The Hyperbolic Paraboloid

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Introduction:

A contour plot of the hyperbolic paraboloid is shown at right. Clearly, the level curves of the surface are hyperbolas. Because of this fact, in addition to the fact that the surface has parabolic cross-sections when sliced along the plane y = x, the surface is called a hyperbolic paraboloid. It is sometimes also called a saddle surface, because a person could sit comfortably at the origin. Like the monkey saddle, it is a special case of the generalized monkey saddle.


Definition:

The surface plotted above is parameterized by:

A similar surface, also referred to as a hyperbolic paraboloid or saddle surface, is parameterized by:

The properties listed below are for the first surface.

Properties:

Partial Derivatives:

Because z is a function of x and y, we can take partial derivatives:

     

     

Tangent Planes:

At u = u0, v = v0, the tangent plane to the surface is parameterized by:

Infinitesimal Area:

The infinitesimal area of a patch on the surface is given by

Gaussian Curvature:

The Gaussian curvature of the hyperbolic paraboloid is always less than zero. In fact, except for a very small region around the origin, it is practically zero.

Gaussian curvature of the surface.
Surface colored by Gaussian curvature.

Mean Curvature:

Mean curvature of the surface.
Surface colored by Mean curvature.