What Is a Catenary?
A catenary is the curve a flexible chain or cable assumes when hanging under its own weight, suspended from two points. Its equation is the hyperbolic cosine:
y = a·cosh(x/a), where cosh(u) = (eu + e−u)/2 and a = H/w (horizontal tension over weight per unit length).
The catenary looks deceptively like a parabola, and for centuries it was assumed to be one, until Leibniz, Huygens, and Johann Bernoulli independently derived its true hyperbolic-cosine form in 1691 in response to a challenge by Jacob Bernoulli. The difference matters: a hanging chain (loaded by its own length) is a catenary, while a suspension bridge cable (loaded uniformly along the horizontal deck) is a genuine parabola. The catenary has a remarkable property that makes it a jewel of calculus: its arc length integral simplifies exactly, because the hyperbolic identity 1 + sinh² = cosh² collapses the usual square root. This calculator computes the sag, the exact arc length, and the tension of a hanging cable, and compares it against the parabola of the same span and sag.
How to Use the Catenary Calculator
Enter the catenary parameter a (which sets the curve's tightness, larger a means a flatter, more taut cable) and the span (horizontal distance between the supports). The steps report the physical origin of the curve, the sag (how far the ends rise above the lowest point), the exact arc length via the cosh identity, the parabola comparison, the tension distribution, and the historical resolution of the catenary-versus-parabola question. The graph draws the catenary against a parabola through the same three points, showing how they differ. This applies the arc length formula in a case where it simplifies exactly, and connects to the second-order differential equation that defines the curve.
Worked Example
Take a = 3 with a span of 8, so the cable runs from x = −4 to x = 4 (measuring from the lowest point at the vertex). The sag is the height gain from the vertex to the supports:
sag = a[cosh(4/a) − 1] = 3[cosh(4/3) − 1] ≈ 3(2.037 − 1) ≈ 3.11
The exact arc length uses the catenary's signature simplification. Because y′ = sinh(x/a), the arc-length integrand √(1 + y′²) becomes √(1 + sinh²(x/a)) = cosh(x/a) exactly, with no messy square root left, so
arc length = ∫−44 cosh(x/3) dx = 2·3·sinh(4/3) ≈ 10.86
The cable is about 10.86 units long to span 8 units horizontally. The parabola through the same endpoints and vertex would have a slightly different length, and the graph shows the catenary sitting a touch flatter near the bottom and steeper at the ends. This exact-arc-length property is unique among common curves, almost every other curve's arc length requires numerical integration, but the catenary's yields a clean closed form, a direct gift of the hyperbolic identity.
Why the Chain Is Not a Parabola
The distinction between the catenary and the parabola is one of the most instructive in applied calculus, because it hinges entirely on how the load is distributed. A hanging chain carries its weight along its own length, so heavier, longer sections of the curve pull down more, and the resulting balance of forces gives the differential equation y″ = (w/H)√(1 + y′²), whose solution is the hyperbolic cosine. A suspension-bridge cable, by contrast, carries a roadway whose weight is spread uniformly along the horizontal, not along the cable, giving the simpler equation y″ = constant, whose solution is a parabola.
Common Mistakes to Avoid
- Assuming a hanging cable is a parabola. It is a catenary (hyperbolic cosine) when loaded by its own weight. Only a cable carrying a uniform horizontal load, like a bridge deck, is a parabola. The two look similar but are genuinely different curves.
- Confusing the parameter a with the sag. The parameter a controls the curve's tightness (a = H/w); the sag is a separate quantity, a[cosh(half-span/a) − 1], that also depends on the span. Larger a means a flatter cable and, for a fixed span, less sag.
- Forgetting the arc-length simplification. The catenary's arc length has a clean closed form because √(1 + sinh²) = cosh exactly. Trying to integrate it numerically works but misses the elegant exact answer that makes the catenary special.
- Mislocating the origin. The standard catenary is centered at its lowest point (the vertex), where the curve is symmetric. Measuring the span from one end instead of from the center shifts the formulas and confuses the sag calculation.
- Ignoring that tension varies along the cable. Tension is minimal and horizontal at the bottom and maximal at the supports. Assuming uniform tension misrepresents the physics and underestimates the load the anchor points must bear.
Real-World Applications
The catenary is one of the most physically important curves in engineering and architecture, appearing wherever a flexible element hangs or a rigid arch must stand. Power transmission lines and telegraph wires hang in catenaries, and electrical engineers use the catenary equation to calculate the sag and tension for a given span and temperature, since the wire lengthens and sags more when hot, sag that must clear the ground and vegetation while the tension must not exceed the cable's strength, a genuine design constraint governing tower spacing and conductor selection. Suspension and cable-stayed bridges involve both catenaries and parabolas: the main cables under a uniform deck are parabolic, but the cables during construction (before the deck is hung) and free-hanging cables are catenaries, and mooring lines and anchor chains for ships and offshore platforms hang in catenaries whose shape determines the restoring forces.
Frequently Asked Questions
What is a catenary?
The curve a flexible chain or cable forms when hanging under its own weight between two supports. Its equation is y = a·cosh(x/a), a hyperbolic cosine. It appears in power lines, hanging chains, and, inverted, in the optimal shape for self-supporting arches.
Why is a hanging chain not a parabola?
Because a chain carries its weight along its own length, not uniformly along the horizontal. This gives the differential equation y″ = (w/H)√(1 + y′²), solved by cosh. A cable carrying a uniform horizontal load (like a bridge deck) instead satisfies y″ = constant and is a true parabola.
What does the parameter a represent?
It is the ratio a = H/w of the horizontal tension H to the weight per unit length w, and it sets the curve's tightness. Large a means a taut, flat cable with high tension and little sag; small a means a deep, loosely hanging cable. It also locates the curve's lowest point at height a.
Why does the catenary have an exact arc length?
Because its derivative is sinh(x/a), and the arc-length integrand √(1 + sinh²) equals cosh exactly by the identity 1 + sinh² = cosh². The troublesome square root disappears, so the integral of cosh gives a clean closed form, a rare and elegant property among curves.
How is the catenary related to the arch?
An inverted catenary is the ideal arch shape. A hanging chain is in pure tension; flipping it produces a shape in pure compression with no bending stress, the most stable form for a self-supporting arch. This is why the Gateway Arch and Gaudí's arches use inverted catenaries.
What is the sag of a catenary?
The vertical distance from the lowest point (vertex) to the level of the supports, equal to a[cosh(half-span/a) − 1]. It grows with the span and with smaller a. Engineers design power lines so the sag clears the ground while keeping tension within safe limits.
Where is the tension greatest in a hanging cable?
At the supports (the highest points), where the cable bears the most weight below it. Tension is minimal at the lowest point, where it is purely horizontal and equals H = wa. The tension at any point is w times the height y, so it increases with sag.
Who discovered the catenary equation?
Leibniz, Christiaan Huygens, and Johann Bernoulli independently derived it in 1691, answering a challenge posed by Jacob Bernoulli. Galileo had earlier guessed the hanging chain was a parabola, a natural error given their similarity; the new calculus was needed to prove it is a hyperbolic cosine.
Are suspension bridge cables catenaries?
The main cables supporting a uniform roadway are parabolas, because the load is spread evenly along the horizontal deck, not the cable. However, a free-hanging cable before the deck is attached, and mooring or anchor chains, are true catenaries. The load distribution decides which curve forms.
What is a catenoid?
The surface of revolution generated by rotating a catenary about a horizontal axis. It is a minimal surface, the shape a soap film takes between two rings, minimizing surface area. This links the catenary to the calculus of variations and minimal-surface theory, another arena where the curve appears naturally.