Catenary Curve Calculator

Hang a chain from two nails and it settles into a curve so familiar that most people never ask what it is. It looks like a parabola. It is not one. The gap between the two is small enough to fool the eye and large enough to matter, and the story of how that gap was found is also the story of why one number is enough to describe every hanging chain there has ever been.

What Galileo actually got wrong

The usual telling says Galileo believed a hanging chain was a parabola and was simply mistaken. The sources do not quite agree with each other on that. The MacTutor archive at St Andrews says plainly that Joachim Jungius disproved Galileo’s claim that a chain hanging under gravity would be a parabola. The encyclopedia account is gentler: in the Two New Sciences of 1638 Galileo wrote that a hanging cord is only approximately a parabola, and he noticed that the approximation improves as the curve gets flatter.

Both accounts agree on what came next. Jungius showed that the curve is not a parabola, and the result appeared in 1669, after his death. Knowing what a curve is not is a long way from knowing what it is.

The answer came a generation later. Jacob Bernoulli set the problem as a challenge, and in 1691 three people answered it: Gottfried Leibniz, Christiaan Huygens and Johann Bernoulli. Their solutions were printed in the journal Acta Eruditorum in June of that year. MacTutor adds that Huygens was the first to use the word catenary, in a letter to Leibniz in 1690. The name comes from the Latin word for chain.

So the honest version of Galileo’s mistake is this. He had the right instinct, that the parabola is a good stand-in when the chain hangs shallow, and he could not say what the true curve was. Nobody could, for another fifty-three years.

Why the weight follows the curve

The reason a chain refuses to be a parabola is easy to state. Each link carries the weight of the chain below it, and that weight is spread evenly along the chain itself, not along the ground beneath it. Where the chain hangs steeply, more chain is packed above each metre of floor, so the steep parts are loaded more heavily than the flat part in the middle.

Cut the chain in your mind at its lowest point and at any other point. The piece between is held by three forces: a horizontal pull at the bottom, a pull along the chain at the top, and its own weight. Balance them and one fact drops out. The slope of the chain at any point equals the length of chain back to the lowest point, divided by a fixed length. That fixed length is written a, and a = H / w, the horizontal pull divided by the weight of one unit of chain.

That statement is a differential equation, because the length along a curve depends on its slope through the arc length formula. Solving it gives a curve built from the hyperbolic cosine, which is half the sum of a growing and a shrinking exponential. The height of the chain is a times the hyperbolic cosine of the distance from the lowest point divided by a. The length of chain out to that distance is a times the hyperbolic sine.

Now change the loading. Hang a heavy, level deck from a light cable, as a suspension bridge does, and the load is even along the ground instead of along the cable. The slope then grows in step with horizontal distance, and the curve that results is a parabola. A bare cable hangs as a catenary; the same cable carrying a much heavier deck pulls into a parabola. A cable whose own weight is not negligible sits somewhere between the two.

One length decides everything

Every catenary is the same shape as every other, in the way that every circle is. Changing a only enlarges or shrinks the curve. A long a means a tight, shallow line and a large horizontal pull for the weight; a short a means a deep, slack one.

Tension has an unexpectedly simple form. At any point it equals the weight per unit length times the height of that point above a horizontal line lying a below the lowest point. The pull is therefore smallest at the bottom, where it is purely horizontal, and largest at the higher support. The horizontal part of the pull is the same everywhere along the line.

Here is a worked case that can be reproduced in the calculator above. Take supports 6 m apart at the same height, and let the line sag 1.5 m. Solving gives a of 3.223 m and a line length of 6.905 m. The line leaves each support at about 47 degrees below the horizontal.

Give the line a mass of 0.35 kg for each metre. Mass is not weight, so it has to be multiplied by gravity, and the calculator uses the standard value of 9.80665 m/s^2, which is exact by definition. That makes 3.432 newtons per metre. The horizontal pull is then 11.06 N, and the tension at each support is 16.21 N, which is just 3.432 times the 4.723 m that separates the support from that reference line.

There is a catch in the arithmetic. The equation that links span, length and a cannot be rearranged to give a directly; it has to be solved by trial, in practice by a routine such as Newton’s method. Newton’s method is fast near the answer and can misbehave far from it, so the calculator keeps the answer trapped between two bounds and halves the interval whenever a step looks unsafe. Only one case needs no such search: when the sag and the length are known for level supports, a simple formula gives a at once.

Supports at different heights add a surprise. The lowest point of the curve slides toward the lower support, and if the line is tight enough it slides past it. The line then climbs all the way from one support to the other, and the mathematical low point sits outside the span, on a part of the curve that does not exist. The lower support is then being pulled upward by the line.

How close is the parabola

For the worked case, draw the parabola through the same two supports and the same low point. The two curves never part by more than 26 mm, on a sag of 1500 mm. That is under two percent of the sag, and it happens out toward the supports, not in the middle.

The familiar shortcuts come from that parabola. One says the length is the span times one plus eight thirds of the square of the sag ratio. Here it gives 7.000 m, about 95 mm too long, an error of 1.4 percent. The other says the horizontal pull is the weight per length times the span squared, divided by eight times the sag. Here it comes out about 7 percent low.

Flatten the line and both shortcuts improve quickly. At a sag of one twentieth of the span, the length rule is off by about three thousandths of one percent and the tension rule by a third of one percent. This is what Galileo noticed without being able to measure it. For a taut line the parabola is good enough for almost any purpose; for a deep swag it is not, and the calculator shows the error at whatever ratio you enter.

The chain turned upside down

Robert Hooke saw a second use for the curve. In 1675 he published a Latin anagram hiding what he called the true form of all manner of arches for building. It was deciphered only after his death, and it reads, in translation: as hangs a flexible cable so, inverted, stand the touching pieces of an arch.

The reasoning is the same balance run backward. A hanging chain is in pure tension; flip it over and the same line carries pure compression, with no tendency to bend. For an arch of even thickness that carries only its own weight, the catenary is the ideal curve. Jacques Heyman, writing in the Royal Society’s history journal, notes that Hooke knew the curve was not a simple parabola but was unable to solve the problem mathematically, and that Christopher Wren’s design for the dome of St Paul’s made use of the hanging chain idea.

Real arches rarely meet that condition exactly. The Gateway Arch in St. Louis is sometimes said to be a catenary, and the encyclopedia entry is firm that this is incorrect: it is narrower near the top, so its weight is not even along its length, and its curve is a weighted catenary instead.

That example is a fair warning, because the ideal chain is ideal in several ways at once. It is perfectly flexible, it cannot stretch, and it carries nothing but its own even weight in still air. A modern study of arch shapes puts the limits bluntly: the parabola assumes the structure weighs nothing, the catenary assumes it carries nothing but itself, and both assumptions are unrealistic.

A real cable stretches under load and with heat. Wind pushes it sideways, ice thickens it, and anything hung from it bends the curve at that point. None of that is in the equation. The numbers from an ideal catenary describe a shape and a balance of forces, and they are a fine way to plan a string of lights, sketch the outline of a model arch, or check a homework problem. They are not a design for a power line, a zip line, rigging or anything that lifts or holds a load. That work belongs to an engineer and the code that governs it.

What survives all those cautions is the elegance of the result. A chain has no plan and no knowledge of mathematics, yet it finds, every time, the one curve in which every link is at rest. It took people until 1691 to write that curve down. The chain only needed to be let go.