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:

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

To rotate the graph, right click and drag.

Joseph Lo