Editing Projective geometry

{{Short description|Type of geometry}}
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In [[mathematics]], '''projective geometry''' is the study of geometric properties that are invariant with respect to [[projective transformation]]s. This means that, compared to elementary [[Euclidean geometry]], projective geometry has a different setting (''[[projective space]]'') and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than [[Euclidean space]], for a given dimension, and that [[geometric transformation]]s are permitted that transform the extra points (called "[[Point at infinity|points at infinity]]") to Euclidean points, and vice versa.

Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a [[transformation matrix]] and [[translation (geometry)|translation]]s (the [[affine transformation]]s). The first issue for geometers is what kind of geometry is adequate for a novel situation. Unlike in [[Euclidean geometry]], the concept of an [[angle]] does not apply in projective geometry, because no measure of angles is invariant with respect to projective transformations, as is seen in [[perspective drawing]] from a changing perspective. One source for projective geometry was indeed the theory of perspective. Another difference from elementary geometry is the way in which [[parallel (geometry)|parallel line]]s can be said to meet in a [[point at infinity]], once the concept is translated into projective geometry's terms. Again this notion has an intuitive basis, such as railway tracks meeting at the horizon in a perspective drawing. See ''[[Projective plane]]'' for the basics of projective geometry in two dimensions.

While the ideas were available earlier, projective geometry was mainly a development of the 19th century. This included the theory of [[complex projective space]], the coordinates used ([[homogeneous coordinates]]) being complex numbers. Several major types of more abstract mathematics (including [[invariant theory]], the [[Italian school of algebraic geometry]], and [[Felix Klein]]'s [[Erlangen programme]] resulting in the study of the [[classical groups]]) were motivated by projective geometry. It was also a subject with many practitioners for its own sake, as [[synthetic geometry]]. Another topic that developed from axiomatic studies of projective geometry is [[finite geometry]].

The topic of projective geometry is itself now divided into many research subtopics, two examples of which are projective algebraic geometry (the study of [[Algebraic variety#Projective varieties|projective varieties]]) and [[projective differential geometry]] (the study of [[differential geometry|differential invariants]] of the projective transformations).

== 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.

== Description ==
{{Unfocused|date=March 2023|section=y|reason="Description" is either vague or too broad.|talk="Description"}}

Projective geometry is less restrictive than either [[Euclidean geometry]] or [[affine geometry]]. It is an intrinsically non-[[Metric (mathematics)|metrical]] geometry, meaning that facts are independent of any metric structure. Under the projective transformations, the [[incidence structure]] and the relation of [[projective harmonic conjugate]]s are preserved. A [[projective range]] is the one-dimensional foundation. Projective geometry formalizes one of the central principles of perspective art: that [[parallel (geometry)|parallel]] lines meet at [[infinity]], and therefore are drawn that way. In essence, a projective geometry may be thought of as an extension of Euclidean geometry in which the "direction" of each line is subsumed within the line as an extra "point", and in which a "horizon" of directions corresponding to coplanar lines is regarded as a "line". Thus, two parallel lines meet on a horizon line by virtue of their incorporating the same direction.

Idealized directions are referred to as points at infinity, while idealized horizons are referred to as lines at infinity. In turn, all these lines lie in the plane at infinity. However, infinity is a metric concept, so a purely projective geometry does not single out any points, lines or planes in this regard—those at infinity are treated just like any others.

Because a [[Euclidean geometry]] is contained within a projective geometry—with projective geometry having a simpler foundation—general results in Euclidean geometry may be derived in a more transparent manner, where separate but similar theorems of Euclidean geometry may be handled collectively within the framework of projective geometry. For example, parallel and nonparallel lines need not be treated as separate cases; rather an arbitrary projective plane is singled out as the ideal plane and located "at infinity" using [[homogeneous coordinates]].

Additional properties of fundamental importance include [[Desargues' Theorem]] and the [[Pappus's hexagon theorem|Theorem of Pappus]]. In projective spaces of dimension 3 or greater there is a construction that allows one to prove [[Desargues' Theorem]]. But for dimension 2, it must be separately postulated.

