# Hyperbolic Paraboloid

**Equation:**

(where A and B have

**DIFFERENT**signs)

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With just the flip of a sign, say x to ^{2} + y^{2}x,^{2} - y^{2}we can change from an elliptic paraboloid to a much more complex surface. Because it's such a neat surface, with a fairly simple equation, we use it over and over in examples. Hyperbolic paraboloids are often referred to as "saddles,"
for fairly obvious reasons. Their official name stems from the fact
that their vertical cross sections are parabolas, while the
horizontal cross sections are hyperbolas. But even the vertical cross
sections are more complicated than with an elliptic paraboloid. Look
at the picture on the left, which shows the surface
Notice that the parabolas open in different directions; the
green parabolas open The second picture lets you explore what happens when you adjust the coefficients of the equation z = Ax^{2} + By^{2}(Here we're assuming - What does the horizontal cross section given by
*z=0*look like? Check on the first picture, and also look at the equation when*z=0*. Is this still a hyperbola? - How would
*z = y*look different than^{2}- x^{2}*z = x*?^{2}- y^{2}
Be very careful; if you hear somebody refer simply to a
"paraboloid," they generally mean an |