# Catenary

When a chain or cable, treated as idealized, hangs with support only at its two ends, the shape it takes under gravity alone is called a catenary. The word comes from the Latin *catena*, chain; the curve is described by the hyperbolic cosine, and in a suitable coordinate system its equation is \( y = a \cosh(x/a) \) with \( a > 0 \).<sup>[1](https://mathshistory.st-andrews.ac.uk//Curves/Catenary/)</sup><sup> • </sup><sup>[2](https://liberzon.csl.illinois.edu/teaching/cvoc/node23.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | The curve of a perfectly flexible, uniform chain hanging between two supports under gravity<sup>[1](https://mathshistory.st-andrews.ac.uk//Curves/Catenary/)</sup> |
| Equation | \( y = a \cosh(x/a) \), up to translation of the origin; \( a \) is the catenary's parameter<sup>[2](https://liberzon.csl.illinois.edu/teaching/cvoc/node23.html)</sup><sup> • </sup><sup>[3](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Catenary)</sup> |
| Solution history | Equation obtained by Leibniz, Huygens, and Johann Bernoulli in 1691, answering a challenge by Jacob Bernoulli<sup>[1](https://mathshistory.st-andrews.ac.uk//Curves/Catenary/)</sup> |
| Physics | The horizontal component of the tension is constant along the chain; the curve minimizes gravitational potential energy for a given chain length<sup>[4](https://courses.smp.uq.edu.au/MATH3401/Lectures/2000wbook.pdf)</sup><sup> • </sup><sup>[5](https://www.stonybrook.edu/laser/_jennifer/math/catenary.html)</sup> |
| Minimal surface | Rotating a catenary about its directrix (asymptote) generates the catenoid, the only minimal surface of revolution, shown by Euler in 1744<sup>[1](https://mathshistory.st-andrews.ac.uk//Curves/Catenary/)</sup><sup> • </sup><sup>[6](https://archive.lib.msu.edu/crcmath/math/math/c/c107.htm)</sup> |
| Architecture | An inverted catenary arch carries its own weight in pure compression without bending moments; used by Gaudí and, earlier, proposed by Hooke in 1676<sup>[7](https://link.springer.com/article/10.1007/s00004-025-00816-0)</sup><sup> • </sup><sup>[8](https://www.tandfonline.com/doi/abs/10.1080/15583058.2017.1290851)</sup> |
| Railways | The overhead contact wire of an electrified railway hangs in a catenary shape between its supports<sup>[9](https://www.sciencedirect.com/topics/engineering/wire-tension)</sup> |

## Mechanical derivation

The curve follows from static equilibrium of a small element of the chain. Let \( T(x) \) be the tensile force in a chain of constant density \( \rho \), with horizontal component \( H \) and vertical component \( V(x) \). Because the load has no horizontal component, \( H \) is the same at every point of the chain. Vertical equilibrium of an arc element of length \( \delta s \), whose weight is \( \rho g \, \delta s \), gives \( dV/dx = \rho g \, ds/dx \), and geometry gives \( V = H \, dy/dx \) and \( ds/dx = \sqrt{1 + (dy/dx)^2} \). Combining these yields the differential equation of the catenary,<sup>[4](https://courses.smp.uq.edu.au/MATH3401/Lectures/2000wbook.pdf)</sup>

\[ \frac{d^2 y}{dx^2} = \frac{\rho g}{H} \sqrt{1 + \left( \frac{dy}{dx} \right)^2}. \]

Solving by two successive integrations, with the vertex at the origin where the slope is zero, gives \( y = a \cosh(x/a) \), where \( a = H/(\rho g) \) sets the scale of the curve.<sup>[10](https://ttc.stu.edu.iq/wp-content/uploads/2024/10/g.pdf)</sup><sup> • </sup><sup>[2](https://liberzon.csl.illinois.edu/teaching/cvoc/node23.html)</sup> A chain suspended too low touches the ground, and the formula then no longer describes the whole span.<sup>[2](https://liberzon.csl.illinois.edu/teaching/cvoc/node23.html)</sup>

The same curve solves a problem in the calculus of variations: among all curves of a given length joining two points, the catenary is the one with the least gravitational potential energy, which is why a freely hanging chain settles into it.<sup>[5](https://www.stonybrook.edu/laser/_jennifer/math/catenary.html)</sup><sup> • </sup><sup>[11](https://www.cfm.brown.edu/people/dobrush/am33/Mathematica/ch2/auto.html)</sup> The variational treatment sets up the Lagrangian for the surface-of-revolution area and applies the [Euler–Lagrange equation](https://www.edgepedia.org/euler-lagrange-equation), or equivalently the Beltrami identity for the hanging-chain energy functional.<sup>[11](https://www.cfm.brown.edu/people/dobrush/am33/Mathematica/ch2/auto.html)</sup><sup> • </sup><sup>[12](https://ecommons.udayton.edu/epumd/vol8/iss1/1/)</sup>

