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 = u_{0}, v = v_{0},
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.


