quality:corruption
Classical examples

The real projective plane RP arises when K is taken to be the real numbers, R. As a closed, non-orientable real 2- manifold, it serves as a fundamental example in topology.

In this construction, consider the unit sphere centered at the origin in R. Each of the R lines in this construction intersects the sphere at two antipodal points. Since the R line represents a point of RP, we will obtain the same model of RP by identifying the antipodal points of the sphere. The lines of RP will be the great circles of the sphere after this identification of antipodal points. This description gives the standard model of elliptic geometry.

The complex projective plane CP arises when K is taken to be the complex numbers, C. It is a closed complex 2-manifold, and hence a closed, orientable real 4-manifold. It and projective planes over other fields (known as pappian planes) serve as fundamental examples in algebraic geometry.

The quaternionic projective plane HP is also of independent interest.