The Whitney Umbrella

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Introduction:

The Whitney Umbrella is a well-known example of a non-regular surface, often referred to as a self-intersecting rectangle in three dimensions. The end of the line of self-intersection (the "plus" shape) is a pinch point (also called a Whitney singularity or branch point). At a pinch point, every neighborhood of the point is self-intersecting.

Definition:

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.