Editing Incidence geometry (section)

== Incidence structures ==
{{main|incidence structure}}

An ''incidence structure'' {{math|(''P'', ''L'', I)}} consists of a set {{math|''P''}} whose elements are called ''points'', a disjoint set {{math|''L''}} whose elements are called ''lines'' and an ''incidence relation'' {{math|I}} between them, that is, a subset of {{math|''P'' × ''L''}} whose elements are called ''flags''.<ref>Technically this is a rank two incidence structure, where rank refers to the number of types of objects under consideration (here, points and lines). Higher ranked structures are also studied, but several authors limit themselves to the rank two case, and we shall do so here.</ref> If {{math|(''A'', ''l'')}} is a flag, we say that {{math|''A''}} is ''incident with'' {{math|''l''}} or that {{math|''l''}} is incident with {{math|''A''}} (the terminology is symmetric), and write {{math|''A'' I ''l''}}. Intuitively, a point and line are in this relation if and only if the point is ''on'' the line. Given a point {{math|''B''}} and a line {{math|''m''}} which do not form a flag, that is, the point is not on the line, the pair {{math|(''B'', ''m'')}} is called an ''anti-flag''.

===Distance in an incidence structure===
There is no natural concept of distance (a [[Metric (mathematics)|metric]]) in an incidence structure. However, a combinatorial metric does exist in the corresponding [[Levi graph|incidence graph (Levi graph)]], namely the length of the shortest [[Path (graph theory)|path]] between two vertices in this [[bipartite graph]]. The distance between two objects of an incidence structure – two points, two lines or a point and a line – can be defined to be the distance between the corresponding vertices in the incidence graph of the incidence structure.

Another way to define a distance again uses a graph-theoretic notion in a related structure, this time the ''collinearity graph'' of the incidence structure. The vertices of the collinearity graph are the points of the incidence structure and two points are joined if there exists a line incident with both points. The distance between two points of the incidence structure can then be defined as their distance in the collinearity graph.

When distance is considered in an incidence structure, it is necessary to mention how it is being defined.