Hyperbolic paraboloid

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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:

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