The Monkey Saddle

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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.