Catenary Curve

What the catenary is

Hang a necklace, a power line, or a length of chain between two hooks and let it settle. The smooth dip it forms is the catenary. It is the unique curve that an idealized chain — perfectly flexible (no bending stiffness), inextensible (fixed length), and of uniform weight per unit length — adopts in equilibrium under gravity. Every link simply hangs from its neighbors, and the only forces in play are tension along the chain and the chain's own weight pulling straight down.

Its equation is the hyperbolic cosine:

y = a · cosh(x / a)     where   cosh(u) = (eᵘ + e⁻ᵘ) / 2

with the lowest point at the origin shifted so the vertex sits at y = a. The single parameter a controls everything: a large a gives a shallow, almost-straight curve; a small a gives a deep, sharp sag.

Where the equation comes from

Consider a small element of chain. Three forces act on it: tension at each end and gravity. Resolving horizontal and vertical equilibrium gives two facts. First, the horizontal component of tension is constant all along the chain — call it H. Second, the vertical component balances the accumulated weight. Writing w = ρg for weight per unit length and using arc length s, equilibrium gives:

dy/dx = (w / H) · s        (slope ∝ chain weight hung so far)

Differentiating and substituting ds = √(1 + (dy/dx)²) dx produces the governing differential equation, with a = H/w:

d²y/dx² = (1/a) · √(1 + (dy/dx)²)

Its solution — choosing the vertex at x = 0 — is exactly y = a·cosh(x/a). The constant a = H/w is the physical heart of the curve: it is the ratio of how hard you pull sideways to how heavy the chain is per metre.

How tension is distributed

The horizontal pull H never changes, but the total tension does. A short derivation gives the elegant results:

Quantity	Expression	Where it is largest / smallest
Horizontal tension	H = w·a (constant)	Same everywhere
Total tension at point (x, y)	T = H·cosh(x/a) = w·y	Min at vertex (T = H), max at supports
Arc length from vertex	s = a·sinh(x/a)	Grows fast toward the ends
Slope	dy/dx = sinh(x/a) = s/a	Zero at vertex, steep at supports

The result T = w·y is striking: the tension at any point equals the weight of a hypothetical piece of chain whose length equals the height y above the curve's natural baseline. The chain "knows" how high it is. This is why suspension cables are thickest and most stressed at the towers, not at mid-span.

Catenary vs parabola — the famous mix-up

Galileo, in his 1638 Two New Sciences, asserted a hanging chain is a parabola. It was a good eye but wrong physics. In 1646 the seventeen-year-old Christiaan Huygens proved it could not be a parabola; the full hyperbolic-cosine solution arrived in 1691 from Huygens, Leibniz, and Johann Bernoulli responding to a challenge from Jakob Bernoulli.

The distinction is physical, not cosmetic. The shape depends on how the load is distributed:

Curve	Equation	Load distributed along…	Real example
Catenary	y = a·cosh(x/a)	the cable's own length (uniform per metre of chain)	Power line, necklace, free-hanging cable
Parabola	y = x²/(2a)	the horizontal span (uniform per metre of deck)	Suspension-bridge main cable (Golden Gate)

The Golden Gate Bridge's main cables really are close to parabolas, because the heavy roadway deck — hung by vertical suspenders — dominates the cable's own weight and is spread evenly along the horizontal. A bare, unloaded chain instead sags as a catenary. Expanding the catenary as a Taylor series shows why the two are confused:

a·cosh(x/a) = a + x²/(2a) + x⁴/(24a³) + …

The first two terms are the parabola y ≈ a + x²/(2a). Only the x⁴ and higher terms make the catenary fatter and steeper at the edges. For a shallow sag the difference is a few percent; for a deep sag it becomes obvious.

The inverted catenary: the perfect arch

A hanging chain can only pull — it carries pure tension. Now imagine freezing it solid and flipping it upside down. Every tension force reverses into a compression directed along the curve. The result is an arch in which the line of thrust runs exactly through the centre of the material at every point, so there is no bending moment anywhere. Stone and masonry are enormously strong in compression but crack easily in tension, so an arch shaped like an inverted catenary stays crack-free and self-supporting.

