The Eight Surface

Click for VRML

Introduction:

The surface pictured above is called an eight surface because it is a surface of revolution of a figure eight. The surface comes to a point at its very center, which causes problems in the VRML version of the surface (you may receive a series of warnings when you view it). Note also that the mean curvature becomes infinitely large at the center of the surface, and the surface can thus not be colored by mean curvature.

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:

Note that, although the numerator of the mean curvature function contains the imaginary number i, the function is always real or infinite because cos(4v)-1 is less than or equal to zero, so its square root is always imaginary or zero.

Mean curvature of the surface.