Hyperboloids of One Sheet

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

It is fun to visualize the slices of a three-dimensional math object. In this lesson we use the idea of slices to understand hyperboloids of one sheet.

Hyperboloids of One Sheet

The next time you drive past a power plant, look at the towers. These are cooling towers. Power plants generate heat as well as electricity. The heat produces hot water which is allowed to cool before escaping into the atmosphere as steam. The tower's special shape aids the cooling process. A wide bottom gives a large area to hold the hot water. It's like a steaming hot cup of coffee; the wider the cup, the faster the liquid will evaporate as steam. As the steam rises, the narrowing of the tower makes the steam rise and cool even faster. By the time the tower has widened again, the steam is much cooler and ready for the atmosphere.

Cooling Tower at a Power Plant
Cooling tower.

This fascinating math shape has a name: a hyperboloid of one sheet.

Getting the Hyperboloid along an Axis

Imagine your job is to install one of these cooling towers. It arrives at the site lying on its side.

Hyperboloid along the x-axis
Hyperboloid along x-axis

Being an aspiring mathematician, you decide to write an equation. As a good start, we define x, y, and z directions. Then, for the opening along the x-axis, we write:

Hyperboloid along the x-axis
Equation of hyperboloid along x-axis.

At this point, we notice the negative goes with the x term. Okay, what if the tower is pointing along the y-axis?

Hyperboloid along the y-axis
Hyperboloid along y-axis.

You probably answered correctly! The y term has the negative sign. Here's the equation:

Hyperboloid along the y-axis
Equation of hyperboloid along y-axis

Enough playing games with this tower! Let's stand it up with the opening along the z-axis.

Hyperboloid along the z-axis
Hyperboloid along z-axis

The minus sign is now in front of the z term, just like we expected:

Hyperboloid along the z-axis
Equation of hyperboloid along z-axis

Here are some observations: the hyperboloid equation describes a symmetric figure going to infinity for both positive and negative z. We are showing only a portion of this infinite figure in the drawings. Also, an actual cooling tower is not symmetric. The part of the tower where it narrows is sometimes called the throat. More tower is situated below the throat than above it. We will return to these observations later when we design a tower.

For now, let's have some real fun by slicing up this hyperboloid as if it were a cake. A hyperboloid cake is not your basic cake!

Slicing the Hyperboloid

Let's say we have the hyperboloid in its standing-up direction. The origin is at the center of the hyperboloid. The origin is where x, y, and z are all equal to zero. Imagine taking a horizontal slice at the narrowest part. This slice amounts to setting z equal to 0:

Equation of hyperboloid along z-axis
Equation of hyperboloid along z-axis

Setting z equal to 0 gives us:

z equals 0
Equation of hyperboloid along z-axis

This is the equation of an ellipse in the xy-plane. Imagine you are suspended above the hyperboloid, a little ways to the side, and you are looking down. Do you see the ellipse? This type of hyperboloid is called an elliptical hyperboloid.

Hyperboloid ellipse in the xy-plane
Hyperboloid ellipse in the xy plane

Typically, though, the horizontal slices for towers are circular. In our hyperboloid equation, we just set a equal to b. This type of hyperboloid is called a circular hyperboloid (also known as a hyperboloid of revolution).

Hyperboloid circle in the xy-plane
Hyperboloid circle in the xy plane

Now let's get creative with our slices. What about a vertical slice? If we take a vertical slice of the hyperboloid, we get a hyperbola.

Hyperbola in the yz-plane
Hyperboloid cross-sections

What happens in the equation? We set x equal to 0 and to get the hyperbola slice in the yz-plane; now our equation reads:

x equals 0
Equation of hyperboloid along z-axis

By the way, the vertical slice in the xz-plane is also a hyperbola. In the equation, this happens by setting y equal to 0.

Now the names make more sense. Recall, a 'hyperboloid of revolution' is the other name for a 'circular hyperboloid.' If we revolved the hyperbola slice around the z-axis, the figure which appears has circles for its horizontal slices. Aha! It's a circular hyperboloid.

We can do lots of slices! Circles (or ellipses) slice parallel to the ground and hyperbolas slice vertically. A nice way to look at the hyperboloid might include many of these slices:

Hyperboloid showing multiple slices
Hyperboloid three cross-sections

Enough slicing! Let's see what it takes to design a cooling tower.

Putting Numbers into the Equation

What if we take the hyperboloid equation and use it to design a cooling tower?

Let's give the throat a 180-foot diameter and place it 340 feet above the ground. At the base, the diameter is 320 feet. The height of 400 feet is a typical number for these structures. The rest of our numbers come from measuring a cooling tower photo and scaling these numbers to the 400-foot height.

Here's a two-dimensional drawing of our cooling tower:

Cooling tower design drawing
Design for the tower

How does all of this relate to the equation and the figure of the 'Hyperboloid along the z-axis' ? As designers, we need to find a, b, and c.

First, let's play with the equation. The base is a circle, which tells us a = b. Also, x^2 + y^2 = R^2 where R is the radius of the circle. So,

Equation using the radius R
Equation using the radius R for the tower.

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