Real projective plane

In mathematics, the real projective plane, denoted ⁠${\displaystyle \mathbf {RP} ^{2}}$⁠ or ⁠${\displaystyle \mathbb {P} _{2}}$⁠, is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the setting for planar projective geometry, in which the relationships between objects are not considered to change under projective transformations. The name projective comes from 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 angle, is a projective transformation.

The fundamental objects in the projective plane are points and straight lines, 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 planes such as the complex projective plane and finite projective planes.

One common model of the real projective plane is the space of lines in three-dimensional Euclidean space which pass through a particular 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 § Homogeneous coordinates below.)

The fundamental polygon of the projective plane – A is identified with A and B is identified with B, each with a twist

The Möbius strip – because of the twist between the identified red A sides of the square, the dotted line is a single edge

In topology, the name real projective plane is applied to any surface which is topologically equivalent to the real projective plane. Topologically, the real projective plane is compact and non-orientable (one-sided). It cannot be embedded in three-dimensional Euclidean space without intersecting itself. It has Euler characteristic 1, hence a 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. 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.)