Editing Projective geometry (section)

== 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.