Using [[Desargues' Theorem]], combined with the other axioms, it is possible to define the basic operations of arithmetic, geometrically. The resulting operations satisfy the axioms of a field – except that the commutativity of multiplication requires [[Pappus's hexagon theorem]]. As a result, the points of each line are in one-to-one correspondence with a given field, {{mvar|F}}, supplemented by an additional element, ∞, such that {{math|1={{var|r}} ⋅ ∞ = ∞}}, {{math|1=−∞ = ∞}}, {{math|1={{var|r}} + ∞ = ∞}}, {{math|1={{var|r}} / 0 = ∞}}, {{math|1={{var|r}} / ∞ = 0}}, {{math|1=∞ − {{var|r}} = {{var|r}} − ∞ = ∞}}, except that {{math|0 / 0}}, {{math|∞ / ∞}}, {{math|∞ + ∞}}, {{math|∞ − ∞}}, {{math|0 ⋅ ∞}} and {{math|∞ ⋅ 0}} remain undefined.

Projective geometry also includes a full theory of [[conic sections]], a subject also extensively developed in Euclidean geometry. There are advantages to being able to think of a [[hyperbola]] and an [[ellipse]] as distinguished only by the way the hyperbola ''lies across the line at infinity''; and that a [[parabola]] is distinguished only by being tangent to the same line. The whole family of circles can be considered as ''conics passing through two given points on the line at infinity'' — at the cost of requiring [[complex number|complex]] coordinates. Since coordinates are not "synthetic", one replaces them by fixing a line and two points on it, and considering the ''linear system'' of all conics passing through those points as the basic object of study. This method proved very attractive to talented geometers, and the topic was studied thoroughly. An example of this method is the multi-volume treatise by [[H. F. Baker]].

== 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>

== Classification ==
There are many projective geometries, which may be divided into discrete and continuous: a ''discrete'' geometry comprises a set of points, which may or may not be ''finite'' in number, while a ''continuous'' geometry has infinitely many points with no gaps in between.

The only projective geometry of dimension 0 is a single point. A projective geometry of dimension 1 consists of a single line containing at least 3 points. The geometric construction of arithmetic operations cannot be performed in either of these cases. For dimension 2, there is a rich structure in virtue of the absence of [[Desargues' Theorem]].

[[File:Fano plane.svg|thumb|The [[Fano plane]] is the projective plane with the fewest points and lines.]]
The smallest 2-dimensional projective geometry (that with the fewest points) is the [[Fano plane]], which has 3 points on every line, with 7 points and 7 lines in all, having the following collinearities:
{{div col}}
* [ABC]
* [ADE]
* [AFG]
* [BDG]
* [BEF]
* [CDF]
* [CEG]
{{colend}}
with [[homogeneous coordinates]] {{math|1=A = (0,0,1)}}, {{math|1=B = (0,1,1)}}, {{math|1=C = (0,1,0)}}, {{math|1=D = (1,0,1)}}, {{math|1=E = (1,0,0)}}, {{math|1=F = (1,1,1)}}, {{math|1=G = (1,1,0)}}, or, in affine coordinates, {{math|1=A = (0,0)}}, {{math|1=B = (0,1)}}, {{math|1=C = (∞)}}, {{math|1=D = (1,0)}}, {{math|1=E = (0)}}, {{math|1=F = (1,1) }}and {{math|1=G = (1)}}. The affine coordinates in a Desarguesian plane for the points designated to be the points at infinity (in this example: C, E and G) can be defined in several other ways.

In standard notation, a [[finite projective geometry]] is written {{math|PG(''a'', ''b'')}} where:
: {{mvar|a}} is the projective (or geometric) dimension, and
: {{mvar|b}} is one less than the number of points on a line (called the ''order'' of the geometry).

Thus, the example having only 7 points is written {{math|PG(2, 2)}}.

The term "projective geometry" is used sometimes to indicate the generalised underlying abstract geometry, and sometimes to indicate a particular geometry of wide interest, such as the metric geometry of flat space which we analyse through the use of [[homogeneous coordinates]], and in which [[Euclidean geometry]] may be embedded (hence its name, [[Projective plane#Some examples|Extended Euclidean plane]]).

