that forces do not result in bending moments). In the offshore oil and gas industry, "catenary" refers to a steel catenary riser, a pipeline suspended between a production platform and the seabed that adopts an approximate catenary shape. In the rail industry it refers to the overhead wiring that transfers power to trains. (This often supports a contact wire, in which case it does not follow a true catenary curve.)

In optics and electromagnetics, the hyperbolic cosine and sine functions are basic solutions to Maxwell's equations.^[5] The symmetric modes consisting of two evanescent waves would form a catenary shape.^{[6] [7] [8]}

History

The word "catenary" is derived from the Latin word catēna, which means "chain". The English word "catenary" is usually attributed to Thomas Jefferson,^{[9] [10]} who wrote in a letter to Thomas Paine on the construction of an arch for a bridge:

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°.^[11] 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.^[12]

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.^[13] Some much older arches approximate catenaries, an example of which is the Arch of Taq-i Kisra in Ctesiphon.^[14]

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^[15] in an appendix to his Description of Helioscopes,^[16] 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^[17] 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;^[12] their solutions were published in the Acta Eruditorum for June 1691. David Gregory wrote a treatise on the catenary in 1697^[12] in which he provided an incorrect derivation of the correct differential equation.

Euler proved in 1744 that the catenary is the curve which, when rotated about the -axis, gives the surface of minimum surface area (the catenoid) for the given bounding circles.^[1] Nicolas Fuss gave equations describing the equilibrium of a chain under any force in 1796.^[18]

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.^{[19] [20]}

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, which is a catenary if . 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.^{[21] [22]}

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 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.^[30]

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.^[31]

Mathematical description

Equation

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

$$y\; =\; a\; \backslash cosh\; \backslash left(\backslash frac\backslash right)\; =\; \backslash frac\backslash left(e^\backslash frac\; +\; e^\backslash right),$$where is the hyperbolic cosine function, and where is the distance of the lowest point above the x axis.^[32] All catenary curves are similar to each other, since changing the parameter is equivalent to a uniform scaling of the curve.

The Whewell equation for the catenary is^[28] $$\backslash tan\; \backslash varphi\; =\; \backslash frac,$$where

\varphi is the tangential angle and the arc length.

Differentiating gives$$\backslash frac\; =\; \backslash frac,$$and eliminating

\varphi gives the Cesàro equation^[33] $$\backslash kappa=\backslash frac,$$where \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 -axis.^[34]

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.^[35] 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.^[36]

Geometrical properties

Over any horizontal interval, the ratio of the area under the catenary to its length equals, 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 -axis.^[37]

Science

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).^[38]

The surface of revolution with fixed radii at either end that has minimum surface area is a catenary revolved about the -axis.^[39]

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.^[40] 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.^[41] An underlying principle is that the chain may be considered a rigid body once it has attained equilibrium.^[42] 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 where represents arc length and is the position vector. This is the natural parameterization and has the property that

$$\backslash frac=\backslash mathbf$$

where is a unit tangent vector.

A differential equation for the curve may be derived as follows.^[43] Let be the lowest point on the chain, called the vertex of the catenary.^[44] The slope of the curve is zero at since it is a minimum point. Assume is to the right of since the other case is implied by symmetry. The forces acting on the section of the chain from to are the tension of the chain at, the tension of the chain at, and the weight of the chain. The tension at is tangent to the curve at and is therefore horizontal without any vertical component and it pulls the section to the left so it may be written where is the magnitude of the force. The tension at is parallel to the curve at and pulls the section to the right. The tension at can be split into two components so it may be written, where is the magnitude of the force and is the angle between the curve at and the -axis (see tangential angle). Finally, the weight of the chain is represented by where is the weight per unit length and is the length of the segment of chain between and .

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

$$T\; \backslash cos\; \backslash varphi\; =\; T\_0$$and$$T\; \backslash sin\; \backslash varphi\; =\; ws\backslash ,,$$

and dividing these gives

$$\backslash frac=\backslash tan\; \backslash varphi\; =\; \backslash frac\backslash ,.$$

It is convenient to write

$$a\; =\; \backslash frac$$

which is the length of chain whose weight is equal in magnitude to the tension at .^[45] Then

$$\backslash frac=\backslash frac$$

is an equation defining the curve.

