In mathematics, the real projective plane, denoted R P 2 {\displaystyle \mathbf {RP} ^{2}} or P 2 {\displaystyle \mathbb {P} _{2}} , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the setting for planar projective geometry, in which the relationships between objects are not considered to change under projective transformations. The name projective comes from perspective drawing: projecting an image from one plane onto another as viewed from a point outside either plane, for example by photographing a flat painting from an oblique angle, is a projective transformation. The fundamental objects in the projective plane are points and straight lines, and as in Euclidean geometry, every pair of points determines a unique line passing through both, but unlike in the Euclidean case in projective geometry every pair of lines also determines a unique point at their intersection (in Euclidean geometry, parallel lines never intersect). In contexts where there is no ambiguity, it is simply called the projective plane; the qualifier "real" is added to distinguish it from other projective planes such as the complex projective plane and finite projective planes. One common model of the real projective plane is the space of lines in three-dimensional Euclidean space which pass through a particular origin point; in this model, lines through the origin are considered to be the "points" of the projective plane, and planes through the origin are considered to be the "lines" in the projective plane. These projective points and lines can be pictured in two dimensions by intersecting them with any arbitrary plane not passing through the origin; then the parallel plane which does pass through the origin (a projective "line") is called the line at infinity. (See § Homogeneous coordinates below.)
In topology, the name real projective plane is applied to any surface which is topologically equivalent to the real projective plane. Topologically, the real projective plane is compact and non-orientable (one-sided). It cannot be embedded in three-dimensional Euclidean space without intersecting itself. It has Euler characteristic 1, hence a demigenus (non-orientable genus, Euler genus) of 1. The topological real projective plane can be constructed by taking the (single) edge of a Möbius strip and gluing it to itself in the correct direction, or by gluing the edge to a disk. Alternately, the real projective plane can be constructed by identifying each pair of opposite sides of the square, but in opposite directions, as shown in the diagram. (Performing any of these operations in three-dimensional space causes the surface to intersect itself.)
Examples Projective geometry is not necessarily concerned with curvature and the real projective plane may be twisted up and placed in the Euclidean plane or 3-space in many different ways. Some of the more important examples are described below. The projective plane cannot be embedded (that is without intersection) in three-dimensional Euclidean space. The proof that the projective plane does not embed in three-dimensional Euclidean space goes like this: Assuming that it does embed, it would bound a compact region in three-dimensional Euclidean space by the generalized Jordan curve theorem. The outward-pointing unit normal vector field would then give an orientation of the boundary manifold, but the boundary manifold would be the projective plane, which is not orientable. This is a contradiction, and so our assumption that it does embed must have been false.
The projective sphere Consider a sphere, and let the great circles of the sphere be "lines", and let pairs of antipodal points be "points". It is easy to check that this system obeys the axioms required of a projective plane:
any pair of distinct great circles meet at a pair of antipodal points; and any two distinct pairs of antipodal points lie on a single great circle. 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. This means that the projective plane is the quotient space of the sphere obtained by partitioning the sphere into equivalence classes under the equivalence relation ~, where x ~ y if y = x or y = −x. This quotient space of the sphere is homeomorphic with the collection of all lines passing through the origin in R3. The quotient map from the sphere onto the real projective plane is in fact a two sheeted (i.e. two-to-one) covering map. It follows that the fundamental group of the real projective plane is the cyclic group of order 2; i.e., integers modulo 2. One can take the loop AB from the figure above to be the generator.
Projective hemisphere
Because the sphere covers the real projective plane twice, the plane may be represented as a closed hemisphere around whose rim opposite points are identified.
Boy's surface – an immersion The projective plane can be immersed (local neighbourhoods of the source space do not have self-intersections) in 3-space. Boy's surface is an example of an immersion. Polyhedral examples must have at least nine faces.
Roman surface
Steiner's Roman surface is a more degenerate map of the projective plane into 3-space, containing a cross-cap.
A polyhedral representation is the tetrahemihexahedron, which has the same general form as Steiner's Roman surface, shown here.
Hemi polyhedra Looking in the opposite direction, certain abstract regular polytopes – hemi-cube, hemi-dodecahedron, and hemi-icosahedron – can be constructed as regular figures in the projective plane; see also projective polyhedra.
Planar projections Various planar (flat) projections or mappings of the projective plane have been described. In 1874 Klein described the mapping:
… excerpt ends here. Continue reading the full article.
