The Funnel Surface

Click for VRML

Introduction:

The funnel surface is the surface of revolution of the curve log x. It can also be plotted with asymptotic curves:

Each curve on the surface of this parameterization of the funnel either has zero curvature or has unit normals perpendicular to the surface's normals.

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.