The horizontal component of the tension, is constant and the vertical component of the tension, is proportional to the length of chain between and the vertex.^[46]

Derivation of equations for the curve

The differential equation

dy/dx = s/a , given above, can be solvedto produce equations for the curve.^[47] We will solve the equation using the boundary condition thatthe vertex is positioned at s _{0=0} and (x,y) = (x _{0,y} _{0)} .

First, invoke the formula forarc lengthto get$$\backslash frac\; =\; \backslash sqrt\; =\; \backslash sqrt\backslash ,,$$then separate variablesto obtain$$\backslash frac\; =\; dx\backslash ,.$$

A reasonably straightforward approach to integrate this is to usehyperbolic substitution,which gives$$a\; \backslash sinh^\backslash frac\; +\; x\_0\; =\; x$$(where

x _{0} is a constant of integration),and hence$$\backslash frac\; =\; \backslash sinh\backslash frac\backslash ,.$$

But $s/a\; =\; dy/dx$, so$$\backslash frac\; =\; \backslash sinh\backslash frac\backslash ,,$$which integrates as$$y\; =\; a\; \backslash cosh\backslash frac\; +\; \backslash delta$$(with

\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$$y\; =\; a\; \backslash cosh\backslash frac\; \backslash quad\backslash square$$

For completeness, the

y \leftrightarrow s relation can be derived bysolving each of the x \leftrightarrow y and x \leftrightarrow s relations for x/a , giving:$$\backslash cosh^\backslash frac\; =\; \backslash frac\; =\; \backslash sinh^\backslash frac\backslash ,,$$so$$y-\backslash delta\; =\; a\backslash cosh\backslash left(\backslash sinh^\backslash frac\backslash right)\backslash ,,$$which can be rewritten as$$y-\backslash delta\; =\; a\backslash sqrt\; =\; \backslash sqrt\backslash ,.$$

Alternative derivation

The differential equation can be solved using a different approach.^[48] From

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

it follows that

$$\backslash frac\; =\; \backslash frac\backslash frac=\backslash cos\; \backslash varphi\; \backslash cdot\; a\; \backslash sec^2\; \backslash varphi=\; a\; \backslash sec\; \backslash varphi$$and$$\backslash frac\; =\; \backslash frac\backslash frac=\backslash sin\; \backslash varphi\; \backslash cdot\; a\; \backslash sec^2\; \backslash varphi=\; a\; \backslash tan\; \backslash varphi\; \backslash sec\; \backslash varphi\backslash ,.$$

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 and -axes can be shifted so and can be taken to be 0. Then

$$\backslash sec\; \backslash varphi\; +\; \backslash tan\; \backslash varphi\; =\; e^\backslash frac\backslash ,,$$and taking the reciprocal of both sides$$\backslash sec\; \backslash varphi\; -\; \backslash tan\; \backslash varphi\; =\; e^\backslash ,.$$

Adding and subtracting the last two equations then gives the solution$$y\; =\; a\; \backslash sec\; \backslash varphi\; =\; a\; \backslash cosh\backslash left(\backslash frac\backslash right)\backslash ,,$$and$$s\; =\; a\; \backslash tan\; \backslash varphi\; =\; a\; \backslash sinh\backslash left(\backslash frac\backslash right)\backslash ,.$$

Determining parameters

In general the parameter is the position of the axis. The equation can be determined in this case as follows:^[49]

Relabel if necessary so that is to the left of and let be the horizontal and be the vertical distance from to . Translate the axes so that the vertex of the catenary lies on the -axis and its height is adjusted so the catenary satisfies the standard equation of the curve $$y\; =\; a\; \backslash cosh\backslash left(\backslash frac\backslash right)$$

and let the coordinates of and be and respectively. The curve passes through these points, so the difference of height is

$$v\; =\; a\; \backslash cosh\backslash left(\backslash frac\backslash right)\; -\; a\; \backslash cosh\backslash left(\backslash frac\backslash right)\backslash ,.$$

and the length of the curve from to is

$$L\; =\; a\; \backslash sinh\backslash left(\backslash frac\backslash right)\; -\; a\; \backslash sinh\backslash left(\backslash frac\backslash right)\backslash ,.$$

When is expanded using these expressions the result is

$$L^2-v^2=2a^2\backslash left(\backslash cosh\backslash left(\backslash frac\backslash right)-1\backslash right)=4a^2\backslash sinh^2\backslash left(\backslash frac\backslash right)\backslash ,,$$so$$\backslash frac\; 1H\; \backslash sqrt=\backslash fracH\; \backslash sinh\backslash left(\backslash frac\backslash right)\backslash ,.$$

This is a transcendental equation in and must be solved numerically. Since

\sinh(x)/x is strictly monotonic on x > 0 ,^[50] there is at most one solution with and so there is at most one position of equilibrium.

