We report a corrected version of the algorithms to compute ternary projective relations between regions appeared in E. Clementini and R. Billen, "Modeling and computing ternary projective relations between regions," IEEE Transactions on Knowledge and Data Engineering, vol. 18, pp. 799-814, 2006.

This paper introduces a reasoning system based on a previously developed model for ternary projective relations between spatial objects. The model applies to spatial objects of the kind point and region is based on basic projective invariants and takes into account the size and shape of the three objects that are involved in a relation. The reasoning system proposes a set of permutation and composition rules, which allow the inference of unknown relations from given ones.

Current spatial database systems offer limited querying capabilities beyond binary topological relations. This paper introduces a model for projective relations between regions to support other qualitative spatial queries. The relations are ternary because they are based on the collinearity invariant of three points under projective geometry. The model is built on a partition of the plane into separate zones that are obtained from projective properties of two reference objects: then, by considering the empty/nonempty intersections of a primary object with these zones, the model is able to distinguish between 34 different projective relations. Then, the paper proposes original algorithms for computing the relations under the assumption that regions of the plane are stored as vector polygons in a spatial database. These algorithms run in optimal O(nlogn) time

A set of topological invariants for relations between lines embedded in the 2-dimensional Euclidean space is given. The set of invariants is proven to be necessary and sufficient to characterize topological equivalence classes of binary relations between simple lines. The topology of arbitrarily complex geometric scenes is described with a variation of the same set of invariants. Polynomial time algorithms are given to assess topological equivalence of two scenes. Invariants and efficient algorithms is due to application areas of spatial database systems where a model for describing topological relations between planar features is sought.

The 4-intersection, a model for the representation of topological relations between two-dimensional objects with connected boundaries and connected interiors, is extended to cover topological relations between two-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.

Two levels of data abstraction, namely conceptual and logical, are essential in the process of designing an object-oriented geographic database. The aim of the present article is to establish a consistent framework for this bilevel design. At the conceptual level, we develop an object-oriented model useful for organizing the designer's knowledge about a geographic application in terms of the basic concepts of the object-oriented paradigm. Both conceptual abstraction primitives (namely, classification, generalization, and aggregation) and a spatial abstraction primitive (namely, location) are discussed. The dynamic part of the model is due to the methods associated with objects. At the logical level, we propose an object-oriented structure of classes and instances, which is uniform with the conceptual model and, therefore, facilitates the conceptual logical mapping. On the whole, the model-based approach adopted in this article offers object orientation, conceptual abstraction, and system extendibility. © 1994 John Wiley & Sons, Inc.

In geographic maps, objects are represented in a context extremely rich of relationships among them, either one-to-one or many-to-many. Unfortunately, all these relationships can not be directly represented in the database scheme. Nevertheless, it should be possible to show them to the user during the interaction. To manage this problem, the authors refer to a two-level database architecture with an object-oriented data model at the bottom level (the logic level), and an interaction model at the top level. The spatial data model given is an essential part of the interaction model. The spatial data model is a conceptual tool that enhances the object-oriented model with spatial relationships between objects. The basic objectives of the spatial data model are outlined: to point out the relationships among objects to be shown in the views, to define data structures suitable for navigating among database objects, and to provide a simple and powerful calculus for posing spatial queries against the database.>