holesregions

Holesregions is a term used in geometry and related fields to describe planar regions that contain one or more holes. In this sense, a holesregion generalizes simple disks and annuli by allowing an arbitrary number of disjoint holes within a single connected component.

Definition: A holesregion in the plane is a connected subset whose boundary consists of one outer boundary and k inner boundaries, with k ≥ 0. The region can be described as the set difference between a bounding polygon (the exterior boundary) and the interiors of k disjoint holes. In three dimensions, the concept extends to regions with tunnels and cavities, where the boundary may include multiple voids.

Properties: For planar holesregions with k holes, the Euler characteristic is χ = 1 − k, and the region is not simply connected when k > 0. Its fundamental group is a free group on k generators. Boundaries are commonly represented as a polygon with holes or through boundary representations used in computer-aided design. Holesregions can be triangulated or meshed for numerical and visualization purposes, with special attention paid to inner boundaries to preserve topology.

Algorithms and representations: Holesregions are often stored as polygon-with-holes data structures, boundary representations, or planar graphs. Operations such as union, intersection, difference, offsetting, and meshing must handle inner boundaries correctly to avoid topological errors.

Applications: They appear in geographic information systems for land parcels with enclaves or exclaves, in CAD for parts with holes (for example, washers and gaskets), in computer graphics for modeling objects with internal cavities, and in topology as simple, non-simply connected domains for study.

See also: polygon with holes, planar graph, Euler characteristic, topology, boundary representation.