However, if both ends of the curve (and) are at the same level, it can be shown that^[51] $$a\; =\; \backslash frac\; \backslash ,$$where L is the total length of the curve between and and is the sag (vertical distance between, and the vertex of the curve).

It can also be shown that$$L\; =\; 2a\; \backslash sinh\; \backslash frac\; \backslash ,$$and$$H\; =\; 2a\; \backslash operatorname\; \backslash frac\; \backslash ,$$where H is the horizontal distance between and which are located at the same level .

The horizontal traction force at and is, where is the weight per unit length of the chain or cable.

Tension relations

There is a simple relationship between the tension in the cable at a point and its - and/or - 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 asBut, as shown above,y = \sqrt{a ^{2} + s ^{2}} (assuming that y _{0=a} ), so we get the simple relations^[52] $$T\; =\; wy\; =\; wa\; \backslash cosh\backslash frac\backslash ,.$$

Variational formulation

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$$U\; =\; \backslash int\_0^D\; w\; y\backslash sqrt\; dx$$(where is the weight per unit length) and is subject to the constraint $$\backslash int\_0^D\; \backslash sqrt\; dx\; =\; L\backslash ,.$$

The modified Lagrangian is therefore $$\backslash mathcal\; =\; (w\; y\; -\; \backslash lambda)\backslash sqrt$$where

λ is the Lagrange multiplier to be determined. As the independent variable x does not appear in the Lagrangian, we can use the Beltrami identity $$\backslash mathcal-y\text{'}\; \backslash frac\; =\; C$$where C is an integration constant, in order to obtain a first integral$$\backslash frac\; =\; -C$$

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.

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.^[53]

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

$$\backslash int\_\backslash mathbf^\backslash mathbf\; w\backslash ,\; ds\backslash ,,$$

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

$$T\; \backslash cos\; \backslash varphi\; =\; T\_0$$and$$T\; \backslash sin\; \backslash varphi\; =\; \backslash int\_\backslash mathbf^\backslash mathbf\; w\backslash ,\; ds\backslash ,,$$and therefore$$\backslash frac=\backslash tan\; \backslash varphi\; =\; \backslash frac\; \backslash int\_\backslash mathbf^\backslash mathbf\; w\backslash ,\; ds\backslash ,.$$

Differentiation then gives

$$w=T\_0\; \backslash frac\backslash frac\; =\; \backslash frac\backslash ,.$$

In terms of and the radius of curvature this becomes

$$w=\; \backslash frac\backslash ,.$$

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.^[54] If the weight of the roadway per unit length is 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 to is where is the horizontal distance between and . Proceeding as before gives the differential equation

$$\backslash frac=\backslash tan\; \backslash varphi\; =\; \backslash fracx\backslash ,.$$

This is solved by simple integration to get

$$y=\backslash fracx^2\; +\; \backslash beta$$

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.^[55]

Catenary of equal strength

In a catenary of equal strength, the cable is strengthened according to the magnitude of the tension at each point, so its resistance to breaking is constant along its length. Assuming that the strength of the cable is proportional to its density per unit length, the weight,, per unit length of the chain can be written, where is constant, and the analysis for nonuniform chains can be applied.^[56]

In this case the equations for tension are

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 is the radius of curvature.

The solution to this is

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

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

$$x\; =\; c\backslash varphi\backslash ,,\backslash quad\; s\; =\; \backslash ln\backslash left(\backslash tan\backslash left(\backslash frac\backslash right)\backslash right)\backslash ,.$$

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".^[57]

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

$$s=\backslash left(1+\backslash frac\backslash right)p\backslash ,,$$

where is a constant equal to, where is the stiffness of the spring.^[58] In the catenary the value of is variable, but ratio remains valid at a local level, so^[59] $$\backslash frac=1+\backslash frac\backslash ,.$$The curve followed by an elastic spring can now be derived following a similar method as for the inelastic spring.^[60]

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

$$\backslash frac=\backslash tan\; \backslash varphi\; =\; \backslash frac\backslash ,,\backslash quad\; T=\backslash sqrt\backslash ,,$$

where is the natural length of the segment from to and is the weight per unit length of the spring with no tension. Write$$a\; =\; \backslash frac$$so$$\backslash frac=\backslash tan\; \backslash varphi\; =\; \backslash frac\; \backslash quad\backslash text\backslash quad\; T=\backslash frac\backslash sqrt\backslash ,.$$

