The 9+-Intersection: A Universal Framework for Modeling Topological Relations

The 9-intersection is an extension of the 9-intersection, which distinguishes the topological relations between various spatial objects by the pattern of a nested matrix. This paper develops a small set of constraints on this matrix, which is applicable to arbitrary pairs of spatial objects in various spaces. Based on this set of universal constraints, the sets of matrix patterns, each representing a candidate for topological relations, are derived for every possible pair of basic objects (points, directed/non-directed line segments, regions, and bodies) embedded in R, R, R, S, and S. The derived sets of candidates are consistent with the sets of topological relations ever identified, as well as yield the identification of some missing sets of topological relations. Finally, the topological relations between a region and a region with a hole in R and S are identified to demonstrate the applicability of our approach to deriving topological relations between more complicated objects.