The fundamental property that singles out all projective geometries is the ''elliptic'' [[incidence (mathematics)|incidence]] property that any two distinct lines {{mvar|L}} and {{mvar|M}} in the [[projective plane]] intersect at exactly one point {{mvar|P}}. The special case in [[analytic geometry]] of ''parallel'' lines is subsumed in the smoother form of a line ''at infinity'' on which {{mvar|P}} lies. The ''line at infinity'' is thus a line like any other in the theory: it is in no way special or distinguished. (In the later spirit of the [[Erlangen programme]] one could point to the way the [[group (mathematics)|group]] of transformations can move any line to the ''line at infinity'').

The parallel properties of elliptic, Euclidean and hyperbolic geometries contrast as follows:
: Given a line {{mvar|l}} and a point {{mvar|P}} not on the line,
::; ''[[Elliptic geometry|Elliptic]]'' :  there exists no line through {{mvar|P}} that does not meet {{mvar|l}}
::; ''[[Euclidean geometry|Euclidean]]'' :  there exists exactly one line through {{mvar|P}} that does not meet {{mvar|l}}
::; ''[[Hyperbolic geometry|Hyperbolic]]'' :  there exists more than one line through {{mvar|P}} that does not meet {{mvar|l}}

The parallel property of elliptic geometry is the key idea that leads to the principle of projective duality, possibly the most important property that all projective geometries have in common.

== Duality ==
{{further|Duality (projective geometry)}}

In 1825, [[Joseph Gergonne]] noted the principle of [[duality (projective geometry)|duality]] characterizing projective plane geometry: given any theorem or definition of that geometry, substituting ''point'' for ''line'', ''lie on'' for ''pass through'', ''collinear'' for ''concurrent'', ''intersection'' for ''join'', or vice versa, results in another theorem or valid definition, the "dual" of the first. Similarly in 3 dimensions, the duality relation holds between points and planes, allowing any theorem to be transformed by swapping ''point'' and ''plane'', ''is contained by'' and ''contains''. More generally, for projective spaces of dimension N, there is a duality between the subspaces of dimension ''R'' and dimension {{nowrap|''N'' − ''R'' − 1}}. For {{nowrap|1=''N'' = 2}}, this specializes to the most commonly known form of duality—that between points and lines.
The duality principle was also discovered independently by [[Jean-Victor Poncelet]].

To establish duality only requires establishing theorems which are the dual versions of the axioms for the dimension in question. Thus, for 3-dimensional spaces, one needs to show that (1*) every point lies in 3 distinct planes, (2*) every two planes intersect in a unique line and a dual version of (3*) to the effect: if the intersection of plane P and Q is coplanar with the intersection of plane R and S, then so are the respective intersections of planes P and R, Q and S (assuming planes P and S are distinct from Q and R).

In practice, the principle of duality allows us to set up a ''dual correspondence'' between two geometric constructions. The most famous of these is the polarity or reciprocity of two figures in a [[conic]] curve (in 2 dimensions) or a quadric surface (in 3 dimensions). A commonplace example is found in the reciprocation of a symmetrical [[polyhedron]] in a concentric sphere to obtain the dual polyhedron.

Another example is [[Brianchon's theorem]], the dual of the already mentioned [[Pascal's theorem]], and one of whose proofs simply consists of applying the principle of duality to Pascal's. Here are comparative statements of these two theorems (in both cases within the framework of the projective plane):
* '''Pascal:''' If all six vertices of a hexagon lie on a [[Conic section#In the real projective plane|conic]], then the intersections of its opposite sides ''(regarded as full lines, since in the projective plane there is no such thing as a "line segment")'' are three collinear points. The line joining them is then called the '''Pascal line''' of the hexagon.
* '''Brianchon:''' If all six sides of a hexagon are tangent to a conic, then its diagonals (i.e. the lines joining opposite vertices) are three concurrent lines. Their point of intersection is then called the '''Brianchon point''' of the hexagon.
: (If the conic degenerates into two straight lines, Pascal's becomes [[Pappus's hexagon theorem|Pappus's theorem]], which has no interesting dual, since the Brianchon point trivially becomes the two lines' intersection point.)