Then from which

Integrating gives the parametric equations

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

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

Other generalizations

Chain under a general force

With no assumptions being made regarding the force acting on the chain, the following analysis can be made.^[61]

First, let be the force of tension as a function of . 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, must be parallel to the chain. In other words,

$$\backslash mathbf\; =\; T\; \backslash mathbf\backslash ,,$$

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

Second, let be the external force per unit length acting on a small segment of a chain as a function of . The forces acting on the segment of the chain between and are the force of tension at one end of the segment, the nearly opposite force at the other end, and the external force acting on the segment which is approximately . These forces must balance so

$$\backslash mathbf(s+\backslash Delta\; s)-\backslash mathbf(s)+\backslash mathbf\backslash Delta\; s\; \backslash approx\; \backslash mathbf\backslash ,.$$

Divide by and take the limit as to obtain

$$\backslash frac\; +\; \backslash mathbf\; =\; \backslash mathbf\backslash ,.$$

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, where the chain has weight per unit length.

See also

• Catenary arch
• Chain fountain or self-siphoning beads
• 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

Bibliography

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

Further reading

• Book: Swetz, Frank. Learn from the Masters. 1995. MAA. 978-0-88385-703-8. 128–9 .
• Book: Venturoli, Giuseppe. Trans. Daniel Cresswell. Elements of the Theory of Mechanics. https://books.google.com/books?id=kHhBAAAAYAAJ&pg=PA67. 1822. J. Nicholson & Son. Chapter XXIII: On the Catenary .

External links

• • • Catenary curve calculator
• Catenary at The Geometry Center
• "Catenary" at Visual Dictionary of Special Plane Curves
• The Catenary - Chains, Arches, and Soap Films.
• Cable Sag Error Calculator – Calculates the deviation from a straight line of a catenary curve and provides derivation of the calculator and references.
• 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.
• The straight line, the catenary, the brachistochrone, the circle, and Fermat Unified approach to some geodesics.
• Ira Freeman "A General Form of the Suspension Bridge Catenary" Bulletin of the AMS

