Scherk's Minimal Surface

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Introduction:

Scherk's minimal surface arises from the solution to a differential equation that describes a minimal monge patch (i.e., a patch that maps [u, v] to [u, v, f(u, v)]). The full surface is obtained by putting a large number of the small units pictured above next to each other in a chessboard pattern, as shown below. The plots below were made by plotting the implicit definition of the surface.

 

Definition:

The definition of Scherk's surface as a monge patch is

or, implicitly,

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.

Mean Curvature:

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