⛓️ Catenary (3D)
A uniformly dense chain hanging freely between two points takes the shape y = a·cosh(x/a) — the catenary. Drag the anchor points and watch the exact chain curve update in real time. Compare with a parabola. Drag the background to orbit the camera.
The Catenary Curve
Galileo thought the hanging chain was a parabola — he was wrong. The true shape is the catenary, described by the hyperbolic cosine: y = a·cosh(x/a). The constant a = T₀/(wg), where T₀ is the horizontal component of tension and w is weight per unit length.
The catenary appears in suspension bridges, power lines, arch design (inverted), and even the shape of a soap film between two rings. Drag the yellow anchor points to explore how the curve responds to different spans and sag values.
About Catenary
This simulation models the catenary curve: the precise shape taken by a flexible, inextensible chain of uniform weight hanging freely between two fixed anchor points. The curve is described by the equation y = a·cosh(x/a), where the parameter a equals the ratio of horizontal tension to weight per unit length. You can drag the yellow anchor points, adjust the cable length, and toggle the parabola overlay to see how the catenary differs from the commonly assumed parabolic shape.
Catenaries appear throughout engineering and architecture: suspension bridge cables, overhead power lines, and the inverted catenary arches of Gothic cathedrals and the Gateway Arch in St. Louis all exploit the same elegant equilibrium. The word itself comes from the Latin catena, meaning "chain".
Frequently Asked Questions
What is a catenary?
A catenary is the curve formed by a perfectly flexible, uniform chain or cable hanging freely under gravity between two support points. Unlike a parabola, which describes a projectile path, the catenary arises from the balance between the chain's weight and the tension that varies continuously along it. Its equation is y = a·cosh(x/a), where cosh is the hyperbolic cosine function.
How do I use the simulation?
Drag the two yellow anchor circles to reposition the support points anywhere on the canvas. Use the Cable length slider to add or remove chain length (more length means deeper sag), and the Density slider to change the effective weight per unit length. Toggle "Compare parabola" to overlay the best-fit parabola and see where the two curves diverge, and toggle "Tension vectors" to visualise how the tension direction and magnitude vary along the chain.
Why is the catenary not a parabola?
A parabola arises when a load is distributed uniformly along the horizontal span, as in the stiffening deck of a suspension bridge. A catenary arises when the load is distributed uniformly along the arc length of the cable itself. Because the arc length per horizontal unit increases toward the supports, the catenary curves more steeply there than a parabola does. For small sag-to-span ratios the two shapes are very close, but for deep sags the difference becomes clearly visible in the simulation.
What is the parameter a in y = a·cosh(x/a) and how is it calculated?
The parameter a equals T₀/(wg), where T₀ is the constant horizontal component of tension, w is the linear mass density of the chain, and g is gravitational acceleration. In the simulation the value of a is solved numerically: given the horizontal span and the total cable length L, the equation 2a·sinh(D/2a) = L is solved iteratively with a Newton method, where D is the horizontal distance between anchors. Once a is found the sag, arc-length, and tension at every point follow directly from the hyperbolic functions cosh and sinh.
Where do catenaries appear in the real world?
Power transmission lines between pylons hang in catenaries, and engineers deliberately allow a controlled sag to reduce the tension load on the towers. The main cables of long-span suspension bridges approximate catenaries before the deck is attached; once the deck is added the distributed deck load shifts the shape toward a parabola. Inverted catenaries form structurally efficient compression arches: the famous Gateway Arch in St. Louis is a weighted catenary, and Robert Hooke used an inverted chain model to design the dome of St. Paul's Cathedral in London.
Is it true that Galileo got the catenary wrong?
Yes. Galileo believed the hanging chain formed a parabola and stated this in his 1638 Discourses and Mathematical Demonstrations Relating to Two New Sciences. He was wrong, and Joachim Jungius later showed by geometrical means that it could not be a parabola. The correct equation was derived in 1691 independently by Christiaan Huygens, Gottfried Wilhelm Leibniz, and Johann Bernoulli in response to a challenge posed by Jakob Bernoulli. This was one of the earliest applications of the newly developed calculus of Leibniz.
Who discovered and named the catenary?
The word "catenary" was coined by Christiaan Huygens in a 1690 letter, from the Latin catena (chain). The curve itself was first correctly described mathematically in 1691 when Huygens, Leibniz, and Johann Bernoulli each independently published solutions to the challenge set by Jakob Bernoulli. Leibniz introduced the hyperbolic cosine representation that remains standard today. Robert Hooke had previously noted, in a 1675 anagram, that an inverted catenary gives a perfect compression arch, anticipating the structural insight by 16 years.
How does tension vary along the chain?
The tension T at any point along the catenary is T(x) = wg·a·cosh(x/a), which is minimal at the lowest point (where T = wg·a, the pure horizontal tension) and increases symmetrically toward the anchors. The horizontal component of tension is constant everywhere along the chain, while the vertical component equals wg times the arc length from the lowest point. In the simulation the yellow tension vectors are proportional to cosh(x/a) and are tangent to the curve, showing both magnitude and direction at once.
What is the surface of revolution of a catenary, and why does it matter?
When a catenary is rotated about its axis of symmetry it generates a catenoid, the minimal surface spanning two parallel rings. A soap film stretched between two coaxial circular wire rings spontaneously adopts the catenoid shape because it minimises its total surface area, a result from the calculus of variations. The catenoid was the first minimal surface to be discovered beyond the flat plane, identified by Leonhard Euler in 1744. In engineering, catenoid-like forms appear in thin-shell roof structures and in nanotube geometry.
How is the catenary used in engineering and modern technology?
In railway electrification systems the overhead contact wire is deliberately tensioned into a catenary (the system is literally called "catenary wire") to maintain uniform contact with the pantograph at high speed. Structural engineers use inverted catenary profiles for masonry arches and shell roofs because pure compression with no bending moment is achieved, maximising strength per unit of material. In aerospace, catenary-based cable networks are used in deployable antenna reflectors and gossamer space structures. Computational architects use catenary meshes as form-finding tools for complex free-form shells.
Are there open research questions related to the catenary?
Classical catenary theory is fully solved, but several active research areas build on it. Elastic catenaries, where the cable can stretch and its length changes with tension, are studied for mooring lines of offshore platforms and for high-altitude tethered balloons, requiring coupled elasticity and geometry equations. Rotating catenaries (whirling chains) introduce centrifugal terms that produce bifurcations and chaotic dynamics under certain conditions. In soft robotics and morphing structures, catenary-based actuator designs are being explored to achieve programmable shape changes, while in nanophysics the equilibrium of suspended graphene ribbons follows a modified catenary equation that includes van der Waals adhesion terms.