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Section 6.2 Hyperbolic Paraboloid

Now change a single sign: keep \(z\) to the first power, but let the squared terms in \(x\) and \(y\) carry opposite signs.
 2 
In equation (6.1), take \(A = 1/4\text{,}\) \(B = -1/9\text{,}\) \(C = 0\text{,}\) \(D = -1\text{,}\) and \(E = 0\text{.}\)
Equation (6.1) becomes
\begin{equation*} z = \frac{x^2}{4} - \frac{y^2}{9}, \end{equation*}
a hyperbolic paraboloid. That one sign is enough to change the surface completely.
As in the previous example (Section 6.1), fixing the value of \(x\text{,}\) \(y\text{,}\) or \(z\) produces traces that are conic sections. Again setting \(x = 2\text{,}\) \(y = -3\text{,}\) and \(z = 1\) in turn gives the following traces, as shown in Figure 6.5:
\begin{align*} z \amp= 1 - \frac{y^2}{9} \amp\amp (parabola)\\ z \amp= \frac{x^2}{4} + 1 \amp\amp (parabola)\\ 1 \amp= \frac{x^2}{4} - \frac{y^2}{9} \amp\amp (hyperbola) \end{align*}
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Figure 6.4. The traces of the hyperbolic paraboloid \(z = \frac{x^2}{4} - \frac{y^2}{9}\) in the planes \(x = 2\text{,}\) \(y = -3\text{,}\) and \(z = 1\text{.}\)

Instructions.
Use the buttons to slice the surface \(z = \frac{x^2}{4} - \frac{y^2}{9}\) with planes \(x=c\text{,}\) \(y=c\text{,}\) or \(z=c\text{,}\) and drag the slider to vary \(c\text{.}\) The equation of each cross-section is displayed above the figure. Drag the figure to view it from a different angle, or press the “Rotate” button to spin it automatically.
Figure 6.5. Slicing the hyperbolic paraboloid \(z = \frac{x^2}{4} - \frac{y^2}{9}\) with planes \(x = c\text{,}\) \(y = c\text{,}\) and \(z = c\text{.}\)
The animation in Figure 6.6 shows these slices being taken one at a time.
Figure 6.6. Slicing \(z = \frac{x^2}{4} - \frac{y^2}{9}\) with planes \(z = c\text{,}\) \(x = c\text{,}\) and \(y = c\text{.}\)
Similar to the elliptical paraboloid, with traces of two parabolas, but with one hyperbola, it is natural to call this quadric surface a hyperbolic paraboloid.
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