The Helicoid

Click for VRML

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.