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.
Incidence structures
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 terminology 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.
Distance in an incidence structure There is no natural concept of distance (a metric) in an incidence structure. However, a combinatorial metric does exist in the corresponding incidence graph (Levi graph), namely the length of the shortest 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.
Partial linear spaces Incidence structures that are most studied are those that satisfy some additional properties (axioms), such as projective planes, affine planes, generalized polygons, partial geometries and near polygons. 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:
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 k. RPr: Each point is incident with the same number of lines. If finite this number is often denoted by r. The second axiom of a partial linear space implies that k > 1. Neither regularity condition implies the other, so it has to be assumed that r > 1. A finite partial linear space satisfying both regularity conditions with k, r > 1 is called a tactical configuration. Some authors refer to these simply as configurations, or projective configurations. If a tactical configuration has n points and m lines, then, by double counting the flags, the relationship nr = mk is established. A common notation refers to (nr, mk)-configurations. In the special case where n = m (and hence, r = k) the notation (nk, nk) is often simply written as (nk).
A linear space is a partial linear space such that:
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. This is used to rule out some very small examples (mainly when the sets P or 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 design.
Fundamental geometric examples Some of the basic concepts and terminology arises from geometric examples, particularly projective planes and affine planes.
Projective planes
A projective plane is a linear space in which:
Every pair of distinct lines meet in exactly one point, and that satisfies the non-degeneracy condition:
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