Deriving the Composition of Binary Topological Relations

Abstract

A new formalism is presented to derive knowledge about the composition of two binary topological relations over a common object. The formalism is based on a topological data model and compares the nine empty and non-empty intersections of interiors, boundaries and exteriors between two objects. Based upon the transitivity of set inclusion, the intersections of the composed topological relations are derived. These intersections are then matched with the intersections of the eight fundamental topological relations, giving an interpretation to the composition of topological relations. The result of this study is the composition table of the eight binary topological relations that exist between n-dimensional point sets with a co-dimension of 0. While the combined topological relations are unique for some compositions, more than half of all possible compositions are disjunctions of possible relations. Geometric prototypes are shown for the two-dimensional case. The composition table enables topological reasoning at the conceptual level of relations, rather than having to calculate all relations from the representation of the spatial objects. Its practical value is that it can serve as in a computational model for an assessment of whether a set of topological predicates is consistent or not and in spatial query processing when no explicit information about spatial relations is available.