== Axioms of projective geometry ==
Any given geometry may be deduced from an appropriate set of [[axiom]]s. Projective geometries are characterised by the "elliptic parallel" axiom, that ''any two planes always meet in just one line'', or in the plane, ''any two lines always meet in just one point''. In other words, there are no such things as parallel lines or planes in projective geometry.

Many alternative sets of axioms for projective geometry have been proposed (see for example Coxeter 2003, Hilbert & Cohn-Vossen 1999, Greenberg 1980).

=== Whitehead's axioms ===
These axioms are based on [[Alfred North Whitehead|Whitehead]], "The Axioms of Projective Geometry". There are two types, points and lines, and one "incidence" relation between points and lines. The three axioms are:
* G1: Every line contains at least 3 points
* G2: Every two distinct points, A and B, lie on a unique line, AB.
* G3: If lines AB and CD intersect, then so do lines AC and BD (where it is assumed that A and D are distinct from B and C).

The reason each line is assumed to contain at least 3 points is to eliminate some degenerate cases. The spaces satisfying these
three axioms either have at most one line, or are projective spaces of some dimension over a [[division ring]], or are [[non-Desarguesian plane]]s.

=== Additional axioms ===
One can add further axioms restricting the dimension or the coordinate ring. For example, Coxeter's ''Projective Geometry'',{{sfn|Coxeter|2003|pp=14–15}} references Veblen{{sfn|Veblen|Young|1938|pp=16,18,24,45}} in the three axioms above, together with a further 5 axioms that make the dimension 3 and the coordinate ring a commutative field of characteristic not&nbsp;2.

=== Axioms using a ternary relation ===
One can pursue axiomatization by postulating a ternary relation, [ABC] to denote when three points (not all necessarily distinct) are collinear. An axiomatization may be written down in terms of this relation as well:
* C0: [ABA]
* C1: If A and B are distinct points such that [ABC] and [ABD] then [BDC]
* C2: If A and B are distinct points then there exists a third distinct point C such that [ABC]
* C3: If A and C are distinct points, and B and D are distinct points, with [BCE] and [ADE] but not [ABE], then there is a point F such that [ACF] and [BDF].
For two distinct points, A and B, the line AB is defined as consisting of all points C for which [ABC]. The axioms C0 and C1 then provide a formalization of G2; C2 for G1 and C3 for G3.

The concept of line generalizes to planes and higher-dimensional subspaces. A subspace, AB...XY may thus be recursively defined in terms of the subspace AB...X as that containing all the points of all lines YZ, as Z ranges over AB...X. Collinearity then generalizes to the relation of "independence". A set {{nowrap|{{mset|A, B, ..., Z}}}} of points is independent, [AB...Z] if {{nowrap|{{mset|A, B, ..., Z}}}} is a minimal generating subset for the subspace AB...Z.

The projective axioms may be supplemented by further axioms postulating limits on the dimension of the space. The minimum dimension is determined by the existence of an independent set of the required size. For the lowest dimensions, the relevant conditions may be stated in equivalent
form as follows. A projective space is of:
* (L1) at least dimension 0 if it has at least 1 point,
* (L2) at least dimension 1 if it has at least 2 distinct points (and therefore a line),
* (L3) at least dimension 2 if it has at least 3 non-collinear points (or two lines, or a line and a point not on the line),
* (L4) at least dimension 3 if it has at least 4 non-coplanar points.

The maximum dimension may also be determined in a similar fashion. For the lowest dimensions, they take on the following forms. A projective space is of:
* (M1) at most dimension 0 if it has no more than 1 point,
* (M2) at most dimension 1 if it has no more than 1 line,
* (M3) at most dimension 2 if it has no more than 1 plane,
and so on. It is a general theorem (a consequence of axiom (3)) that all coplanar lines intersect—the very principle that projective geometry was originally intended to embody. Therefore, property (M3) may be equivalently stated that all lines intersect one another.

