Real projective plane

Central projection of the projective hemisphere onto a plane yields the usual infinite projective plane, described below.

These equations are similar to those of a torus. Figure 1 shows a closed cross-capped disk.

A cross-capped disk 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 cross-capped disk is homeomorphic to a self-intersecting disk, as shown in Figure 3.

The self-intersecting disk is homeomorphic to an ordinary disk. The parametric equations of the self-intersecting disk are:

Projecting the self-intersecting 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 article on the fundamental polygon describes the higher non-orientable surfaces.