The Generalized Monkey Saddle

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Introduction:

The hyperbolic paraboloid and monkey saddle are really special cases of this general class of surfaces. Because the standard definition of the function involves taking the real part of a complex number (see first equations below), it is difficult to differentiate the function, and thus to compute partials, Gaussian curvature, etc. The Gaussian and mean curvature have been specially evaluated for the particular case of n = 5, plotted above, as they were for the hyperbolic paraboloid and the monkey saddle (n = 2 and n = 3, respectively).

It is somewhat easier to compute a general formula for Gaussian and mean curvature from the polar form of the generalized monkey saddle (see second equations below.) Because there are no complex numbers involved, differentiation is straightforward.

Definition:

,

where Re[z] denotes the real portion of the complex number z, I denotes , and n is some integer power. Geometrically, n represents the number of dips in the saddle (2 for legs, and n - 2 for tails).

An alternate representation of the generalized monkey saddle that does not involve complex numbers is:

Note that the equations are a polar definition of the monkey saddle. The polar version of the surface looks like this when plotted with n = 5:

Properties:

Gaussian Curvature:

With the polar representation of the monkey saddle, it is possible to compute a general formula for Gaussian curvature:

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

Mean Curvature:

Mean curvature can also be computed from the polar representation:

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