Richmond's Minimal Surface

Click for VRML

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).

Click for VRML

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.