It is generally assumed that projective spaces are of at least dimension 2. In some cases, if the focus is on projective planes, a variant of M3 may be postulated. The axioms of (Eves 1997: 111), for instance, include (1), (2), (L3) and (M3). Axiom (3) becomes vacuously true under (M3) and is therefore not needed in this context.

=== Axioms for projective planes ===
{{main|Projective plane}}
In [[incidence geometry]], most authors<ref>{{harvnb|Bennett|1995|p=4}}, {{harvnb|Beutelspacher|Rosenbaum|1998|p=8}}, {{harvnb|Casse|2006|p=29}}, {{harvnb|Cederberg|2001|p=9}}, {{harvnb|Garner|1981|p=7}}, {{harvnb|Hughes|Piper|1973|p=77}}, {{harvnb|Mihalek|1972|p=29}}, {{harvnb|Polster|1998|p=5}} and {{harvnb|Samuel|1988|p=21}} among the references given.</ref> give a treatment that embraces the [[Fano plane]] {{nowrap|PG(2, 2)}} as the smallest finite projective plane.  An axiom system that achieves this is as follows:
* (P1) Any two distinct points lie on a line that is unique.
* (P2) Any two distinct lines meet at a point that is unique.
* (P3) There exist at least four points of which no three are collinear.

Coxeter's ''Introduction to Geometry''{{sfn|Coxeter|1969|pp=229–234}} gives a list of five axioms for a more restrictive concept of a projective plane that is attributed to Bachmann, adding [[Pappus's hexagon theorem|Pappus's theorem]] to the list of axioms above (which eliminates [[non-Desarguesian plane]]s) and excluding projective planes over fields of characteristic 2 (those that do not satisfy [[Fano's axiom]]). The restricted planes given in this manner more closely resemble the [[real projective plane]].

== Perspectivity and projectivity ==
Given three non-[[collinear]] points, there are three lines connecting them, but with four points, no three collinear, there are six connecting lines and three additional "diagonal points" determined by their intersections. The science of projective geometry captures this surplus determined by four points through a quaternary relation and the projectivities which preserve the [[complete quadrangle]] configuration.

An [[harmonic quadruple]] of points on a line occurs when there is a complete quadrangle two of whose diagonal points are in the first and third  position of the quadruple, and the other two positions are points on the lines joining two quadrangle points through the third diagonal point.{{sfn|Halsted|1906|pp=15,16}}

A spatial [[perspectivity]] of a [[projective configuration]] in one plane yields such a configuration in another, and this applies to the configuration of the complete quadrangle. Thus harmonic quadruples are preserved by perspectivity. If one perspectivity follows  another the configurations follow along. The composition of two perspectivities is no longer a perspectivity, but a '''projectivity'''.

While corresponding points of a perspectivity all converge at a point, this convergence  is ''not'' true for a  projectivity that is ''not'' a perspectivity. In projective geometry the intersection of lines formed by  corresponding points of a projectivity in a plane are of particular interest. The set of such intersections is called a '''projective conic''', and in acknowledgement of the work of [[Jakob Steiner]], it is  referred to as a [[Steiner conic]].

Suppose a projectivity  is formed by two perspectivities centered on points ''A'' and ''B'', relating ''x'' to ''X'' by an intermediary ''p'':
: <math>x \ \overset{A}{\doublebarwedge}\ p \ \overset{B}{\doublebarwedge} \ X.</math>
The  projectivity is then <math>x \ \barwedge \ X .</math> Then given the projectivity <math>\barwedge</math> the induced conic is
: <math>C(\barwedge) \ = \ \bigcup\{xX \cdot yY : x \barwedge X \ \ \land \ \  y \barwedge Y \} .</math>

Given a conic ''C'' and a point ''P'' not on it, two distinct [[secant line]]s through ''P'' intersect ''C'' in four points. These four points determine a quadrangle of which ''P'' is a diagonal point. The line through the other two diagonal points is called the [[pole and polar|polar of ''P'']] and ''P ''is the '''pole''' of this line.{{sfn|Halsted|1906|p=25}} Alternatively, the polar line of ''P'' is the set of [[projective harmonic conjugate]]s of ''P'' on a variable secant line passing through ''P'' and ''C''.

