Computation of topological relations with 3-SRM.

Topological relation models are fundamental to spatial databases and GIS, providing a basis for reasoning about how spatial objects relate. Existing binary frameworks such as RCC-8 and the 9-Intersection Model effectively describe relations between two regions but cannot capture the global structure of configurations involving three spatial entities. To overcome this limitation, we propose a formally defined ternary intersection calculus, the Three-Simple-Region Model (3-SRM), for computing topological relations among three simple regions in 2D space. The model is constructed on the basis of three 3x3 matrices [Formula: see text], [Formula: see text], and [Formula: see text]. The configuration of [Formula: see text], [Formula: see text], and [Formula: see text] results in a total of 16 topological relations. The identified topological relations in 2D space among three spatial regions are disjoint, meet, covers, covered-by, equal, contain, inside, overlap, between, in-between, outer, inner, meet-inside, inside-meet, exterior meet, and boundary exterior meet. The model characterizes each triadic relation by rigorously evaluating the emptiness patterns of all interior–boundary–exterior intersections among the three regions, providing a natural extension of traditional binary frameworks while maintaining their fundamental topological semantics.