## 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:

1. Make sure you have the latest version of Java 7 installed in your computer. Tablets and smartphones are not supported.