Reasoning about binary topological relations

Abstract

A new formalism is presented to reason about topological relations. It is applicable as a foundation for an algebra over topological relations. The formalism is based upon the nine intersections of boundaries, interiors, and complements between two objects. Properties of topological relations are determined by analyzing the nine intersections to detect, for instance, symmetric topological relations and pairs of converse topological relations. Based upon the standard rules for the transitivity of set inclusion, the intersections of the composition of two binary topological relations are determined. 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 complete set of binary toplogical relations which result from the composition of two topological relations between n-dimensional point sets embedded in an n-dimensional space. While the combined topological relations are unique for some compositions, more than half of all possible compositions are underdetermined. Geometric prototypes are shown for the 2-dimensional case.