Editing Projective geometry (section)

== Overview ==
Projective geometry is an elementary non-[[Metric (mathematics)|metrical]] form of geometry, meaning that it does not support any concept of distance. In two dimensions it begins with the study of [[Configuration (geometry)|configurations]] of [[Point (geometry)|point]]s and [[Line (geometry)|line]]s. That there is indeed some geometric interest in this sparse setting was first established by [[Girard Desargues|Desargues]] and others in their exploration of the principles of [[perspective (graphical)|perspective art]].{{sfn|Ramanan|1997|p=88}} In [[higher dimension]]al spaces there are considered [[hyperplane]]s (that always meet), and other linear subspaces, which exhibit [[#Duality|the principle of duality]]. The simplest illustration of duality is in the projective plane, where the statements "two distinct points determine a unique line" (i.e. the line through them) and "two distinct lines determine a unique point" (i.e. their point of intersection) show the same structure as propositions. Projective geometry can also be seen as a geometry of constructions with a [[straightedge|straight-edge]] alone, excluding [[Compass (drafting)|compass]] constructions, common in [[straightedge and compass construction]]s.{{sfn|Coxeter|2003|p=v}} As such, there are no circles, no angles, no measurements, no parallels, and no concept of [[wikt:intermediacy|intermediacy]] (or "betweenness").{{sfn|Coxeter|1969|p=229}} It was realised that the theorems that do apply to projective geometry are simpler statements. For example, the different [[conic section]]s are all equivalent in (complex) projective geometry, and some theorems about circles can be considered as special cases of these general theorems.

During the early 19th century the work of [[Jean-Victor Poncelet]], [[Lazare Carnot]] and others established projective geometry as an independent field of [[mathematics]].{{sfn|Coxeter|1969|p=229}} Its rigorous foundations were addressed by [[Karl von Staudt]] and perfected by Italians [[Giuseppe Peano]], [[Mario Pieri]], [[Alessandro Padoa]] and [[Gino Fano]] during the late 19th century.{{sfn|Coxeter|2003|p=14}} Projective geometry, like [[affine geometry|affine]] and [[Euclidean geometry]], can also be developed from the [[Erlangen program]] of Felix Klein; projective geometry is characterized by [[Invariant (mathematics)|invariants]] under [[Transformation (geometry)|transformations]] of the [[projective group]].

After much work on the very large number of theorems in the subject, therefore, the basics of projective geometry became understood. The [[incidence structure]] and the [[cross-ratio]] are fundamental invariants under projective transformations. Projective geometry can be modeled by the [[affine geometry|affine plane]] (or affine space) plus a line (hyperplane) "at infinity" and then treating that line (or hyperplane) as "ordinary".{{sfn|Coxeter|1969|pp=93,261}} An algebraic model for doing projective geometry in the style of [[analytic geometry]] is given by homogeneous coordinates.{{sfn|Coxeter|1969|pp=234–238}}{{sfn|Coxeter|2003|pp=111–132}} On the other hand, axiomatic studies revealed the existence of [[non-Desarguesian plane]]s, examples to show that the axioms of incidence can be modelled (in two dimensions only) by structures not accessible to reasoning through homogeneous coordinate systems.
[[File:Growth measure and vortices.jpg|thumb|upright=1.3|Growth measure and the polar vortices. Based on the work of Lawrence Edwards]]
In a foundational sense, projective geometry and [[ordered geometry]] are elementary since they each involve a minimal set of [[axioms]] and either can be used as the foundation for [[affine geometry|affine]] and [[Euclidean geometry]].{{sfn|Coxeter|1969|pp=175–262}}{{sfn|Coxeter|2003|pp=102–110}} Projective geometry is not "ordered"{{sfn|Coxeter|1969|p=229}} and so it is a distinct foundation for geometry.