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In Mathematics , the real Projective Plane is the space of lines in '''R'''3 passing through the origin. It is a non- Orientable two-dimensional Manifold , that is, a Surface , that has basic applications to Geometry , but which cannot be Embedded in our usual three-dimensional space. It has Euler Characteristic of 1 giving a Genus of 1. It is often described intuitively, in relation with a × [0,1 with sides identified by the relations: :(0, ''y'') ~ (1, 1 − ''y'') for 0 ≤ ''y'' ≤ 1 and :(''x'', 0) ~ (1 − ''x'',1) for 0 ≤ ''x'' ≤ 1, as in the diagram on the right. FORMAL CONSTRUCTION Consider a Sphere , and let the Great Circle s of the sphere be "lines", and let pairs of Antipodal Point s be "points". It is easy to check that it obeys the axioms required of a Projective Plane :
This is the real projective plane. If we identify each point on the sphere with its antipodal point, then we get a representation of the real projective plane in which the "points" of the projective plane really are points. The resulting surface, a 2-dimensional Compact Non-orientable manifold, is a little hard to visualize, because it cannot be embedded in 3-dimensional Euclidean Space without intersecting itself. TRYING TO EMBED THE REAL PROJECTIVE PLANE IN THREE-SPACE The projective plane cannot strictly be Embedded (that is without intersection) in three-dimensional space. However, it can be Immersed (local neighbourhoods do not have self-intersections). Boy's Surface is an example of an immersion. The Roman Surface is another interesting example, but this contains Cross-cap s so it is not technically an immersion. The same goes for a sphere with a cross-cap. A Polyhedral representation is the Tetrahemihexahedron . HOMOGENEOUS COORDINATES The set of lines in the plane can be represented using , and the set of coordinates (''a'':''b'':0) defines a line at infinity. SEE ALSO |
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