On the equivalence of topological relations

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Abstract
The 4-intersection, a model for binary topological relations, is based on the intersections of the boundaries and interiors of two point sets in a topological space, considering the content invariant (i.e., emptiness/non-emptiness) of the intersections. If the 4-intersections of two pairs of point sets have different contents, then their topological relations are different as well; however, the reverse cannot be stated as there may be different topological relations that map onto a 4-intersection with the same content. This paper refines the model of empty/non-empty 4-intersections with further topological invariants to account for more details about topological relations. The invariants used are the dimension of the components, their types (touching, crossing, and different refinements of crossings), their relationships with respect to the exterior neighbourhoods, and the sequence of the components. These invariants, applied to non-empty boundary-boundary intersections, comprise a classification invariant for binary topological relations between homogeneously 2-dimensional, connected point sets (disks) in the plane such that if two different 4-intersections with the necessary invariants are equal, then their topological relations are identical. The model presented applies to processing G1S queries about whether or not two pairs of spatial objects have the same topological relation and gives rise to the formal definition of topological similarity.

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