The Pseudo Cross Cap

Click for VRML

Introduction:

Although it looks quite similar to the cross cap, the pseudo cross cap (shown in a cut view throughout this page) is actually significantly different. Although it is not regular, it is still considered to be orientable. Note that if you take a normal and rotate it around on the surface, it is impossible to back to where you started, but have the normal pointing in the opposite direction, without passing through the region where the surface reduces to a point.

Definition:

Geometrically, the surface is formed by taking a figure eight and shrinking it down to two points on the top and bottom.

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

dA = [click for formula]

Gaussian Curvature:

K[u, v] = [click for formula]

Gaussian curvature of the surface.
Surface colored by Gaussian curvature.

Mean Curvature:

H[u, v] = [click for formula]

Mean curvature of the surface.