intersectan

Intersectan is a term used in theoretical and computational geometry to denote a class of combinatorial structures that encode the pattern of intersections between members of two or more families of geometric objects. In its common usage, an intersectan I is an incidence structure consisting of a finite set of points P, a finite collection of geometric objects L drawn from two base families (for example F and G), and an incidence relation R ⊆ P × L that records which points lie on which objects. The defining feature is that the incidence pattern reflects the actual intersection behavior of the underlying object families.

A typical construction views P as the set of all intersection points formed by pairs (f,g) with f ∈ F and g ∈ G, and L as the union of the originating objects from both families. An element p ∈ P is incident with a member l ∈ L exactly when p lies on l. This framework allows for multiple incidences: a single point can lie on several objects if several intersections occur at the same location.

Examples include grids formed by intersecting vertical and horizontal lines, which yield a regular intersectan with a highly symmetric incidence pattern, and more complex arrangements such as families of curves where intersections reflect a prescribed equation or constraint.

Properties of intersectans include their planarity, symmetry (automorphism groups), and relationships to incidence graphs and dual structures. They are studied for insights in computational geometry, graph drawing, mesh generation, and pattern recognition.

History notes that the term intersectan appears in discussions of incidence geometry and related fields as a descriptive concept rather than a single standardized formalism, with varying definitions across authors. See also incidence geometry, intersection graphs, and incidence structures.