The self-intersection of a one-sided Surface. It can be described as a circular Hole which, when entered, exits from its opposite point (from a topological viewpoint, both singular points on the cross-cap are equivalent). The cross-cap has a segment of double points which terminates at two ``Pinch Points'' known as Whitney Singularities.

The cross-cap can be generated using the general method for Nonorientable Surfaces using the polynomial function

(1) |

(2) | |||

(3) | |||

(4) |

for and . To make the equations slightly simpler, all three equations are normally multiplied by a factor of 2 to clear the arbitrary scaling constant. Three views of the cross-cap generated using this equation are shown above. Note that the middle one looks suspiciously like Maeder's Owl Minimal Surface.

Another representation is

(5) |

(6) | |||

(7) | |||

(8) |

(Geometry Center) where, for aesthetic reasons, the - and -coordinates have been multiplied by 2 to produce a squashed, but topologically equivalent, surface. Nordstrand gives the implicit equation

(9) |

(10) |

Taking the inversion of a cross-cap such that (0, 0, ) is sent to gives a Cylindroid, shown above (Pinkall 1986).

The cross-cap is one of the three possible Surfaces obtained by sewing a Möbius Strip to the edge of a Disk. The other two are the Boy Surface and Roman Surface.

**References**

Fischer, G. (Ed.). Plate 107 in
*Mathematische Modelle/Mathematical Models, Bildband/Photograph Volume.*
Braunschweig, Germany: Vieweg, p. 108, 1986.

Geometry Center. ``The Crosscap.'' http://www.geom.umn.edu/zoo/toptype/pplane/cap/.

Pinkall, U. *Mathematical Models from the Collections of Universities and Museums* (Ed. G. Fischer).
Braunschweig, Germany: Vieweg, p. 64, 1986.

© 1996-9

1999-05-25