Catenary Curves
The shape a hanging chain makes under its own weight. Galileo thought it was a parabola. It took the Bernoullis, Leibniz, and Huygens to prove him wrong. Drag the anchors and see the difference.
About this lab
The catenary equation
A catenary is the curve formed by a uniform chain or cable hanging freely under its own weight, supported only at its endpoints. The equation is:
y = a · cosh(x / a) = a · (e^(x/a) + e^(-x/a)) / 2
The parameter a is the ratio of horizontal tension to the weight per
unit length of the chain. A heavier chain (or less horizontal tension) produces a
smaller a and a deeper sag. The catenary is the unique curve that
minimizes gravitational potential energy for a given length of chain.
History: Galileo’s error
In 1638, Galileo claimed that a hanging chain formed a parabola. The conjecture was intuitive — the catenary and parabola look extremely similar for shallow sags. But in 1669, Joachim Jungius showed by careful measurement that Galileo was wrong: the hanging chain is not quite parabolic.
In 1691, Johann Bernoulli posed the catenary as a challenge problem. Within a year, three independent solutions arrived: from Leibniz, Huygens, and Johann Bernoulli himself. The solution required the newly invented calculus, and the catenary became one of the first real triumphs of the new mathematics.
The word “catenary” comes from the Latin catena, meaning chain. Huygens coined the term in a 1690 letter to Leibniz.
Catenary vs. parabola
Toggle the parabola overlay to see the difference directly. For small sags (chain barely longer than the span), the catenary and parabola are nearly identical. As the chain gets longer and the sag increases, the catenary falls more sharply near the middle and flattens more gradually at the sides.
Mathematically, the distinction is in the Taylor series:
cosh(x) = 1 + x²/2! + x⁴/4! + x⁶/6! + ... parabola = 1 + x²/2
The parabola captures the first two terms exactly. The catenary diverges in the fourth-order term and beyond. A cable carrying a uniform horizontal load (like a suspension bridge deck) really does form a parabola; the catenary describes a cable carrying only its own weight.
Architectural applications
The Gateway Arch in St. Louis is a weighted catenary — an inverted catenary that is slightly wider at the base to account for its varying cross-section. The arch is in pure compression everywhere, with no bending moments, making it structurally optimal.
Antoni Gaudí used hanging chain models extensively. He suspended weighted strings from the ceiling, let them form natural catenaries, then turned the model upside down to get arch shapes in pure compression. The result was the Sagrada Família and Colonia Güell, with their organic, structurally optimized forms.
Power lines hang in catenaries between transmission towers. Engineers must account for the catenary sag when designing clearances, especially since the cable’s length changes with temperature (longer when hot, more sag in summer).
Suspension bridges present an interesting contrast. The main cables of a bridge like the Golden Gate hang under the combined weight of the cable and the deck. If the deck is much heavier than the cable (as it usually is), the shape approaches a parabola. If the cable alone were hanging, it would be a catenary.