#### The hyperboloid of one sheet

Equation: $\displaystyle\frac{x^2}{A^2}+\frac{y^2}{B^2} - \frac{z^2}{C^2} = 1$

The hyperboloid of one sheet is possibly the most complicated of all the quadric surfaces. For one thing, its equation is very similar to that of a hyperboloid of two sheets, which is confusing. (See the page on the two-sheeted hyperboloid for some tips on telling them apart.) For another, its cross sections are quite complex.

Having said all that, this is a shape familiar to any fan of the Simpsons, or even anybody who has only seen the beginning of the show. A hyperboloid of one sheet looks an awful lot like a cooling tower at the Springfield Nuclear Power Plant.

Below, you can see the cross sections of a simple one-sheeted hyperboloid with $A=B=C=1$. The horizontal cross sections are ellipses -- circles, even, in this case -- while the vertical cross sections are hyperbolas. The reason I said they are so complex is that these hyperbolas can open up and down or sideways, depending on what values you choose for $x$ and $y$. Check the example and see for yourself. Yikes! If you do these cross sections by hand, you have to check an awful lot of special cases.

*Hyperboloid of one sheet cross sections.* The hyperboloid of one sheet $x^2+y^2-z^2=1$ is plotted along with its cross sections. You can drag the blue points on the sliders to change the location of the different types of cross sections.

The constants $A$, $B$, and $C$ once again affect how much the hyperboloid stretches in the x-, y-, and z-directions. You can see this for yourself in the second applet. Notice how quickly the hyperboloid grows, particularly in the $z$-direction. When $C=2$, a relatively small number, the surface already stretches from -8 to +8 on the $z$-axis.

*Hyperboloid of one sheet coefficients.* The hyperboloid of one sheet $\frac{x^2}{A^2}+\frac{y^2}{B^2} - \frac{z^2}{C^2} = 1$ is plotted. You can drag the blue points on the sliders to change the coefficients $A$, $B$, and $C$.

One caveat: the applet only shows a small portion of the hyperboloid, but it continues on forever. So adjusting the value of $C$ doesn't really make the surface taller -- it's already "infinitely" tall -- but it certainly does affect the shape and slope of the surface. If you know something about partial derivatives, you could investigate how quickly $z$ changes with respect to $x$ and $y$ for different values of $C$. You could also explore why adjusting $C$ seems to have a more dramatic effect than changing $A$ and $B$.

Here are a few more points for you to consider.

- Once again, the sliders don't go all the way to 0. Why not? Make all of them as small as possible and zoom in to see the resulting hyperboloid.
- Look at the equation. What should happen when $x=A$ or $x=-A$? Check this in the first applet; recall that $A=1$ there.
- Does there always have to be a “hole” through the hyperboloid, or could the sides touch at the origin? In other words, could the cross section given by $z=0$ ever be a point instead of an ellipse? Experiment with the second applet; be sure to look directly from the top and zoom in before just assuming that the hole is gone.

#### List of quadric surfaces

- Elliptic paraboloid
- Hyperbolic paraboloid
- Ellipsoid
- Double cone
- Hyperboloid of one sheet
- Hyperboloid of two sheets