Topological relationships between complex spatial objects

TLDR

Topological relationships among spatial objects have long been studied across AI, cognitive science, linguistics, robotics, and GIS, yet they have only been defined for simple abstractions such as points, lines, and regions. This work addresses the gap created by complex spatial data types by defining and enumerating the topological relationships that apply to them. The authors introduce versatile data types for complex points, lines, and regions, use the 9‑intersection model to derive all mutually exclusive relationships for every type combination, and propose cluster predicates and predicate groups to manage the resulting large set. Completeness and mutual exclusivity of the relationships are proven using a proof‑by‑constraint‑and‑drawing technique.

Abstract

For a long time topological relationships between spatial objects have been a focus of research in a number of disciplines like artificial intelligence, cognitive science, linguistics, robotics, and spatial reasoning. Especially as predicates they support the design of suitable query languages for spatial data retrieval and analysis in spatial databases and geographical information systems (GIS). Unfortunately, they have so far only been defined for and applicable to simplified abstractions of spatial objects like single points, continuous lines, and simple regions. With the introduction of complex spatial data types an issue arises regarding the design, definition, and number of topological relationships operating on these complex types. This article closes this gap and first introduces definitions of general and versatile spatial data types for complex points , complex lines , and complex regions . Based on the well known 9-intersection model, it then determines the complete sets of mutually exclusive topological relationships for all type combinations. Completeness and mutual exclusion are shown by a proof technique called proof-by-constraint-and-drawing . Due to the resulting large numbers of predicates and the difficulty of handling them, the user is provided with the concepts of topological cluster predicates and topological predicate groups , which permit one to reduce the number of predicates to be dealt with in a user-defined and/or application-specific manner.