The Helicoid

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Introduction:

The helicoid is the only known ruled minimal surface, other than a plane. It also has the interesting property that, while its topology is finite, its Gaussian curvature is infinite (when v = 0).

Definition:

Geometrically, the helicoid is defined by simultaneously rotating and translating a line at constant speed about an axis to which it is perpendicular.

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.