A Critical Comparison of the 4-Intersection and 9-Intersection Models for Spatial Relations: Formal Analysis*

Abstract

Two formalisms for binary topological spatial relations are compared for their expressive power. The 4-intersection considers the two objects’ interiors and boundaries and analyzes the intersections of these four object parts for their content (i.e., emptiness and non-emptiness). The 9-intersection adds to the 4-intersection the intersections with the two objects’ complements. The major results are (1) for objects with co-dimension 0, the 4-intersection and the 9-intersection with the content invariant provide the same results; and (2) for objects with co-dimension > 0, the 9intersection with the content invariant provides more details than the 4-intersection. These additional details are crucial to determine when two objects are equal. It is also demonstrated that the additional details can provide crucial information when specifying the semantics of spatial relations in GIS query languages. * This work was partially supported through the NCGIA by NSF grant No. SES-8810917. Additionally, Max Egenhofer’s work is also supported by NSF grant No. IRI-9309230, a grant from Intergraph Corporation, and a University of Maine Summer Faculty Research Grant. Some of the ideas were refined while on a leave of absence at the Universita di L’Aquila, Italy, partially supported by the Italian National Council of Research (CNR) under grant No. 92.01574.PF69. Jayant Sharma is partially supported by a University of Maine Graduate Research Assistantship (UGRA). Max J. Egenhofer, Jayant Sharma, and David Mark A Critical Comparison of the 4-Intersection and 9-Intersection Models for Spatial Relations: Formal Analysis In: R. McMaster and M. Armstrong (eds), Autocarto 11.