Real projective plane


Real projective plane

In mathematics, the real projective plane is the space of lines in R3 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 without intersecting itself. It has Euler characteristic of 1 giving a genus of 1.

The real projective plane is sometimes described in terms of a construction based on the Möbius strip: if one could glue the (single) edge of the Möbius strip to itself in the correct direction, one would obtain the projective plane. Equivalently, gluing a disk along the boundary of the Möbius strip gives the projective plane.

Since the Möbius strip, in turn, can be constructed from a square by gluing two of its sides together, the real projective plane can thus be represented as a unit square ( [0,1] × [0,1] ) with its sides identified by the following equivalence 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.

Construction

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 it 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.

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 projective plane is the quotient space of the sphere obtained by partitioning the sphere into equivalence classes under the equivalence relation ~, where x ~ y iff y = −x. This quotient space of the sphere is homeomorphic with the collection of all lines passing through the origin in R3.

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.

The quotient map from the sphere onto the real projective plane is in fact a (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.

Immersing the real projective plane in three-space

The projective plane cannot 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 a more degenerate map of the projective plane into 3-space, containing a cross-cap. The same goes for a sphere with a cross-cap.

The proof that the projective plane does not embed in three-dimensional Euclidean space goes like this: Assume 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 projective space, which is not orientable. This is a contradiction, and so our assumption that it does embed must have been false.

A polyhedral representation is the tetrahemihexahedron.

Looking in the opposite direction, the hemi-cube, hemi-dodecahedron, and hemi-icosahedron, abstract regular polychora, can be constructed as a regular figure in the "projective plane".

Homogeneous coordinates

The set of lines in the plane can be represented using homogeneous coordinates. A line "ax"+"by"+"c"=0 can be represented as ("a":"b":"c"). These coordinates have the equivalence relation ("a":"b":"c") = ("da":"db":"dc") for all non zero values of "d". Hence a different representation of the same line "dax"+"dby"+"dc"=0 has the same coordinates. The set of coordinates ("a":"b":1) gives the usual real plane, and the set of coordinates ("a":"b":0) defines a line at infinity.

Embedding into 4-dimensional space

The projective plane embeds into 4-dimensional Euclidean space. Consider mathbb RP^2 to be the quotient of the two-sphere S^2 = {(x,y,z) in mathbb R^3 : x^2+y^2+z^2 = 1} by the antipodal relation (x,y,z)sim (-x,-y,-z),. Consider the function mathbb R^3 o mathbb R^4 given by (x,y,z)longmapsto (xy,xz,y^2-z^2,2yz). This map restricts to a map whose domain is S^2 and, since it is a purely quadratic polynomial, it can be factorised to give a map mathbb RP^2 o mathbb R^4. Moreover, this map is an embedding. Notice that this embedding admits a projection into R^3 which is the Roman surface.

Higher genus

The article on the fundamental polygon provides a description of the real projective planes of higher genus.

ee also

*Projective space
*Pu's inequality for real projective plane

External links

*


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