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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(9I_A\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(9I_B\)</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(9I_C\)</EquationSource> </InlineEquation>. The configuration of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(9I_A\)</EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(9I_B\)</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(9I_C\)</EquationSource> </InlineEquation> 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.</p>

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Computation of topological relations with 3-SRM

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月23日

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  • Nivedita P. Totad,
  • Girish M. Sajjanshettar,
  • Prakash K. Aithal

摘要

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 \(9I_A\) , \(9I_B\) , and \(9I_C\) . The configuration of \(9I_A\) , \(9I_B\) , and \(9I_C\) 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.