## Geometric properties

With the vertex at \( t = 0 \), the parametric form is \( x = t \), \( y = a \cosh(t/a) \). The arc length measured from the vertex is \( s = a \sinh(t/a) \), so the slope is proportional to the arc length from the center of symmetry, and the intrinsic equation is \( s = c \tan \psi \). The curvature is \( \kappa = -(1/a) \operatorname{sech}^2(t/a) \), and the Cesàro equation, expressing curvature in terms of arc length, is \( (s^2 + a^2) \kappa = -a \).<sup>[6](https://archive.lib.msu.edu/crcmath/math/math/c/c107.htm)</sup><sup> • </sup><sup>[3](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Catenary)</sup>

The catenary is the evolute of the tractrix, and conversely the tractrix, the involute of the catenary, has a cusp at the catenary's vertex and the property that the length of its tangent is constant; it describes the path of a weight dragged by a string along a straight line.<sup>[1](https://mathshistory.st-andrews.ac.uk//Curves/Catenary/)</sup><sup> • </sup><sup>[3](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Catenary)</sup> The catenary is also the locus of the focus of a parabola rolling along a straight line.<sup>[1](https://mathshistory.st-andrews.ac.uk//Curves/Catenary/)</sup>

## History

Galileo investigated the hanging chain and concluded, incorrectly, that it was a parabola. Jungius detected the error in 1669, but the true form was not found until 1691, when [Jacob Bernoulli](https://www.edgepedia.org/jacob-bernoulli) published the problem in the *Acta Eruditorum* and Leibniz, Huygens, and [Johann Bernoulli](https://www.edgepedia.org/johann-bernoulli) each produced a solution.<sup>[3](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Catenary)</sup><sup> • </sup><sup>[1](https://mathshistory.st-andrews.ac.uk//Curves/Catenary/)</sup> Huygens was the first to use the term, writing *catenaria* in a letter to Leibniz dated 18 November 1690, and David Gregory wrote a treatise on the curve in 1690.<sup>[1](https://mathshistory.st-andrews.ac.uk//Curves/Catenary/)</sup><sup> • </sup><sup>[13](https://mathshistory.st-andrews.ac.uk/Miller/mathword/c/)</sup> The English word appears in 1725 in *Lexicon Technicum*, and the modern English form is usually attributed to [Thomas Jefferson](https://www.edgepedia.org/thomas-jefferson), who used it in a 1788 letter to Thomas Paine about bridge design.<sup>[13](https://mathshistory.st-andrews.ac.uk/Miller/mathword/c/)</sup><sup> • </sup><sup>[11](https://www.cfm.brown.edu/people/dobrush/am33/Mathematica/ch2/auto.html)</sup>

In 1744 Euler showed that a catenary revolved about its asymptote generates the only minimal surface of revolution, the catenoid, the surface of least area for given bounding circles.<sup>[1](https://mathshistory.st-andrews.ac.uk//Curves/Catenary/)</sup><sup> • </sup><sup>[6](https://archive.lib.msu.edu/crcmath/math/math/c/c107.htm)</sup>

## Catenary arches and structures

Inverting the hanging chain gives the natural shape for an arch. [Robert Hooke](https://www.edgepedia.org/robert-hooke) proposed this chain theory of arches in 1676, and [Christopher Wren](https://www.edgepedia.org/christopher-wren) used it in the dome of Saint Paul's Cathedral (1675).<sup>[8](https://www.tandfonline.com/doi/abs/10.1080/15583058.2017.1290851)</sup> An inverted catenary arch under its own weight is in pure compression, without bending moments, so it needs no external buttressing; a semicircular arch, by contrast, pushes outward and must be supported from outside.<sup>[7](https://link.springer.com/article/10.1007/s00004-025-00816-0)</sup><sup> • </sup><sup>[14](https://gaudi.museum/en/discover-gaudi/antoni-gaudi/gaudi-method/)</sup>

**Gaudí's hanging models.** [Antoni Gaudí](https://www.edgepedia.org/antoni-gaudi) took this principle to its literal extreme, defining arches and vaults with hanging models of ropes and small sandbags in his work on the [Sagrada Família](https://www.edgepedia.org/sagrada-familia) temple in Barcelona and the crypt of the Colònia Güell.<sup>[7](https://link.springer.com/article/10.1007/s00004-025-00816-0)</sup> He first used catenary arches in the Obrera Mataronense Cooperative building of 1878 to 1882, twelve arches in a structure 48 by 12 meters, and applied them later in the attic of the Batlló House, described as a series of sixty catenary arches supporting the roof without a central pillar.<sup>[14](https://gaudi.museum/en/discover-gaudi/antoni-gaudi/gaudi-method/)</sup> For short spans the catenary and the parabola are so similar that sources disagree about the same building; the arches of the Milà House are called catenary by one source and parabolic by another.<sup>[14](https://gaudi.museum/en/discover-gaudi/antoni-gaudi/gaudi-method/)</sup> The vertical profile of the Sagrada Família's bell towers, which rise and narrow without external support, follows the inverted catenary.<sup>[15](https://gaudi.museum/en/discover-gaudi/sagrada-familia/architecture-and-symbolism/)</sup>

