Abstract. Analysis of global geographic phenomena requires non-planar models. In the past, models for topological relations have focused either on a twodimensional or a three-dimensional space. When applied to the surface of a sphere, however, neither of the two models suffices. For the two-dimensional planar case, the eight binary topological relations between spatial regions are well known from the 9-intersection model. This paper systematically develops the binary topological relations that can be realized on the surface of a sphere. Between two regions on the sphere there are three binary relations that cannot be realized in the plane. These relations complete the conceptual neighborhood graph of the eight planar topological relations in a regular fashion, providing evidence for a regularity of the underlying mathematical model. The analysis of the algebraic compositions of spherical topological relations indicates that spherical topological reasoning often provides fewer ambiguities than planar topological reasoning. Finally, a comparison with the relations that can be realized for one-dimensional, ordered cycles draws parallels to the spherical topological relations. 1

The 4-intersection, a model for the representation of topological relations between 2-dimensional objects with connected boundaries and connected interiors, is extended to cover topological relations between 2-dimensional objects with arbitrary holes, called regions with holes. Each region with holes is represented by its generalized region---the union of the object and its holes--- and the closure of each hole. The topological relation between two regions with holes, A and B, is described by the set of all individual topological relations between (1) A 's generalized region and B's generalized region, (2) A 's generalized region and each of B's holes, (3) B's generalized region with each of A 's holes, and (4) each of A 's holes with each of B's holes. As a side product, the same formalism applies to the description of topological relations between 1-spheres. An algorithm is developed that minimizes the number of individual topological relations necessary to describe a configuration completely. This model of representing complex topological relations is suitable for a multi-level treatment of topological relations, at the least detailed level of which the relation between the generalized regions prevails. It is shown how this model applies to the assessment of consistency in multiple representations when, at a coarser level of less detail, regions are generalized by dropping holes.

Based on the 9-intersection for binary topological relations, two models of conceptual neighborhoods among topological relations between a line and a region are developed. The snapshot model derives the neighborhoods by comparing pairs of topological relations and selects neighbors based on least noticeable differences, whereas the smooth-transition model develops neighborhoods based on the knowledge of the deformations that may change a topological relation. The resulting similarity diagrams show some differences, which were compared with the results from tests in which human subjects were asked to organized lineregion relations into groups of similar relations. The groupings the subjects made indicate that the smooth-transition model captures more important aspects of the similarity of topological lineregion relations than the snapshot model.

Results from human subjects testing are used to calibrate the meaning of 'the road crosses the park' and three other similar sentences in English. Sixty stimulus maps represent two or more examples of each of the 19 line-region spatial relationships defined by the 9-intersection model. The subjects' mean agreement that the sentence applies to each map is expressed as a weighted sum of binary variables that define which of the 9-intersection categories the relationship between road and park on the map belongs to. Statistically, regression equations based on 9-intersection topology alone account for between 60 percent and 90 percent in the variation in mean subjects' responses. 'Crosses' and 'goes across' appear to be more sensitive to variations in metric properties that are 'goes through' and 'enters'. Results confirm that the method used has potential for defining cognitively meaningful spatial predicates and for comparing the meanings of similar terms in different natural languages. ...

Thirty-two native-speakers of English drew examples of roads that fit the spatial relations to a park, as indicated in 64 English-language sentence. Also, 19 native speakers of Spanish drew examples for 43 Spanish-language sentences. Then, each of the 2856 drawings (2044 English and 812 Spanish) was classified according to the road-park spatial relation into one of 19 categories of spatial relations defined by the 9-intersection model. For each of the 107 sentences, the proportion of subjects drawing each relation was determined. These counts indicate the prototypical spatial relations corresponding to each sentence. Results confirm our previous work on prototypical spatial relations: 2522 of the 2856 drawings (88 percent) fell into just 5 spatial relations, roughly equivalent to 'inside', 'outside' (disjoint), 'enters', 'crosses', and 'goes to'. Evidently, there are many ways in English and Spanish to express relations approximately corresponding to the English inside, o...