Cross-Cap Surface

cross cap close disk 010

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.
cross cap close disk 001
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.
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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.
cross cap natural famil 001
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.
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Cross-Cap Surface Anaglyph. Last steps before the boundary is closed along a self-intersection.
cross cap st
cross cap st
cross cap sw
cross cap sw

Projective Plane Surfaces

  1. Cross-Cap Surface
  2. Boy's Surface
  3. Boy's Surface (Bryant-Kusner)
  4. Steiner Surface
  5. Inverted Boy Surface

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