Incidence geometry

Incidence geometry studies which points, lines, and higher-dimensional subspaces are related by containment or intersection, often without specifying distances or angles.

Incidence structure: A system of objects and a specified relation indicating which objects are incident with one another. This abstract framework captures the point–line relations studied in incidence geometry.
Szemerédi–Trotter theorem: A theorem bounding the number of incidences between a finite set of points and a finite set of lines in the real plane. It is a landmark quantitative result for planar point-line incidences.
Erdős distinct distances problem: The problem of determining the minimum number of distinct pairwise distances among a finite set of planar points. Incidence bounds help control point pairs at specified distances.
Fano plane: The projective plane with seven points and seven lines, each line containing three points. It is the smallest finite projective plane and a compact incidence model.
Euclidean geometry: The geometry of points and shapes governed by distance, angle, and parallelism in flat space. Unlike Euclidean geometry, incidence geometry can ignore distances and angles entirely.
Projective plane: A point-line incidence structure in which every two points determine one line and every two lines meet at one point. Projective planes are central models built from incidence axioms.
Crossing lemma: A graph-theoretic inequality relating the number of edges in a drawing to its number of crossings. It helps derive upper bounds for incidences by drawing graphs from geometric configurations.
Additive combinatorics: The study of additive patterns in sets, using tools from number theory, combinatorics, and harmonic analysis. Geometric incidence estimates can bound additive structures through point-line transformations.
Desargues configuration: A configuration of ten points and ten lines in which each line contains three points and each point lies on three lines. It illustrates a highly symmetric pattern of point-line incidences.
Metric geometry: The study of spaces equipped with a distance function satisfying metric axioms. Metric geometry organizes spaces by distances rather than by incidence relations.

Linked from 32 pages