Catenary Curve Calculator

Created by Davide Borchia
Reviewed by Anna Szczepanek, PhD and Rijk de Wet
Last updated: Apr 13, 2022

Suspended ropes have a simple yet fascinating mathematical definition: discover it with our catenary curve calculator!

The catenary curve is the graph generated by the catenary function. It describes the ideal behavior of a rope hanging in a gravitational field under its own weight. Catenary curves find applications in many fields, so it's worth learning about them. Keep reading to learn:

  • What a catenary curve is;
  • The catenary curve equation;
  • Applications of the catenary curve in architecture;
  • Where to find the catenary curve in nature; and
  • How to use our catenary curve calculator.

What is a catenary curve?

Take a rope and hang it between two supports so that it sags a little. That's the definition of catenary curve — our job is done!

We are joking, but not entirely: that's exactly how you describe a catenary curve. However, there is more to say! 😄

The word catenary comes from the Latin "catēna", chain ⛓. The curve describes not only ropes, but also chains. In the past, many mathematicians tried to describe the behavior of a hanging rope. They first compared the catenary curve to a parabola, but soon they shifted to more complex functions once they realized the parabola fails to capture the catenary curve perfectly.

🔎 Two famous scientists of the past, Galileo Galilei and Robert Hooke (yes, the guy who came up with the spring constant) tried their luck with catenary curves. Galileo noted that the curve designed by a hanging rope was not a parabola; Hooke is considered the first to have found the mathematical expression for building arches following the catenary.

The catenary curve equation

A catenary curve follows a simple mathematical formula:

y=acoshxay=a\cdot \cosh{\frac{x}{a}}

cosh\cosh is the hyperbolic cosine, a function part of the family of hyperbolic functions. It is defined as:

coshx=ex+ex2\cosh{x}=\frac{e^x+e^{-x}}{2}

This makes possible to write the catenary curve formula as:

y=acoshxa=a2(exa+exa)y=a\cdot \cosh{\frac{x}{a}}=\frac{a}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})

If you want to learn more about hyperbolic functions, check our hyperbolic functions calculator!

It is possible to define a generalization of the catenary curve equation, the weighted catenary. It requires two parameters, instead of one:

y=bcoshxa=b2(exa+exa)y=b\cdot{}\cosh{\frac{x}{a}}=\frac{b}{2}(e^{\frac{x}{a}}+e^{-\frac{x}{a}})

🔎 Weighted catenaries are not that common! However, you can find one in St. Louis, Missouri: the Gateway Arch, which is often mistaken for a parabola!

Applications of the catenary curve in architecture — and not only

The catenary curve is so simple that it has conquered the heart of humanity from the beginning of civilization. Its interesting properties made it a perfect choice for a particular field: here, we introduce the love story between the catenary curve and architecture.

The shape of a catenary curve facilitates unloading the weight of the suspended structure on the lateral supports. Catenary curves appear in suspended bridges, either by coincidence (lay down a rope bridge and it will assume that shape) or by design.

🔎 The ropes of bridges where they act as a support for the road below don't follow a catenary shape, but rather a parabolic one. But don't worry, they are almost identical: for centuries catenaries hid behind parables, their more famous "cousins"!

Since antiquity, architects applied the catenary curve when designing arches and domes. Many examples can be found — from the Tāq Kasrām (the millenary imperial complex still standing in Iraq) to the Clocháin (the tiny stone houses in Skelling Michael) to the Brunelleschi Dome to the mud huts of the Musgun people in Cameroon.

Taq Kasram in Iraq
The Tāq Kasrām in Iraq is a long standing example of the strength of catenary arches.

But obviously, it doesn't stop here: catenary curves find many other applications. If you take a train (an electric one is the best way to commute), the electric lines hanging overhead are catenary curves. And speaking of electric lines, almost every landscape is populated by power lines that are — as you now know — catenary curves.

A string of helium-filled balloons (a balloon arch) hangs as an upside-down catenary curve too!

Nature is the world's best engineer, and the catenary curve appears in many more or less unexpected places. Take a closer look at a spider web: the ropes anchored to other strings are catenaries. Many arches found in nature assume catenary shape due to erosion, too.

But opening the fridge would give you the easiest example of catenary curves in nature: eggs! An egg's particular strength is associated to their ability to dissipate forces applied on them, thanks to their double catenary shape.

A mathematical curiosity!

Catenary curves are the only shapes on which regular polygons can roll smoothly. Choose a point on the polygon and trace its movement: you will obtain a curve called a roulette: we talked about it in our involute function calculator.

In the particular case of polygons rolled on catenaries, we call those curves unduloids.

Square unduloid
The generation of the unduloid of the square (in red). The repeated blue curve at the bottom is the catenary.

How to use our catenary curve calculator?

Our catenary curve calculator is easy to use and still offers you many functionalities.

First, you have to choose the type of catenary you want to use in the tool: weighted or not We show the respective catenary curve formula there, don't worry!

We offer you four modes of operation:

  • Value;
  • Graph;
  • Table; and
  • Graph and table.

Choose the one you need. In the last three, you can change the function domain and the sampling frequency by changing the parameters hidden in the advanced mode.

🙋 If you are using the calculator in value mode, remember that you can input the value of the catenary function to find the associated value of xx!

FAQ

What is a catenary curve?

The definition of catenary curve is: "a mathematical curve representing the shape of a rope hanging between two supports under the sole influence of its own weight".

Find more with the catenary curve calculator on omnicalculator.com

How do I calculate the catenary curve?

The equation of the catenary curve makes use of hyperbolic functions. Given the parameter a, the sag of the rope, we find the catenary with the formula y = a * cosh(x/a), where cosh is the hyperbolic cosine.

Which are the applications of the catenary curve?

Architecture and civilian engineering are the most common applications of catenary curves. The fact that a catenary unloads the weight it carries very efficiently on its supports makes these curves a perfect choice to suspend bridges or create the most resistant shape of arches!

What is the difference between catenary curve and parabola?

In a catenary curve, the force is uniform on the length of the rope, while in a parabola, the force is uniform when you consider the horizontal length. This is why the catenary is more common in self-suspending structures, while you can find the parabolic shape when the cable supports something else (think of the Golden Gate Bridge).

Davide Borchia
Type of curve
Catenary
catenary formula
Sag parameter (a)
Type of output
Graph
Graph
Change the domain/step by clicking on advanced mode!
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