Editing Incidence geometry (section)

== Partial linear spaces ==
Incidence structures that are most studied are those that satisfy some additional properties (axioms), such as [[projective plane]]s, [[Affine plane (incidence geometry)|affine planes]], [[generalized polygon]]s, [[partial geometry|partial geometries]] and [[near polygon]]s. Very general incidence structures can be obtained by imposing "mild" conditions, such as:

A [[partial linear space]] is an incidence structure for which the following axioms are true:<ref>{{harvnb|Moorhouse|loc=pg.5}}</ref>
* Every pair of distinct points determines at most one line.
* Every line contains at least two distinct points.

In a partial linear space it is also true that every pair of distinct lines meet in at most one point. This statement does not have to be assumed as it is readily proved from axiom one above.

Further constraints are provided by the regularity conditions:

'''RLk''': Each line is incident with the same number of points. If finite this number is often denoted by {{math|''k''}}.

'''RPr''': Each point is incident with the same number of lines. If finite this number is often denoted by {{math|''r''}}.

The second axiom of a partial linear space implies that {{math|''k'' > 1}}. Neither regularity condition implies the other, so it has to be assumed that {{math|''r'' > 1}}.

A finite partial linear space satisfying both regularity conditions with {{math|''k'', ''r'' > 1}} is called a ''tactical configuration''.<ref>{{harvnb|Dembowski|1968|page=5}}</ref> Some authors refer to these simply as ''[[configuration (geometry)|configurations]]'',<ref>{{citation
| last=Coxeter
| first=H. S. M.
| author-link=H.S.M. Coxeter
| title=Introduction to Geometry
| location=New York
| publisher=John Wiley & Sons
| year=1969
| page=233
| isbn=978-0-471-50458-0}}</ref> or ''projective configurations''.<ref>{{citation
 | last1 = Hilbert | first1 = David | author1-link = David Hilbert
 | last2 = Cohn-Vossen | first2 = Stephan | author2-link = Stephan Cohn-Vossen
 | edition = 2nd
 | isbn = 978-0-8284-1087-8
 | pages = 94–170
 | publisher = Chelsea
 | title = Geometry and the Imagination
 | year = 1952}}</ref> If a tactical configuration has {{math|''n''}} points and {{math|''m''}} lines, then, by double counting the flags, the relationship {{math|1=''nr'' = ''mk''}} is established. A common notation refers to {{math|(''n''<sub>''r''</sub>, ''m''<sub>''k''</sub>)}}-''configurations''. In the special case where {{math|1=''n'' = ''m''}} (and hence, {{math|1=''r'' = ''k''}}) the notation {{math|(''n''<sub>''k''</sub>, ''n''<sub>''k''</sub>)}} is often simply written as {{math|(''n''<sub>''k''</sub>)}}.

[[File:Sample Incidence.jpg|thumb|Simplest non-trivial linear space]]

A [[Linear space (geometry)|''linear space'']] is a partial linear space such that:<ref>{{harvnb|Moorhouse|loc=pg. 5}}</ref>
* Every pair of distinct points determines exactly one line.

Some authors add a "non-degeneracy" (or "non-triviality") axiom to the definition of a (partial) linear space, such as:

* There exist at least two distinct lines.<ref>There are several alternatives for this "non-triviality" axiom. This could be replaced by "there exist three points not on the same line" as is done in {{harvtxt|Batten|Beutelspacher|1993|loc=pg. 1}}. There are other choices, but they must always be ''existence'' statements that rule out the very simple cases to exclude.</ref>

This is used to rule out some very small examples (mainly when the sets {{math|''P''}} or {{math|''L''}} have fewer than two elements) that would normally be exceptions to general statements made about the incidence structures. An alternative to adding the axiom is to refer to incidence structures that do not satisfy the axiom as being ''trivial'' and those that do as ''non-trivial''.

Each non-trivial linear space contains at least three points and three lines, so the simplest non-trivial linear space that can exist is a triangle.

A linear space having at least three points on every line is a [[Sylvester–Gallai configuration|Sylvester–Gallai design]].