On the Internal Topological Structure of Plane Regions

TL;DR

This paper investigates the internal topological structure of complex bounded plane regions, emphasizing semi-algebraic regions that can model intricate spatial entities and are closed under set operations.

Contribution

It introduces the study of semi-algebraic regions' internal topology, addressing limitations of simple regions in representing complex spatial phenomena.

Findings

Semi-algebraic regions can model complex plane regions with holes and islands.

These regions are closed under union, intersection, and difference operations.

The approach enhances the understanding of spatial object topology in various disciplines.

Abstract

The study of topological information of spatial objects has for a long time been a focus of research in disciplines like computational geometry, spatial reasoning, cognitive science, and robotics. While the majority of these researches emphasised the topological relations between spatial objects, this work studies the internal topological structure of bounded plane regions, which could consist of multiple pieces and/or have holes and islands to any finite level. The insufficiency of simple regions (regions homeomorphic to closed disks) to cope with the variety and complexity of spatial entities and phenomena has been widely acknowledged. Another significant drawback of simple regions is that they are not closed under set operations union, intersection, and difference. This paper considers bounded semi-algebraic regions, which are closed under set operations and can closely approximate most plane regions arising in practice.