Notes and References

1. [#MathWorld|MathWorld]
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3. Web site: Shape of a hanging rope. https://web.archive.org/web/20180920110549/http://www2.mae.ufl.edu/~uhk/HANGING-ROPE.pdf . 2018-09-20 . live . Department of Mechanical & Aerospace Engineering - University of Florida . 2017-05-02 . 2020-06-04.
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6. Bourke. Levi. Blaikie. Richard J.. 2017-12-01. Herpin effective media resonant underlayers and resonant overlayer designs for ultra-high NA interference lithography. JOSA A. EN. 34. 12. 2243–2249. 10.1364/JOSAA.34.002243. 29240100. 2017JOSAA..34.2243B. 1520-8532.
7. Pu. Mingbo. Guo. Yinghui. Li. Xiong. Ma. Xiaoliang. Luo. Xiangang. 2018-07-05. Revisitation of Extraordinary Young's Interference: from Catenary Optical Fields to Spin–Orbit Interaction in Metasurfaces. ACS Photonics. 5. 8. 3198–3204. en. 10.1021/acsphotonics.8b00437. 126267453 . 2330-4022.
8. Pu. Mingbo. Ma. XiaoLiang. Guo. Yinghui. Li. Xiong. Luo. Xiangang. 2018-07-23. Theory of microscopic meta-surface waves based on catenary optical fields and dispersion. Optics Express. EN. 26. 15. 19555–19562. 10.1364/OE.26.019555. 30114126. 2018OExpr..2619555P. 1094-4087. free.
9. Web site: "Catenary" at Math Words . Pballew.net . 1995-11-21 . 2010-11-17.
10. Book: Barrow, John D.. 100 Essential Things You Didn't Know You Didn't Know: Math Explains Your World. 2010. W. W. Norton & Company. 978-0-393-33867-6. 27.
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12. [#Lockwood|Lockwood]
13. 532102 . Monuments and Microscopes: Scientific Thinking on a Grand Scale in the Early Royal Society . Notes and Records of the Royal Society of London . 55 . 2 . 289–308 . Lisa . Jardine. 2001 . 10.1098/rsnr.2001.0145 . 144311552 .
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15. [cf.]
16. Web site: Arch Design . Lindahall.org . 2002-10-28 . 2010-11-17 . dead . https://web.archive.org/web/20101113210736/http://www.lindahall.org/events_exhib/exhibit/exhibits/civil/design.shtml . 2010-11-13 .
17. The original anagram was abcccddeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuux: the letters of the Latin phrase, alphabetized.
18. [#Routh|Routh]
19. Book: Minogue. Coll. Sanderson. Robert. Wood-fired Ceramics: Contemporary Practices. 2000. University of Pennsylvania. 978-0-8122-3514-2. 42 .
20. Book: Peterson. Susan. Peterson. Jan. The Craft and Art of Clay: A Complete Potter's Handbook. 2003. Laurence King. 978-1-85669-354-7. 224 .
21. Hicks. Clifford B.. The Incredible Gateway Arch: America's Mightiest National Monument. 120. December 1963. 89. 0032-4558. 6. .
22. and  
23. Book: Owen . Byer . Felix . Lazebnik . Deirdre L. . Smeltzer . Deirdre Smeltzer . Methods for Euclidean Geometry . 2010-09-02 . MAA . 978-0-88385-763-2 . 210.
24. Book: Fernández Troyano, Leonardo. Bridge Engineering: A Global Perspective. 2003. Thomas Telford. 978-0-7277-3215-6. 514 .
25. Book: W. . Trinks. M. H. . Mawhinney. R. A. . Shannon. R. J. . Reed. J. R.. Garvey. Industrial Furnaces. 2003-12-05. Wiley. 978-0-471-38706-0. 132 .
26. Book: Scott, John S. . Dictionary Of Civil Engineering. 1992-10-31. Springer. 978-0-412-98421-1. 433.
27. Cranked stress ribbon design to span Medway. Paul. Finch. . 207 . 19 March 1998 . 51.
28. [#Lockwood|Lockwood]
29. Web site: Hanging With Galileo. June 30, 2006. Paul. Kunkel. Whistler Alley Mathematics. March 27, 2009.
30. Web site: Chain, Rope, and Catenary – Anchor Systems For Small Boats . Petersmith.net.nz . 2010-11-17.
31. Web site: Efficiency of Cable Ferries - Part 2 . Human Power eJournal . 2023-12-08.
32. Web site: Catenary . Weisstein . Eric W. . Eric W. Weisstein . MathWorld--A Wolfram Web Resource . 2019-09-21 . 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 [...] .
33. [#MathWorld|MathWorld]
34. [#Routh|Routh]
35. Yates p. 80
36. Hall . Leon . Wagon . Stan . Stan Wagon. 1992 . Roads and Wheels . Mathematics Magazine . 65 . 5 . 283–301 . 2691240 . 10.2307/2691240.
37. Parker . Edward . 2010 . A Property Characterizing the Catenary . Mathematics Magazine . 83 . 63–64 . 10.4169/002557010X485120 . 122116662 .
38. Book: Landau, Lev Davidovich. The Classical Theory of Fields. 1975. Butterworth-Heinemann. 978-0-7506-2768-9. 56.
39. Book: Yates, Robert C. . Curves and their Properties . NCTM . 1952 . 13.
40. [#Routh|Routh]
41. Book: Church, Irving Porter. Mechanics of Engineering. 1890. Wiley. 387.
42. [#Whewell|Whewell]
43. Following Routh Art. 443 p. 316
44. [#Routh|Routh]
45. [#Whewell|Whewell]
46. [#Routh|Routh]
47. 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).
48. Following Lamb p. 342
49. Following Todhunter Art. 186
50. See Routh art. 447
51. Archived at Ghostarchive and the Wayback Machine: Web site: Chaînette - partie 3 : longueur . YouTube.
52. [#Routh|Routh]
53. Following Routh Art. 450
54. Following Routh Art. 452
55. 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.
56. Following Routh Art. 453
57. Pu. Mingbo. Li. Xiong. Ma. Xiaoliang. Luo. Xiangang. 2015. Catenary Optics for Achromatic Generation of Perfect Optical Angular Momentum. Science Advances. 1. 9. e1500396 . 10.1126/sciadv.1500396. 26601283. 4646797. 2015SciA....1E0396P.
58. [#Routh|Routh]
59. [#Routh|Routh]
60. Following Routh Art. 500
61. Follows Routh Art. 455