# Computation of topological relations with 3-SRM

## TL;DR

This paper introduces a new model for computing topological relations among three spatial regions in 2D space, extending traditional binary frameworks.

## Contribution

The paper proposes the 3-SRM, a formally defined ternary intersection calculus for triadic spatial relations.

## Key findings

- The 3-SRM model identifies 16 distinct topological relations among three spatial regions.
- Each relation is characterized by evaluating the emptiness patterns of interior–boundary–exterior intersections.
- The model extends binary frameworks while preserving their topological semantics.

## Abstract

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 \documentclass[12pt]{minimal}
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\begin{document}$$9I_C$$\end{document} 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.