pointregion

Pointregion is a term used in geometry and related disciplines to describe a region in space that is defined with respect to a distinguished point. The exact meaning varies by context, but it generally denotes a neighborhood, cell, or zone associated with a single reference point.

In formal terms, a pointregion can be seen as a subset R of a metric space (X, d) determined by a rule that references a point p ∈ X. A simple example is a disk around p with radius r: D(p, r) = { x ∈ X : d(x, p) ≤ r }. More generally, pointregions may be defined by comparisons to other reference points. For instance, the Voronoi region of p within a finite site set S is V(p) = { x ∈ X : d(x, p) ≤ d(x, q) for all q ∈ S \ {p} }, effectively partitioning space into cells associated with each site.

Alternative interpretations appear in image processing and geographic information systems. In image analysis, a pointregion might refer to a region-growing zone seeded at a point p, where neighboring pixels are added according to a similarity predicate relative to p. In GIS, a pointfeature can be associated with a buffered area to form a pointregion that represents influence or uncertainty around the location.

Applications of the concept include spatial partitioning, neighborhood analysis, mesh generation, clustering, and region-based image segmentation. The term is not rigorously standardized, and its precise definition is chosen to fit the problem context.

See also: Voronoi diagram, neighborhood, disk, buffer, region growing, metric space.