## Hyperboloids

*Scroll down to the bottom to view the interactive graph.*

This graph illustrates the transition from a hyperboloid of one sheet to a hyperboloid of two sheets. Consider the equation \[x^2 + y^2 - z^2 = C\] In case if \(C > 0\), the level curves \[x^2 + y^2 = C + k^2\] are circles at any level \(z = k\) Therefore, the surface continues from negative \(z\) to positive \(z\).

On the other hand, if \(C = -|C| < 0\), then the level curves \[x^2 + y^2 = -|C| + k^2\] exist only when \(z = k \ge \sqrt{|C|}\) or \(z = k \le -\sqrt{|C|}\). Therefore, the surface consists of an upper and a lower piece.

The transition between a hyperboloid of one sheet and a hyperboloid of two sheets can be illustrated by varying \(C\).

**To view the interactive graph:**

- Make sure you have the latest version of Java 7 installed in your computer. Tablets and smartphones are not supported.
- Download this file, hyperboloid.ggb.
- Click here to open GeoGebra.
- After you open GeoGebra, click "File" in the toolbar, then click "Open".
- Choose the .ggb file you just downloaded and click the "Open" button.
- Now you should be able to view the graph inside GeoGebra.

To rotate the graph, **right click and drag**.

Joseph Lo