== See also ==
{{colbegin|colwidth=22em}}
* [[Projective line]]
* [[Projective plane]]
* [[Incidence (mathematics)]]
* [[Fundamental theorem of projective geometry]]
* [[Projective line over a ring]]
* [[Grassmann–Cayley algebra]]
{{colend}}

== Notes ==
{{reflist|20em}}

== References ==
{{refbegin}}
* {{cite book |first=F. |last=Bachmann |title=Aufbau der Geometrie aus dem Spiegelungsbegriff |url=https://books.google.com/books?id=skGoBgAAQBAJ&pg=PR2 |date=2013 |orig-year=1959 |edition=2nd|publisher=Springer-Verlag |isbn=978-3-642-65537-1 }}
* {{cite book|last=Baer|first=Reinhold|title=Linear Algebra and Projective Geometry|year=2005|publisher=Dover|location=Mineola NY|isbn=0-486-44565-8}}
* {{cite book|last=Bennett|first=M.K.|title=Affine and Projective Geometry|year=1995|publisher=Wiley|location=New York|isbn=0-471-11315-8}}
* {{cite book|last1=Beutelspacher|first1=Albrecht|last2=Rosenbaum|first2=Ute|title=Projective Geometry: From Foundations to Applications|year=1998|publisher=Cambridge University Press|location=Cambridge|isbn=0-521-48277-1}}
* {{cite book|last=Casse|first=Rey|title=Projective Geometry: An Introduction|year=2006|publisher=Oxford University Press |isbn=0-19-929886-6}}
* {{cite book|last=Cederberg |first=Judith N. |title=A Course in Modern Geometries |publisher=Springer-Verlag |year=2001 |isbn=0-387-98972-2}}
* {{cite book |author-link=H. S. M. Coxeter |first=H.S.M. |last=Coxeter |title=The Real Projective Plane |publisher=Springer Verlag |edition=3rd |orig-year=1993 |year=2013 |isbn=  9781461227342 |url=https://books.google.com/books?id=uz3aBwAAQBAJ}}
* {{cite book |first=H.S.M. |last=Coxeter |title=Projective Geometry |publisher=Springer Verlag |edition=2nd |year=2003 |isbn=978-0-387-40623-7 }}
* {{cite book|last=Coxeter |first=H.S.M. |title=Introduction to Geometry |url= https://archive.org/details/introductiontoge0002coxe |url-access=registration |publisher=Wiley |year=1969 |isbn=0-471-50458-0}}
* {{cite book | last1=Dembowski | first1=Peter | title=Finite Geometries | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]], Band 44 | mr=0233275 | year=1968 | isbn=3-540-61786-8 | url-access=registration | url=https://archive.org/details/finitegeometries0000demb }}
* {{cite book |author-link=Howard Eves |first=Howard |last=Eves |title=Foundations and Fundamental Concepts of Mathematics |url=https://books.google.com/books?id=J9QcmFHj8EwC&pg=PP1 |date=2012 |orig-year=1997 |publisher=Courier Corporation |isbn=978-0-486-13220-4 |edition=3rd }}
* {{cite book|last=Garner|first=Lynn E.|title=An Outline of Projective Geometry|year=1981|publisher=North Holland |isbn=0-444-00423-8}}
* {{cite book |first=M.J. |last=Greenberg |title=Euclidean and Non-Euclidean Geometries: Development and History |url=https://books.google.com/books?id=4uw0dwi7bmQC |date=2008 |publisher=W. H. Freeman |isbn=978-1-4292-8133-1 |edition=4th}}
* {{cite book|first=G. B. |last=Halsted |authorlink=G. B. Halsted |year=1906 |url= https://archive.org/details/syntheticproject00halsuoft/page/n3/mode/2up |title=Synthetic Projective Geometry|publisher=New York Wiley }}
* {{cite book |first1=Richard |last1=Hartley |first2=Andrew |last2=Zisserman |title= Multiple view geometry in computer vision|publisher=Cambridge University Press |edition=2nd |year=2003 |isbn=0-521-54051-8 }}
* {{cite book |author-link=Robin Hartshorne |first=Robin |last=Hartshorne |title=Foundations of Projective Geometry |publisher=Ishi Press |edition=2nd |year=2009 |isbn=978-4-87187-837-1}}
* {{cite book |first=Robin |last=Hartshorne |title=Geometry: Euclid and Beyond |url=https://books.google.com/books?id=C5fSBwAAQBAJ |date=2013 |publisher=Springer |isbn=978-0-387-22676-7 |orig-year=2000}}
* {{cite book |author-link=David Hilbert |first1=D. |last1=Hilbert |first2=S. |last2=Cohn-Vossen |title=Geometry and the Imagination |url=https://books.google.com/books?id=7WY5AAAAQBAJ |year=1999 |publisher=American Mathematical Society |edition=2nd |isbn=978-0-8218-1998-2}}
* <cite id=refHughes1973>{{cite book |first1=D.R. |last1=Hughes |first2=F.C. |last2=Piper |title=Projective Planes |url=https://books.google.com/books?id=bKh6QgAACAAJ |year=1973 |publisher=Springer-Verlag |isbn=978-3-540-90044-3}}</cite>
* {{cite book|last=Mihalek|first=R.J.|title=Projective Geometry and Algebraic Structures|year=1972|publisher=Academic Press|location=New York|isbn=0-12-495550-9}}
* {{cite book|last=Polster |first=Burkard |title=A Geometrical Picture Book|url= https://archive.org/details/geometricalpictu0000pols |url-access=registration |publisher=Springer-Verlag |year=1998 |isbn=0-387-98437-2}}
* {{cite journal |doi=10.1007/BF02835009 |first=S. |last=Ramanan |title=Projective geometry |journal=Resonance |publisher=Springer India |issn=0971-8044 |volume=2 |issue=8 |pages=87–94 |date=August 1997 |s2cid=195303696 }}
* {{cite book|last=Samuel|first=Pierre|title=Projective Geometry|url=https://archive.org/details/projectivegeomet0000samu|url-access=registration|year=1988|publisher=Springer-Verlag |isbn=0-387-96752-4}}
* [[Luis Santaló|Santaló, Luis]] (1966) ''Geometría proyectiva'', Editorial Universitaria de Buenos Aires
* {{cite book|first1=Oswald|last1=Veblen|first2=J. W. A.|last2= Young|title=Projective Geometry|year=1938|place=Boston|publisher= Ginn & Co.|url= https://archive.org/details/117714799_001 |isbn=978-1-4181-8285-4}}
{{refend}}

