The Monkey Saddle

Click for VRML

Introduction:

The hyperbolic paraboloid is often called a saddle surface, because a person could sit on it comfortably. A monkey, however, would run into trouble because he would have nowhere to put his tail! The above surface is called a monkey saddle, because it has a convenient dip in the back to accommodate the monkey's tail. The monkey saddle is in fact a special case of the generalized monkey saddle.

Definition:

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:

Gaussian curvature of the surface.
Surface colored by Gaussian curvature.
Note: The jagged edge near the minimum in the graph of Gaussian curvature is not a feature of the graph; rather, it is an artifact of the plotting process.

Mean Curvature:

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