incidencegeometrie

Incidence geometrie, also known as Inzidenzgeometrie, is a branch of geometry that studies the incidence relations between basic geometric objects such as points, lines and planes. The central notion is an incidence structure, consisting of a set of points P, a set of blocks L (typically lines, but also higher‑dimensional subspaces), and an incidence relation I ⊆ P × L that specifies which points lie on which blocks.

Axioms in incidence geometry vary with the chosen framework. Classical examples include projective and affine planes. In a projective plane any two distinct points determine a unique line, and any two distinct lines meet in a unique point. In an affine plane any two distinct points determine a unique line, and given a point not on a line there is a unique line through the point parallel to the given line. More general incidence structures may relax these conditions, leading to partial planes, configurations or other incidence structures.

Duality is a common theme: swapping the roles of points and blocks yields the dual incidence geometry, in which blocks play the role of points and vice versa. Incidence graphs, which are bipartite graphs between points and blocks, are often used to study these structures combinatorially.

Finite incidence geometries include finite projective and affine planes, and more general configurations described by parameters such as how many points lie on each line and how many lines pass through each point. These geometries connect to design theory, coding theory, finite geometry, and matroid theory, among other areas, and are studied for their combinatorial, geometric, and symmetry properties.