Crossingspoints

Crossingspoints is a term used in geometry and graph drawing to denote the points where two or more curves, lines, or trajectories intersect within a common space. The concept focuses on the geometric intersection itself, rather than the labeling of points as vertices in a graph.

Across contexts, crossingspoints can be simple (two paths intersecting at a single point) or multiple (three or more paths meeting at the same location). A key distinction is between crossingspoints and vertices: in many graph drawings a point that is a vertex remains a vertex, while interior intersections of edges that do not correspond to vertices are treated as crossingspoints. Some authors distinguish proper crossings, where the intersecting curves cross transversely, from touch or tangent contacts, where they merely touch.

Planarity and crossings: In a planar embedding of a graph, the number of crossingpoints is zero. Crossingspoints are central to the definition of crossing number, which measures the minimum possible number of such points over all drawings of the graph.

Detection and computation: Detecting crossingspoints typically uses geometric predicates such as orientation tests for line segments, robust arithmetic to handle collinear cases, and sweep-line algorithms for efficiency. In higher dimensions, crossingpoints can refer to intersection of manifolds or curves, with more complex algebraic methods required.

Applications: Crossingspoints matter in graph drawing, VLSI circuit layout, geographic information systems, robotics (motion planning), and computer-aided design, where minimizing or controlling intersections affects readability, performance, or feasibility.

See also: intersection, cross, vertex (graph theory), crossing number, planarity. History and terminology vary by field; the term "crossingspoints" is not universally adopted and may be used as a descriptive label in some texts.