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


:("x", 0) ~ (1 − "x", 1) for 0 ≤ "x" ≤ 1,

as in the diagram on the right.


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


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Projective plane — See real projective plane and complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In mathematics, a projective plane has two possible definitions, one of them coming from linear algebra, and another (which is… …   Wikipedia

  • Real projective line — In real analysis, the real projective line (also called the one point compactification of the real line, or the projectively extended real numbers ), is the set mathbb{R}cup{infty}, also denoted by widehat{mathbb{R and by mathbb{R}P^1.The symbol… …   Wikipedia

  • Real projective space — In mathematics, real projective space, or RP n is the projective space of lines in R n +1. It is a compact, smooth manifold of dimension n , and a special case of a Grassmannian.ConstructionAs with all projective spaces, RP n is formed by taking… …   Wikipedia

  • Complex projective plane — In mathematics, the complex projective plane, usually denoted CP2, is the two dimensional complex projective space. It is a complex manifold described by three complex coordinates where, however, the triples differing by an overall rescaling are… …   Wikipedia

  • Fake projective plane — For Freedman s example of a non smoothable manifold with the same homotopy type as the complex projective plane, see 4 manifold. In mathematics, a fake projective plane (or Mumford surface) is one of the 50 complex algebraic surfaces that have… …   Wikipedia

  • Projective geometry — is a non metrical form of geometry, notable for its principle of duality. Projective geometry grew out of the principles of perspective art established during the Renaissance period, and was first systematically developed by Desargues in the 17th …   Wikipedia

  • Projective space — In mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non zero vectors which are equal up to a multiplication by a non zero scalar. A formal… …   Wikipedia

  • projective geometry — the geometric study of projective properties. [1880 85] * * * Branch of mathematics that deals with the relationships between geometric figures and the images (mappings) of them that result from projection. Examples of projections include motion… …   Universalium

  • Plane at infinity — In projective geometry, the plane at infinity is a projective plane which is added to the affine 3 space in order to give it closure of incidence properties. The result of the addition is the projective 3 space, P^3 . If the affine 3 space is… …   Wikipedia

  • Projective transformation — A projective transformation is a transformation used in projective geometry: it is the composition of a pair of perspective projections. It describes what happens to the perceived positions of observed objects when the point of view of the… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”