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.
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.
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:
At this point, we notice the negative goes with the x term. Okay, what if the tower is pointing along the y-axis?
You probably answered correctly! The y term has the negative sign. Here’s the equation:
Enough playing games with this tower! Let’s stand it up with the opening along the z-axis.
The minus sign is now in front of the z term, just like we expected:
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:
Setting z equal to 0 gives us:
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.
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).
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.
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:
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:
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:
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,
At the throat, z = 0. From our equation using the radius, we get R^2/a^2 – 0 = 1, which means a = R. The design diameter is 180 feet, so the radius R is 180/2 = 90. Great! We have a = 90. What happened to b? Recall, b is equal to a.
This leaves us with finding c. To find this, we need to ask: what are z and R at the base? z is -340 feet and R = 320/2 = 160. Remember, our hyperboloid equation has z = 0 at the throat, so z is -340 at the ground. From R^2/a^2 – z^2/c^2 = 1, we can plug in our numbers and get 160^2/90^2 – (-340)^2/c^2 = 1. Of course, (-340)^2 is the same as (340)^2. To isolate c, we find: (160/90)^2 – 1 = 340^2/c^2 which simplifies to c=340/((160/90)^2-1)^.5, which rounds to 231.
Here’s another thing we need to keep in mind: the hyperboloid equation is laid out to have z = 0 at the throat. We usually think of z = 0 as the ground level. It would be nice to shift the z values by 340 feet so the tower rests on the ground (and not under it!). We can do this by subtracting 340 from z in the equation. If you are uncomfortable with subtracting 340, we can choose from two ways to reason this out. If we solved the hyperboloid equation for z, then we would add 340 to the right-hand side to bring the values up. Transferring the 340 to the left-hand side means z has become z – 340. Another way to think about this is we have 0 at the throat. If the z term is now the term z – 340, then this new term is zero for z equal to 340. Hope that helps.
So, our design equation is:
What about the ‘for’ statement? The hyperboloid equation implies symmetry, and our design equation is valid for all values of z. Once we have a, b, and c, we can truncate the tower at any height we like. In this design, z starts at z = 0 (the ground) and ends at z = 400.
We’ve played with this figure enough – it’s time to cool some steam!
A hyperboloid of one sheet is the typical shape for a cooling tower. A vertical and a horizontal slice through the hyperboloid produce two different but recognizable figures. One of the two slices is always a hyperbola. The other slice is either an ellipse or a circle. If this other slice is an ellipse, we have an elliptical hyperboloid. If the other slice is a circle, we have a circular hyperboloid. Another name for a circular hyperboloid is a hyperboloid of revolution.