The Cross Cap

Click for VRML

Introduction:

Like Steiner's Roman surface, the cross cap is a realization of the real projective plane. Instead of the roman map, though, the cross cap is constructed from the antipodal map crosscapmap(x, y, z) = (yz, 2xy, x2 + y2). The surface plotted above is the image of a sphere under crosscapmap, and is one of several non-orientable surfaces like the Klein bottle. In fact, the image of a torus centered at the origin is topologically a Klein bottle:

Definition:

Properties:

Tangent Planes:

At u = u0, v = v0, the tangent plane to the surface is parameterized by:

[click for formula]

Infinitesimal Area:

The infinitesimal area of a patch on the surface is given by

dA = [click for formula]

Gaussian Curvature:

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

Mean Curvature:

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

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