## A hyperbolic paraboloid

 This figure shows a finite portion of a hyperbolic paraboloid. Its equation is fairly simple, namely  z = xy.   Thus, it is a smooth quadric surface. In multivariable calculus, it appears as graph of the function   f(x,y) = xy.   This is the most basic example of a function which has a critical point where the second derivative test shows that the function has neither a local maximum nor a local minimum. Of course, this is directly related to the "saddle shaped" appearance of the surface.     Another interesting property of this surface is that it is a ruled surface. By definition, a ruled surface contains an infinite family of straight lines. In fact, a smooth quadric surface contains two families of straight lines. In our figure, this property is reflected in the checkerboard pattern used to color the surface.    Click here to see: a drawing of some of these lines a sketch of the surface that shows some of the lines a drawing that shows a twisted cubic curve on the hyperbolic paraboloid      A few buildings have been constructed with a roof in the shape of a hyperbolic paraboloid. Click here to see pictures of one of these.    The other smooth quadric ruled surface, the hyperboloid of one sheet, also contains two 1-parameter families of lines.

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Prof. Joel Roberts
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University of Minnesota
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