The Cross-cap is a representation of the projective plane. It is like a shrinked Torus where there's no middle hole, and the side has been clipped so that they cross. The significance of this surface is only its topological property.
This cross-cap image is done with the following parametric formula:
x = (aa * aa) * (sin(u) * sin(2 * v) / 2)
y = (aa * aa) * (sin(2 * u) * cos(v) * cos(v))
z = (aa * aa) * (cos(2 * u) * cos(v) * cos(v))
where aa is a constant.
The cross-cap was the first surface that represented the projective plane in R^3.
Imagine a half-sphere and connect the opposite points on its boundary. the animation
shows one way to do this in R^3.
The cross-cap is made of a 1-parameter family of circles. The strip between
two neighboring circles is a Moebius strip. The animation moves these
Moebius strips over the surface.
The cross-caps occur as a natural family, the ratio between the largest and smallest circle
of the cap parametrizes the family. For a surface of positive curvature one has at points where
the principal curvatures are not the same a natural cross-cap made out of the normal curvature
circles at the chosen point.
Cross-Cap Surface Anaglyph.
Last steps before the boundary is closed along a self-intersection.
cross cap st
cross cap sw
