Topological Reasoning in Geographic SpaceTopological Reasoning in Geographic SpaceThe course starts with a discussion of Naive Geography as a motivation for dealing with properties of large-scale, geographic space in an intuitive way. Based on this setting, we examine algebraic properties of abstract spatial concepts, such as coincidence, nearness, connectivity, containment, and support, leading to the importance of topology and topological space as a high-level, qualitative framework for spatial reasoning. We review relevant concepts from point-set topology, in particular the notions of interior and boundary.
The core for reasoning about topological information is the set of binary topological relations that are determined by the 4-intersection and 9-intersection. We discuss in detail these methods, including the derivation of the set of binary topological relations that can be realized for certain types of objects (regions, lines, points) and for different kinds of embedding spaces (2-dimensional plane, 3-dimensional space, surface of a sphere). Since the 9-intersection lends itself to a formal analysis of properties of topological relations, we study how spatial information can be derived from examining properties of the 9-intersection matrices.
A key for topological reasoning is the composition over topological relations, deriving the unknown topological relation(s) between A and C from A topologicalRelation1 B and B topologicalRelation2 C. We employ a systematic mechanism that allows for the derivation of all compositions within a comprehensive framework. With these compositions we demonstrate that the set of binary topological relations forms a relation algebra and exploit mechanisms from constraint satisfaction to either derive topological information in a network of incomplete topological relations or to determine whether a qualitative topological relation description is consistent or not. We also demonstrate how these methods can be used to describe the topology (and topological relations) of more complex spatial objects, such as regions with holes or regions with separations.
This week-long short course wraps up with a review of the state of the art of these methods in commercial systems and international standards, and a discussion of open research questions.
[ Max J. Egenhofer | NCGIA-UMaine | Department of Spatial Information Science and Engineering ]