Richmond's Minimal Surface

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Introduction:

Richmond's surface is a minimal surface with one planar end. It can also be expressed as a family of surfaces in polar coordinates (second equations below).

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Definition:

In polar coordinates, the family of minimal surfaces can be parameterized by

,

where n is the degree of the surface, and t is a parameter that creates a family of surfaces.

Properties:

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.