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,
x^{2} + y^{2}). The surface plotted above
is the image of a sphere under crosscapmap, and is one of several
nonorientable 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 = u_{0}, v = v_{0},
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.


