Crosscap
From Academic Kids

In mathematics, a crosscap is a twodimensional surface that is topologically equivalent to a Möbius strip. The term 'crosscap', however, often implies that the surface has been deformed so that its boundary is an ordinary circle.
A crosscap that has been closed up by gluing a disc to its boundary is called a real projective plane. Two crosscaps glued together at their boundaries form a Klein bottle. An important theorem of topology, the classification theorem for surfaces, states that all twodimensional nonorientable manifolds are spheres with some number of 'handles' and at most two crosscaps.
Crosscap model of the real projective plane
A crosscap can also refer synonymously to the closed surface obtained by gluing a disk to a crosscap. This surface can be represented parametrically by the following equations:
 <math> X(u,v) = r \, (1 + \cos v) \, \cos u, <math>
 <math> Y(u,v) = r \, (1 + \cos v) \, \sin u, <math>
 <math> Z(u,v) =  \hbox{tanh} \left( {2 \over 3} (u  \pi) \right) \, r \, \sin v,<math>
where both u and v range from 0 to 2π. These equations are similar to those of a torus. Figure 1 shows a closed crosscap.

A crosscap has a plane of symmetry which passes through its line segment of double points. In Figure 1 the crosscap is seen from above its plane of symmetry z = 0, but it would look the same if seen from below.
A crosscap can be sliced open along its plane of symmetry, while making sure not to cut along any of its double points. The result is shown in Figure 2.

Once this exception is made, it will be seen that the sliced crosscap is homeomorphic to a selfintersecting disk, as shown in Figure 3.
Missing image SelfIntersectingDisk.PNG Image:SelfIntersectingDisk.PNG 

The selfintersecting disk is homeomorphic to an ordinary disk. The parametric equations of the selfintersecting disk are:
 <math> X(u,v) = r \, v \, \cos 2 u, <math>
 <math> Y(u,v) = r \, v \, \sin 2 u, <math>
 <math> Z(u,v) = r \, v \, \cos u, <math>
where u ranges from 0 to 2π and v ranges from 0 to 1.
Projecting the selfintersecting disk onto the plane of symmetry (z = 0 in the parametrization given earlier) which passes only through the double points, the result is an ordinary disk which repeats itself (doubles up on itself).
The plane z = 0 cuts the selfintersecting disk into a pair of disks which are mirror reflections of each other. The disks have centers at the origin.
Now consider the rims of the disks (with v = 1). The points on the rim of the selfintersecting disk come in pairs which are reflections of each other with respect to the plane z = 0.
A crosscap is formed by identifying these pairs of points, making them equivalent to each other. This means that a point with parameters (u,1) and coordinates <math> (r \, \cos 2 u, r \, \sin 2 u, r \, \cos u) <math> is identified with the point (u + π,1) whose coordinates are <math> (r \, \cos 2 u, r \, \sin 2 u,  r \, \cos u) <math>. But this means that pairs of opposite points on the rim of the (equivalent) ordinary disk are identified with each other; this is how a real projective plane is formed out of a disk. Therefore the surface shown in Figure 1 (crosscap with disk) is topologically equivalent to the real projective plane, RP^{2}.
See also: Boy's surface, Roman surface.