Editing Real projective plane (section)

== 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.{{sfn|ps=|Apéry|1987}} Some of the more important examples are described below.

The projective plane cannot be [[embedding|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 [[Jordan curve theorem|generalized Jordan curve theorem]]. The outward-pointing unit normal vector field would then give an [[Orientation (mathematics)|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 circle]]s of the sphere be "lines", and let pairs of [[antipodal point]]s 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 {{nowrap|''x'' ~ ''y''}} if {{nowrap|1=''y'' = ''x''}} or {{nowrap|1=''y'' = &minus;''x''}}. This quotient space of the sphere is [[homeomorphic]] with the collection of all lines passing through the origin in '''R'''<sup>3</sup>.

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 ===
[[File:Sphere symmetry group ci.png|thumb|A hemisphere can represent a real projective plane by attaching opposite points on the equator together.]]
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.{{sfn|ps=|Weeks|2002|p=59}}
{{clear}}

===Boy's surface – an immersion===
The projective plane can be [[Immersion (mathematics)|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.{{sfn|ps=|Brehm|1990|pp=51–56}}

=== Roman surface ===
[[Image:Steiner's Roman Surface.gif|thumb|An animation of the Roman surface]]
Steiner's [[Roman surface]] is a more degenerate map of the projective plane into 3-space, containing a [[cross-cap]].

[[File:Tetrahemihexahedron.png|thumb|The [[tetrahemihexahedron]] is a polyhedral representation of the real projective plane.]]
A [[Polyhedron|polyhedral]] representation is the [[tetrahemihexahedron]],{{sfn|ps=|Richter}} which has the same general form as Steiner's Roman surface, shown here.

=== Hemi polyhedra ===
Looking in the opposite direction, certain [[abstract regular polytope]]s – [[Hemicube (geometry)|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:{{sfn|ps=|Apéry|1987}}
: <math>k (x, y) = \left(1 + x^2 + y^2\right)^\frac{1}{2} \binom xy</math>

Central projection of the projective hemisphere onto a plane yields the usual infinite projective plane, described below.

=== Cross-capped disk ===
A closed surface is obtained by gluing a [[disk (mathematics)|disk]] to a [[cross-cap]]. This surface can be represented parametrically by the following equations:
:<math>\begin{align}
  X(u,v) &= r \, (1 + \cos v) \, \cos u, \\
  Y(u,v) &= r \, (1 + \cos v) \, \sin u, \\
  Z(u,v) &= -\operatorname{tanh}\left(u - \pi \right) \, r \, \sin v,
\end{align}</math>

where both ''u'' and ''v'' range from 0 to 2''π''.

These equations are similar to those of a [[torus]].  Figure&nbsp;1 shows a closed cross-capped disk.

{|
| [[Image:CrossCapTwoViews.PNG|500px]]
|-
| align=center | Figure 1. Two views of a cross-capped disk.
|}

A cross-capped disk has a [[plane of symmetry]] that passes through its line segment of double points.  In Figure&nbsp;1 the cross-capped disk is seen from above its plane of symmetry ''z'' = 0, but it would look the same if seen from below.

A cross-capped disk can be sliced open along its plane of symmetry, while making sure not to cut along any of its double points.  The result is shown in Figure&nbsp;2.

{|
| [[Image:CrossCapSlicedOpen.PNG|500px]]
|-
| align=center | Figure 2. Two views of a cross-capped disk which has been sliced open.
|}

Once this exception is made, it will be seen that the sliced cross-capped disk is [[homeomorphism|homeomorphic]] to a self-intersecting disk, as shown in Figure&nbsp;3.

{|
| [[Image:SelfIntersectingDisk.PNG|500px]]
|-
| align=center | Figure 3. Two alternative views of a self-intersecting disk.
|}

The self-intersecting disk is homeomorphic to an ordinary disk.  The parametric equations of the self-intersecting disk are:
: <math>\begin{align}
  X(u, v) &= r \, v \, \cos 2u, \\
  Y(u, v) &= r \, v \, \sin 2u, \\
  Z(u, v) &= r \, v \, \cos u,
\end{align}</math>
where ''u'' ranges from 0 to 2''π'' and ''v'' ranges from 0 to 1.

Projecting the self-intersecting disk onto the plane of symmetry (''z'' = 0 in the parametrization given earlier) which passes only through the double points, the result is an ordinary disk which repeats itself (doubles up on itself).

The plane ''z'' = 0 cuts the self-intersecting disk into a pair of disks which are mirror [[Reflection (mathematics)|reflection]]s of each other.  The disks have centers at the [[Origin (mathematics)|origin]].

Now consider the rims of the disks (with ''v'' = 1).  The points on the rim of the self-intersecting disk come in pairs which are reflections of each other with respect to the plane ''z'' = 0.

A cross-capped disk is formed by identifying these pairs of points, making them equivalent to each other.  This means that a point with parameters (''u'', 1) and coordinates <math>(r \, \cos 2u, r \, \sin 2u, r \, \cos u)</math> is identified with the point (''u'' + π, 1) whose coordinates are <math> (r \, \cos 2 u, r \, \sin 2 u, - r \, \cos u) </math>.  But this means that pairs of opposite points on the rim of the (equivalent) ordinary disk are identified with each other; this is how a real projective plane is formed out of a disk.  Therefore, the surface shown in Figure&nbsp;1 (cross-cap with disk) is topologically equivalent to the real projective plane ''RP''<sup>2</sup>.