Editing Real projective plane (section)

{{short description|Compact non-orientable two-dimensional manifold}}

In [[mathematics]], the '''real projective plane''', denoted {{tmath|\mathbf{RP}^2}} or {{tmath|\mathbb{P}_2}}, is a [[two-dimensional]] [[projective space]], similar to the familiar [[Euclidean plane]] in many respects but without the concepts of [[Euclidean distance|distance]], [[circle]]s, [[angle measure]], or [[parallel (geometry)|parallelism]]. It is the setting for planar [[projective geometry]], in which the relationships between objects are not considered to change under [[projective transformation]]s. The name ''projective'' comes from [[Perspective (graphical)|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 projection|oblique]] angle, is a projective transformation.

The fundamental objects in the projective plane are [[point (geometry)|points]] and [[straight line]]s, 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 plane]]s such as the [[complex projective plane]] and [[finite projective plane]]s.

One common model of the real [[projective plane]] is the space of lines in three-dimensional [[Euclidean space]] which pass through a particular [[origin (mathematics)|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 {{slink||Homogeneous coordinates}} below.)

{| class=wikitable align=right
| valign=top width=150|[[File:ProjectivePlaneAsSquare.svg|150px]]<br>The [[fundamental polygon]] of the projective plane – A is identified with A and B is identified with B, each with a twist
| valign=top width=150 style="padding-top:8px"|[[File:MöbiusStripAsSquare.svg|132px|center]] <div style="margin-top:6px;">The [[Möbius strip]] – because of the twist between the identified red A sides of the square, the dotted line is a single edge</div>
|}

In [[topology]], the name ''real projective plane'' is applied to any [[Surface (topology)|surface]] which is [[topological equivalence|topologically equivalent]] to the real projective plane. Topologically, the real projective plane is [[Compact space|compact]] and non-[[Orientability|orientable]] (one-sided).  It cannot be [[embedding|embedded]] in three-dimensional Euclidean space without intersecting itself. It has [[Euler characteristic]] 1, hence a [[Genus (mathematics)|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 (mathematics)|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.)