Editing Projective geometry (section)

== History ==
{{further|Mathematics and art}}

The first geometrical properties of a projective nature were discovered during the 3rd century by [[Pappus of Alexandria]].{{sfn|Coxeter|1969|p=229}} [[Filippo Brunelleschi]] (1404–1472) started investigating the geometry of perspective during 1425{{sfn|Coxeter|2003|p=2}} (see ''{{slink|Perspective (graphical)#History}}'' for a more thorough discussion of the work in the fine arts that motivated much of the development of projective geometry). [[Johannes Kepler]] (1571–1630) and [[Girard Desargues]] (1591–1661) independently developed the concept of the "point at infinity".{{sfn|Coxeter|2003|p=3}} Desargues developed an alternative way of constructing perspective drawings by generalizing the use of vanishing points to include the case when these are infinitely far away. He made [[Euclidean geometry]], where parallel lines are truly parallel, into a special case of an all-encompassing geometric system. Desargues's study on conic sections drew the attention of 16-year-old [[Blaise Pascal]] and helped him formulate [[Pascal's theorem]]. The works of [[Gaspard Monge]] at the end of 18th and beginning of 19th century were important for the subsequent development of projective geometry. The work of Desargues was ignored until [[Michel Chasles]] chanced upon a handwritten copy during 1845. Meanwhile, [[Jean-Victor Poncelet]] had published the foundational treatise on projective geometry during 1822.  Poncelet examined the projective properties of objects (those invariant under central projection) and, by basing his theory on the concrete [[pole and polar]] relation with respect to a circle, established a relationship between metric and projective properties. The [[non-Euclidean geometry|non-Euclidean geometries]] discovered soon thereafter were eventually demonstrated to have models, such as the [[Klein model]] of [[hyperbolic space]], relating to projective geometry.

In 1855 [[A. F. Möbius]] wrote an article about permutations, now called [[Möbius transformation]]s, of [[generalised circle]]s in the [[complex plane]]. These transformations represent projectivities of the [[complex projective line]]. In the study of lines in space, [[Julius Plücker]] used [[homogeneous coordinates]] in his description, and the set of lines was viewed on the [[Klein quadric]], one of  the  early contributions of projective geometry to a new field called [[algebraic geometry]], an offshoot of [[analytic geometry]] with projective ideas.

Projective geometry was instrumental in the validation of speculations of Lobachevski and Bolyai concerning [[hyperbolic geometry]] by providing [[model (logic)|model]]s for the [[coordinate systems for the hyperbolic plane|hyperbolic plane]]:<ref>[[John Milnor]] (1982) [https://projecteuclid.org/euclid.bams/1183548588 Hyperbolic geometry: The first 150 years], [[Bulletin of the American Mathematical Society]] via [[Project Euclid]]</ref> for example, the [[Poincaré disc model]] where generalised circles perpendicular to the [[unit circle]] correspond to "hyperbolic lines" ([[geodesic]]s), and the "translations" of this model are described by Möbius transformations that map the [[unit disc]] to itself. The distance between points is given by a  [[Cayley–Klein metric]], known to be invariant under the translations since it depends on [[cross-ratio]], a key projective invariant. The translations are described variously as [[isometries]] in [[metric space]] theory, as [[linear fractional transformation]]s formally, and as projective linear transformations of the [[projective linear group]], in this case {{nowrap|[[SU(1, 1)]]}}.

The work of [[Jean-Victor Poncelet|Poncelet]], [[Jakob Steiner]] and others was not intended to extend analytic geometry. Techniques were supposed to be ''[[synthetic geometry|synthetic]]'': in effect [[projective space]] as now understood was to be introduced axiomatically. As a result, reformulating early work in projective geometry so that it satisfies current standards of rigor can be somewhat difficult. Even in the case of the [[projective plane]] alone, the axiomatic approach can result in [[model theory|model]]s not describable via [[linear algebra]].

This period in geometry was overtaken by research on the general [[algebraic curve]] by [[Clebsch]], [[Bernhard Riemann|Riemann]], [[Max Noether]] and others, which stretched existing techniques, and then by [[invariant theory]]. Towards the end of the century, the [[Italian school of algebraic geometry]] ([[Federigo Enriques|Enriques]], [[Corrado Segre|Segre]], [[Francesco Severi|Severi]]) broke out of the traditional subject matter into an area demanding deeper techniques.

During the later part of the 19th century, the detailed study of projective geometry became less fashionable, although the literature is voluminous. Some important work was done in [[enumerative geometry]] in particular, by Schubert, that is now considered as anticipating the theory of [[Chern class]]es, taken as representing the [[algebraic topology]] of [[Grassmannian]]s.

Projective geometry later proved key to [[Paul Dirac]]'s invention of [[quantum mechanics]].  At a foundational level, the discovery that [[quantum measurement]]s could fail to commute had disturbed and dissuaded [[Werner Heisenberg|Heisenberg]], but past study of projective planes over noncommutative rings had likely desensitized Dirac.  In more advanced work, Dirac used extensive drawings in projective geometry to understand the intuitive meaning of his equations, before writing up his work in an exclusively algebraic formalism.<ref>{{cite journal |last=Farmelo |first=Graham |date=15 September 2005 |title=Dirac's hidden geometry |url=https://www.nature.com/articles/437323a.pdf |department=Essay |journal=[[Nature (journal)|Nature]] |publisher=Nature Publishing Group |volume=437 |issue=7057 |page=323|doi=10.1038/437323a |pmid=16163331 |bibcode=2005Natur.437..323F |s2cid=34940597 }}</ref>