Cross-Cap

A cross-cap is a self-intersecting image in three-dimensional Euclidean space of a Möbius strip, the surface with boundary obtained by removing an open disk from the real projective plane (Hatcher, p. 46). The word "cross-cap" is sometimes also written without the hyphen as the single word "crosscap."

The name is also used for the singular image of the entire real projective plane obtained by capping the Möbius strip with a disk (Geometry Center). The figures above and the equations below describe this capped model. It has a segment of double points which terminates at two pinch points.

Attaching a cross-cap to a surface means removing an open disk and sewing a Möbius strip to the resulting boundary. This can also be visualized by attaching two zips around the puncture in the same direction, distorting the hole so that the zips line up, allowing the surface to intersect itself, and then zipping up. Topologically, a cross-handle is equivalent to attaching two cross-caps (Francis and Weeks 1999).

A sphere with one cross-cap has traditionally been called a real projective plane. While this is appropriate in the study of projective geometry when an affine structure is present, J. H. Conway advocates use of the term cross surface in a purely topological interpretation (Francis and Weeks 1999). Other familiar representations of the real projective plane include the Boy surface and Roman surface.

A sphere with two attached cross-caps is topologically equivalent to a Klein bottle (Francis and Weeks 1999). The surface with three cross-caps is known as Dyck's surface (Francis and Collins 1993, Francis and Weeks 1999).

The cross-cap can be generated using the general method for nonorientable surfaces using the polynomial function

(1)

(Pinkall 1986). Transforming to spherical coordinates gives

			(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 Bour's minimal surface.

Another representation is

(5)

(Gray 1997), giving parametric equations

			(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. It is therefore a quartic surface given by

(9)

The volume enclosed by the surface in this parametrization is

(10)

The moment of inertia tensor for the solid with uniform density and mass is given by

(11)

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