Catenary Tales

Alan Cai

January 20, 2023

As a child holds a jump rope hanging freely from limb to limb, a seemingly normal curve is produced. The same curve is seen for loose caution tape dangling loosely from post to post and in telephone wires stretching from pole to pole. This curve is not a parabola nor a hyperbola, but a special shape called a catenary.

From a physics standpoint, a catenary is a rope or chain supported on both sides and freely dangling across the midspan. When observing catenaries, keep in mind that suspension bridges actually do not follow the trajectory of a catenary because the weight of the bridge which they support exceeds that of the cable itself causing the total contraption to not be freely hanging across the middle. Although rather similar in shape, suspension bridges such as the Golden Gate Bridge in San Francisco are not actually catenaries but rather ordinary parabolas due to the added weight underneath that they must bear.

Conic sections(including non-horizontal/vertical hyperbolas) follow the general equation:

With some or no constants(h, a, b, c, d, or e) being equal to zero. For example, the unit circle is

And the unit hyperbola is

With the unit circle, trigonometric functions like cosine and sine are possible in which cos(j) is the x value of the resulting point of intersection when a radius is rotated j with respect to the x-axis and sin(j) is the y value of the previously mentioned point of intersection. From the Pythagorean theorem, key trigonometric identities such as

can be derived.

With a unit hyperbola, however, an interesting new set of functions can be similarly applied, namely the hyperbolic cosine and hyperbolic sine, from which the rest of the basic functions can be developed. Similar to the regular sine and cosine, cosh(j) is the x value of the resulting point of intersection when a line is rotated j with respect to the x-axis, and sin(j) is the y value of the said point. From the equation of the unit hyperbola, the following base identity may be achieved:

The Catenary curve is precisely a transformation of the graph of cosh(x), which can be neatly rewritten as

These unique identities are what make hyperbolic functions so powerful. In elementary calculus, they can be used as powerful trigonometric substitution tools while in more abstract mathematics, hyperbolic functions can be used in Euler’s formula and beyond.

Any modern child would definitely have seen or played with the catenary curve sometime or another. Yet, the seemingly simple shape has so many complex elements and origins. The next time you see a barrier rope at the airport, perhaps you would benefit from taking note of the intriguing shape.

For your enjoyment, feel free to take a look at a graph of all four shapes previously mentioned(green catenary curve, purple upside down parabola, red unit circle, and blue unit hyperbola) courtesy to Desmos: