Plücker's Conoid

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Introduction:

Plücker's conoid is a well-known ruled surface. As with the catenoid, another ruled surface, Plücker's conoid must be reparameterized to see the rulings:

Note that the apparent gap in the center of the surface is an artifact of the plotting, and not a feature of the surface. The rulings are the horizontal lines that pass though the z axis. A more general form of Plücker's conoid is parameterized below, with n folds instead of just two (second equations).

Definition:

The non-polar parameterization of Plücker's conoid is:

Or, in polar coordinates,

,

where n is the number of folds in the surface. The second surface plotted above is the case n = 2.

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.