Editing Real projective plane (section)

=== 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>.