## Hyperbolic paraboloid

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

Consider the function \[f(x,y) = x^2 - y^2.\] To illustrate the surface \(z = f(x,y)\), we can try to imagine how the traces look like when fixing \(x\) or \(y\). For example, if we fix \(y\), i.e. set \[y = k = \text{constant,}\] then the trace along the surface is defined by \(z = f(x,k) = x^2 -k^2\), which is a parabola opening upward and shifted down by \(k^2\).

The shape of the surface will be apparent if we put all the traces together.

You may see this in 3D if you have a pair of red-cyan 3D glasses.

This surface is called a **hyperbolic paraboloid** because the traces parallel to the \(xz\)- and \(yz\)-planes are parabolas and the level curves (traces parallel to the \(xy\)-plane) are hyperbolas. The following figure shows the hyperbolic shape of a level curve.

**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, hyperbolic_paraboloid.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