**Railway overhead lines.** In electrified railways the overhead contact system takes its name from the same curve: the contact wire hangs under its own weight, supported by a messenger wire and droppers, in a classic catenary shape.<sup>[9](https://www.sciencedirect.com/topics/engineering/wire-tension)</sup> The first catenary-suspended contact wire was installed by AEG in 1903, carrying AC 6 kV at 25 Hz on the Niederschoeneweide–Spindlersfeld line.<sup>[16](https://jernbaneteknologi.no/en/chapters/15-railway-catenary-system/)</sup> Because the catenary equation is nonlinear, computing the equilibrium configuration of such a two-wire system connected by droppers requires numerical solution, by nested Newton–Raphson schemes.<sup>[17](https://www.sciencedirect.com/science/article/abs/pii/S0141029606000460)</sup>

## Generalizations

The hanging chain problem has been extended beyond the Euclidean plane. In the sphere and the [hyperbolic plane](https://www.edgepedia.org/hyperbolic-plane), a catenary is defined as the shape of a hanging chain whose potential energy is measured by the distance to a given geodesic of the space, and the hyperbolic case admits further variants using horocycle distances.<sup>[18](https://arxiv.org/abs/2208.13694)</sup> Variations of the classical problem include chains of variable density, extensible chains, and chains acted on by a force directed toward a fixed center, cases already considered by Jacob Bernoulli.<sup>[3](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Catenary)</sup>

## References

1. [Catenary, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//Curves/Catenary/)
2. [2.1.3 Catenary, Illinois calculus of variations notes](https://liberzon.csl.illinois.edu/teaching/cvoc/node23.html)
3. [Catenary, 1911 Encyclopædia Britannica, Wikisource](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Catenary)
4. [The catenary problem, MATH3401 lecture notes, University of Queensland](https://courses.smp.uq.edu.au/MATH3401/Lectures/2000wbook.pdf)
5. [Catenary Curves, Stony Brook University](https://www.stonybrook.edu/laser/_jennifer/math/catenary.html)
6. [Catenary, Eric Weisstein's Treasure Trove of Mathematics (MSU archive)](https://archive.lib.msu.edu/crcmath/math/math/c/c107.htm)
7. [Estimating the Minimum Thickness of Earthquake-Resistant Vaults and Catenary Arches by Using a Hanging Chain, Nexus Network Journal](https://link.springer.com/article/10.1007/s00004-025-00816-0)
8. [Hooke's Chain Theory and the Construction of Catenary Arches in Spain, International Journal of Architectural Heritage](https://www.tandfonline.com/doi/abs/10.1080/15583058.2017.1290851)
9. [Wire Tension, ScienceDirect Topics](https://www.sciencedirect.com/topics/engineering/wire-tension)
10. [Appendix 1: The Catenary, or Curve of a Hanging Chain, G. F. Simmons, Calculus with Analytic Geometry](https://ttc.stu.edu.iq/wp-content/uploads/2024/10/g.pdf)
11. [Example: Catenary, Mathematica tutorial, Brown University](https://www.cfm.brown.edu/people/dobrush/am33/Mathematica/ch2/auto.html)
12. [Derivation of the Sliding Catenary Curve via Calculus of Variations, Ethan Shade, Electronic Proceedings of Undergraduate Mathematics Day](https://ecommons.udayton.edu/epumd/vol8/iss1/1/)
13. [Earliest Known Uses of Some of the Words of Mathematics (C), MacTutor](https://mathshistory.st-andrews.ac.uk/Miller/mathword/c/)
14. [Gaudí's method: geometry and nature, Gaudí Museum](https://gaudi.museum/en/discover-gaudi/antoni-gaudi/gaudi-method/)
15. [Architecture and symbols of the Sagrada Família, Gaudí Museum](https://gaudi.museum/en/discover-gaudi/sagrada-familia/architecture-and-symbolism/)
16. [Railway Catenary System, Railway Engineering Compendium](https://jernbaneteknologi.no/en/chapters/15-railway-catenary-system/)
17. [Computation of the initial equilibrium of railway overheads based on the catenary equation, ScienceDirect](https://www.sciencedirect.com/science/article/abs/pii/S0141029606000460)
18. [The hanging chain problem in the sphere and in the hyperbolic plane, Rafael López, arXiv](https://arxiv.org/abs/2208.13694)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Differential geometry*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license