Robert Hooke captured this in 1675 with a Latin anagram whose solution reads: "Ut pendet continuum flexile, sic stabit contiguum rigidum inversum" — "As hangs the flexible line, so but inverted will stand the rigid arch." The principle guided architects for centuries:

Structure	Use of the catenary
St Paul's Cathedral dome (Wren, ~1710)	Hidden middle dome shaped to a cubic close to an inverted catenary
Gaudí's Sagrada Família & Colònia Güell	Designed with upside-down hanging-chain models to find pure-compression forms
Gateway Arch, St. Louis (1965)	A weighted (flattened) catenary, 192 m tall and 192 m wide
Brick and stone bridges	Arches kept inside the catenary thrust line resist collapse

The Gateway Arch is technically a weighted catenary: because the cross-section is heavier near the base than the top, the matching hanging shape is a chain weighted more at its ends, giving a slightly pointier, flatter-bottomed curve than the pure y = a·cosh(x/a).

A worked example with real numbers

Take a power line of mass 1.0 kg/m strung between two poles 100 m apart, pulled to a horizontal tension H = 5000 N. Then w = ρg = 1.0 × 9.81 ≈ 9.81 N/m, so:

a = H / w = 5000 / 9.81 ≈ 509.7 m
half-span x = 50 m
sag = a·(cosh(50/509.7) − 1) ≈ 509.7 × 0.004818 ≈ 2.46 m
arc length = 2·a·sinh(50/509.7) ≈ 100.16 m  (only 16 cm longer than the span)
max tension at pole = H·cosh(50/509.7) ≈ 5000 × 1.00482 ≈ 5024 N

So a tautly strung 100 m line dips about 2.5 m, and the extra cable needed beyond the straight-line distance is a mere 16 cm. Slacken the tension to H = 1000 N and a drops to about 102 m, the sag balloons to roughly 12 m, and the line swings into a deep, unmistakable catenary. This is exactly the trade-off engineers manage: more tension means less sag but higher loads on the towers.

Beyond the curve: the catenoid

Spin a catenary about a horizontal axis and you sweep out a catenoid, a saddle-shaped surface of revolution. Euler showed in 1744 that the catenoid is the minimal surface spanning two coaxial circular rings — the surface of least area, exactly the shape a soap film snaps into between two wire loops. It was the first non-planar minimal surface ever found, tying the humble hanging chain to the mathematics of soap bubbles, tents, and tensile architecture.

Where the catenary shows up

• Power and telephone lines. Every span between poles is a catenary; sag is engineered against temperature (cables lengthen and droop when hot).
• Suspension bridges. The main cable is a parabola once the deck is hung, but reverts to a catenary during construction before the deck is added.
• Architecture. Inverted catenaries give the strongest masonry arches and domes — Gaudí, Wren, the Gateway Arch.
• Anchor and mooring lines. Ship and oil-rig "catenary moorings" use the chain's sag to absorb shock and keep pull horizontal at the seabed.
• Railway overhead lines. The "catenary system" suspends the contact wire so a pantograph keeps even contact at speed.
• Soap films and minimal surfaces. Catenoids between rings, and tensile-fabric roofs designed by hanging-model methods.

Common mistakes

• Calling it a parabola. A free chain is a catenary; only a horizontally-uniform load (a bridge deck) gives a true parabola. They agree only to second order.
• Forgetting the chain must be uniform and inextensible. A chain that stretches under load, or whose weight per length varies, gives a different curve (e.g. the weighted catenary of the Gateway Arch).
• Thinking tension is constant. Only the horizontal component H is constant; total tension T = w·y is largest at the supports.
• Ignoring bending stiffness for stiff cables. A real wire rope or rod has flexural rigidity, so near the supports it deviates from the ideal catenary (an "elastica" problem).
• Assuming any arch is a catenary. A semicircular Roman arch is not a catenary; its thrust line can escape the masonry and demands thick haunches or buttressing to stay standing.