Pointsregions

Pointsregions is a concept in computational geometry that refers to a partition of a domain into regions associated with a finite set of anchor points. The idea is to assign to each point an influence region where that point is most relevant according to a chosen rule.

Formally, let P = {p1,...,pk} be points in a domain D. A pointsregions partition is a collection R = {R1,...,Rk} of subsets of D that are pairwise disjoint and cover D, where each R_i consists of points x with s_i(x) ≤ s_j(x) for all j ≠ i for some scoring functions s_i. In the simplest case with s_i(x) = distance(x, p_i) under Euclidean distance, the R_i are Voronoi cells.

Generalizations replace distance with other metric-based scores or introduce weights. Additively weighted regions use s_i(x) = distance(x, p_i) + w_i, while multiplicatively weighted variants use s_i(x) = w_i · distance(x, p_i). Other scoring functions may account for terrain, direction, or obstacles. These variants model influence zones in nonuniform environments and can yield non-convex or disconnected regions.

Boundaries are loci where two scoring functions tie. Depending on the rules, boundaries may be curved or linear and can have singularities. Construction algorithms extend those for Voronoi diagrams, including sweep-line, incremental insertion, and divide-and-conquer methods, with practical complexities near-linear to O(n log n) for standard cases.

Applications include geographic information systems, facility location, robotics, computer graphics, and mesh generation, where pointsregions model influence, proximity, or clustering. They also provide a framework for assigning data points to representative centers in analyses.

See also: Voronoi diagram; power diagram; Apollonius diagram; proximity graph.