Catenary and Parabola Comparison

Date: 04/06/2004 at 13:48:57
From: Curious Student
Subject: Catenary and Parabola

What is the difference between a catenary and a parabola?  I don't 
know the difference in shape.  Why is the St. Louis arch a catenary 
and not a parabola?

Date: 04/06/2004 at 16:48:40
From: Doctor Peterson
Subject: Re: Catenary and Parabola

Hi, Student.

The catenary and the parabola are two different curves that look 
generally similar, in that they are symmetrical and have a cup shape, 
going up infinitely on either side of a minimum.  The parabola, in its 
simplest form, is

  y = x^2

while the catenary is defined by the hyperbolic cosine:

  y = cosh(x) = (e^x + e^-x)/2

Here is a graph of the parabola (blue) and catenary (red) together, 
so you can see the difference:

     

I used y = (cosh(x) - 1)/(cosh(1) - 1) in order to move the vertex 
from (0,1) down to the origin, and to make it agree with the parabola 
when x = 1.  This makes it clear that the catenary is slightly more 
"flat" at the bottom, and that it rises faster than the parabola for 
large values of x.  In fact, if you graphed the two for a larger 
domain, the catenary would be far higher than the parabola.

As for why the St. Louis arch, or a hanging cable, takes the shape of 
a catenary, while the cables on a suspension bridge form a parabola, 
that is just a result of the physics of each situation.  Once you have 
learned calculus, you will be able to see that the catenary is the 
solution to a differential equation that describes a shape that 
directs the force of its own weight along its own curve, so that, if 
hanging, it is pulled into that shape, and, if standing upright, it 
can support itself.  The parabola does not have the same property, but 
is the solution of other important equations that describe other 
situations.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 04/06/2004 at 22:08:34
From: Curious Student
Subject: Thank you (Catenary and Parabola)

Thank you, I now understand and will share this information with my
math class.