A Voronoi-based 9-intersection model for spatial relations

Models of spatial relations are a key component of geographical information science (GIS). EŒorts have been made to formally deŽ ne spatial relations. The foundationmodel for such a formal presentation is the 4-intersection model proposedby Egenhofer and Franzosa (1991). In this model, the topological relations between two simple spatial entities A and B are transformed into pointset topology problems in terms of the intersections of A’s interior and boundary with B’s interior and boundary. Later, Egenhofer and Herring (1991 ) extended this model to 9-intersection by addition of another element, i.e. the exterior of an entity, which is then deŽ ned as its complement.However, the use of its complement as the exterior of an entity causes the linear dependency between its interior, boundary and exterior. Thus such an extension from 4to 9-intersection should be of no help in terms of the number of relations. This can be conŽ rmed by the discovery of Egenhofer et al. (1993). The distinction of additional relations in the case where the co-dimensionis not zero is purely due to the adoption of deŽ nitions of the interior, boundary and exterior of entities in a lower dimensional to a higher dimension of space, e.g. lines in 1-dimensional space to 2-dimensional space. With such adoption, the topological convention that the boundary of a spatial entity separates its interior from its exterior is violated. It is such a change of conventional topological properties that causes the linear dependency between these three elements of a spatial entity (i.e. the interior, boundary and exterior) to disappear, thus making the distinction of additional relations possible in such a case (i.e. the co-dimension is not zero). It has been discussed that the use of Voronoi-regions of an entity to replace its complement as its exterior in the 9-intersection model would solve the problem (i.e. violation of topological convention) or would make this model become more comprehensive. Therefore, a Voronoi-based 9-intersection model is proposed. In addition to the improvement in the theoretical aspect, the Voronoi-based 9-intersection model (V9I ) can also distinguish additional relations which are beyond topological relations, such as high-resolution disjoint relations and relations of complex spatial entities. However, high-resolution disjoint relations deŽ ned by this model are not purely topological. In fact, it is a mixture of topology and metric. International Journal of Geographical Information Science ISSN 1365-8816 print/ISSN 1362-3087 online © 2001 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080 /13658810010017892