== External links ==
{{Commons category}}
* [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.17.1329 Projective Geometry for Machine Vision] — tutorial by Joe Mundy and Andrew Zisserman.
* [http://xahlee.info/projective_geometry/projective_geometry.html Notes] based on Coxeter's ''The Real Projective Plane''.
* [http://lear.inrialpes.fr/people/triggs/pubs/isprs96/isprs96.html Projective Geometry for Image Analysis] — free tutorial by Roger Mohr and Bill Triggs.
* [http://www.geometer.org/mathcircles/projective.pdf Projective Geometry.] — free tutorial by Tom Davis.
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/burali-forti_-_grassman_and_proj._geom..pdf The Grassmann method in projective geometry] A compilation of three notes by Cesare Burali-Forti on the application of exterior algebra to projective geometry
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/burali-forti_-_diff._geom._following_grassmann.pdf C. Burali-Forti, "Introduction to Differential Geometry, following the method of H. Grassmann"] (English translation of book)
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/kummer_-_rectilinear_ray_systems.pdf E. Kummer, "General theory of rectilinear ray systems"] (English translation)
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/pasch_-_focal_and_singularity_surfaces.pdf M. Pasch, "On the focal surfaces of ray systems and the singularity surfaces of complexes"] (English translation)

{{Mathematical art}}

{{Authority control}}

{{DEFAULTSORT:Projective Geometry}}
[[Category:Projective geometry| ]]
[[Category:Geometry|P]]