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 contains an
infinite family of straight lines. In fact, a smooth quadric surface
contains ruled surface families of straight lines. In our figure,
this property is reflected in the checkerboard pattern used to color
the surface. Click here to see:
two- 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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*I made the figure on this page by substituting my own data in a
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Prof. Joel Roberts

School of Mathematics

University of Minnesota

Minneapolis, MN 55455

USA

Office: 351 Vincent Hall

Phone: (612) 625-1076

Dept. FAX: (612) 626-2017

e-mail: `roberts@math.umn.edu
http://www.math.umn.edu/~roberts
`