Editing Projective plane (section)

{{Short description|Geometric concept of a 2D space with a "point at infinity" adjoined}}
{{Use American English|date = January 2019}}
[[File:finite_projective_planes.svg|thumb|upright|Drawings of the finite projective planes of orders 2 (the [[Fano plane]]) and 3, in grid layout, showing a method of creating such drawings for prime orders]]
[[File:Railroad-Tracks-Perspective.jpg|thumb|upright|These parallel lines appear to intersect in the [[vanishing point]] "at infinity". In a projective plane this is actually true.]]
In [[mathematics]], a '''projective plane''' is a geometric structure that extends the concept of a [[plane (geometry)|plane]]. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus ''any'' two distinct lines in a projective plane intersect at exactly one point.

Renaissance artists, in developing the techniques of drawing in [[Perspective (graphical)#Renaissance|perspective]], laid the groundwork for this mathematical topic. The archetypical example is the [[real projective plane]], also known as the '''extended Euclidean plane'''.<ref>The phrases "projective plane", "extended affine plane" and "extended Euclidean plane" may be distinguished according to whether the line at infinity is regarded as special (in the so-called "projective" plane it is not, in the "extended" planes it is) and to whether Euclidean metric is regarded as meaningful (in the projective and affine planes it is not). Similarly for projective or extended spaces of other dimensions.</ref> This example, in slightly different guises, is important in [[algebraic geometry]], [[topology]] and [[projective geometry]] where it may be denoted variously by {{nowrap|PG(2, '''R''')}},  '''RP'''<sup>2</sup>, or '''P'''<sub>2</sub>('''R'''), among other notations. There are many other projective planes, both infinite, such as the [[complex projective plane]], and finite, such as the [[Fano plane]].

A projective plane is a 2-dimensional [[projective space]]. Not all projective planes can be [[embedding|embedded]] in 3-dimensional projective spaces; such embeddability is a consequence of a property known as [[Desargues' theorem]], not shared by all projective planes.