The closed topological Manifold, denoted , which is obtained by projecting the points of a plane from a fixed point (not on the plane), with the addition of the Line at Infinity, is called the real projective plane. There is then a one-to-one correspondence between points in and lines through . Since each line through intersects the sphere centered at and tangent to in two Antipodal Points, can be described as a Quotient Space of by identifying any two such points. The real projective plane is a Nonorientable Surface.

The Boy Surface, Cross-Cap, and Roman Surface are all homeomorphic to the real projective plane and, because is nonorientable, these surfaces contain self-intersections (Kuiper 1961, Pinkall 1986).

**References**

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

Gray, A. ``Realizations of the Real Projective Plane.'' §12.5 in
*Modern Differential Geometry of Curves and Surfaces.*
Boca Raton, FL: CRC Press, pp. 241-245, 1993.

Klein, F. §1.2 in *Vorlesungen über nicht-euklidische Geometrie.* Berlin, 1928.

Kuiper, N. H. ``Convex Immersion of Closed Surfaces in .'' *Comment. Math. Helv.* **35**, 85-92, 1961.

Pinkall, U. *Mathematical Models from the Collections of Universities and Museums* (Ed. G. Fischer).
Braunschweig, Germany: Vieweg, pp. 64-65, 1986.

© 1996-9

1999-05-25