A Formal Definition of Binary Topological Relationships

The exploration of spatial relationships is a multi-disciplinary effort involving researchers from linguistics, cognitive science, psychology, geography, cartography, semiology, computer science, surveying engineering, and mathematics. Terms like and or and are not as clearly understood as the standard relationships between integer numbers. The treatment of relationships among spatial objects is an essential task in geographic data processing and CAD/CAM. Spatial query languages, for example, must offer terms for spatial relationships; spatial database management systems need algorithms to determine relationships. Hence, a formal definition of spatial relationships is necessary to clarify the users’ diverse understanding of spatial relationships and to actually deduce relationships among spatial objects. Based upon such formalisms, spatial reasoning and inference will be possible. The topological relationships are a specific subset of the large variety of spatial relationships. They are characterized by the property to be preserved under topological transformations, such as translation, rotation, and scaling. A model of topological relations is presented which is based upon fundamental concepts of algebraic topology in combination with set theory. Binary topological relationships may be defined in terms of the boundaries and interiors of the two objects to be compared. A formalism is developed which identifies 16 potential relationships. Prototypes are shown for the eight relationships that may exist between two objects of the same dimension embedded in the corresponding space.