The Catenary Curve - Who Knew?

In the continuing spirit of finding real life examples of curves to talk about in class, today I am highlighting the catenary curve.

What is a catenary curve?

A catenary curve looks like a parabola, but it isn't!  An example of a catenary curve is a hanging chain that is supported at both ends and is hanging under its own weight.

A great example of an approximate inverted catenary curve is the St. Louis Arch.

Both the chain and the St. Louis arch actually look they could be parabolas, but let's examine the equations of these curves a little closer.

Mathematically, the catenary curve is the hyperbolic cosine function. y = a cosh(x/a)

As you can see from this picture of the graph of a parabola and y = cosh(x), when you have a relatively small window, the parabola and hyperbolic cosine function look very similar.

But, when you enlarge the window, or zoom out, you can see that the curves look very different.

According to Wolfram MathWorld, Galileo originally believed that these types of curves were parabolas.  However, in 1691, Leibniz, Huygens, and Bernoulli showed that these are not parabolas.  They needed calculus to show it though, so who can blame Galileo :)

A great lesson for students that things are not always what they seem!