Incidence geometry

In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An incidence structure is what is obtained when all other concepts are removed and all that remains is the data about which points lie on which lines. Even with this severe limitation, theorems can be proved and interesting facts emerge concerning this structure. Such fundamental results remain valid when additional concepts are added to form a richer geometry. It sometimes happens that authors blur the distinction between a study and the objects of that study, so it is not surprising to find that some authors refer to incidence structures as incidence geometries.

Incidence structures arise naturally and have been studied in various areas of mathematics. Consequently there are different terminologies to describe these objects. In graph theory they are called hypergraphs, and in combinatorial design theory they are called block designs. Besides the difference in terminology, each area approaches the subject differently and is interested in questions about these objects relevant to that discipline. Using geometric language, as is done in incidence geometry, shapes the topics and examples that are normally presented. It is, however, possible to translate the results from one discipline into the terminology of another, but this often leads to awkward and convoluted statements that do not appear to be natural outgrowths of the topics. In the examples selected for this article we use only those with a natural geometric flavor.

A special case that has generated much interest deals with finite sets of points in the Euclidean plane and what can be said about the number and types of (straight) lines they determine. Some results of this situation can extend to more general settings since only incidence properties are considered.

An incidence structure (P, L, I) consists of a set P whose elements are called points, a disjoint set L whose elements are called lines and an incidence relation I between them, that is, a subset of P × L whose elements are called flags. If (A, l) is a flag, we say that A is incident with l or that l is incident with A (the relation is symmetric), and write 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 B and a line m which do not form a flag, that is, the point is not on the line, the pair (B, m) is called an anti-flag.

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