Object representation and reasoning in vector based geographic information systems (GIS) is based on Euclidean geometry. Euclidean geometry is built upon Euclid's first postulate, stating that two points uniquely determine a line. This postulate makes geometric constructions unambiguous and thereby provides the foundation for consistent geometric reasoning. It holds for exact coordinate points and lines, but is violated, if points and lines are allowed to have extension. As an example for a point that has extension consider a point feature that represents the city of Vienna in a small scale GIS map representation. Geometric constructions with such a point feature easily produce inconsistencies in the data. The present paper addresses the issue of consistency by formalizing Euclid's first postulate for geometric primitives that have extension.

It has been pointed out repeatedly that spatial operations must be extended to include support for 3D and moving objects. The attempt to code by hand each spatial operation for each data type (e.g., static 2D, moving 2D, static 3D, and moving 3D) is forbidding and has led to specific solutions for particular purposes. In this paper, we have advocated an operation-independent approach to extend 2D spatial operations to 3D and moving objects. The approach is based on implementation of the concepts of n-dimensional geometry through definition of transformations between domains called "lifting". It is explained via some sample spatial operations and then the implementation results for convex hull computation are represented.

Consistency constraints placed on a database to assure, that values incorporated in the database are consistent, are a well known foundation of Geographical Information Systems. Unfortunately in real situations rules for consistency constraints are not so clear, and inconsistent ontologies are common place, not least in geographical information, covering as it does a much wider realm than many other information systems I have suggested elsewhere 5-tiers of ontology for GIS. Such an ontology can integrate different ontological approaches in a unified system. In this paper the relation of the 5-tier ontology and consistency constraints is explored, and it is shown that different constraints are appropriate to different tiers.

Categorical coverages are popular for the presentation of thematic data in cartographic form and have gained widespread use. Routines for their preparation are included in most GIS software. Categorical coverages are simple to implement and easy to understand. The popularity of the categorical coverage is closely related to the intuitive rules that determine its behaviour. This paper presents a formalization of these rules based on the concepts of a partition of space and the refinement of such partitions of space. A family of categorical coverages is defined as all categorical coverages that can be produced from a data set through aggregations of categories. It is shown that categorical coverages are partially ordered by refinement of partitions. This ordering is preserved in the mapping from the ordering of the partition of the attribute domain to the partition of space. This formalization is not only useful for implementors of GIS, but the list of possible operations and their properties is the base for the design of a user interface to produce and manipulate categorical coverages. The thematic definition of categorical coverages is directly related to the cartographic rendering. Changes in the thematic definition, i.e., changes of the map legend, can be automatically propagated to the rendering. Limitation of cartographic rendering, e.g., resolution of the screen, limits the presentation of the result. These influences from the rendering cannot automatically be translated into changes in the subdivisions of the categories. This may explain why good graphical presentation is still an art and the production of thematic maps is not fully automated.

. Geographers use spatial reasoning extensively in large-scale spaces, i.e., spaces that cannot be seen or understood from a single point of view. Spatial reasoning differentiates several spatial relations, e.g. topological or metric relations, and is typically formalized using a Cartesian coordinate system and vector algebra. This quantitative processing of information is clearly different from the ways human draw conclusions about spatial relations. Formalized qualitative reasoning processes are shown to be a necessary part of Spatial Expert Systems and Geographical Information Systems. Addressing a subset of the total problem, namely reasoning with cardinal directions, a completely qualitative method, without recourse to analytical procedures, is introduced and a method for its formal comparison with quantitative formula is defined. The focus is on the analysis of cardinal directions and their properties. An algebraic method is used to formalize the meaning of directions. The standard directional symbols (N, W, etc.) are supplemented with a symbol corresponding to an undetermined direction between points too close to each other which greatly increases the power of the inference rules. Two specific systems to determine and reason with cardinal directions are discussed in some detail. From this example and some other previous work, a comprehensive set of research steps is laid out, following a mathematically based taxonomy. It includes the extension of distance and direction reasoning to extended objects and the definitions of other metric relations that characterize situations when objects are not disjointed. The conclusions compare such an approach with other concepts.

At the Department of Geoinformation of the Technical University of Vienna, a European Community funded project to develop an International Post-Graduate Course on GIS is underway. This intensive course is intended for participants with a variety of disciplinary backgrounds and experience who need a broad theoretical overview of GIS coupled with the necessary knowledge and skills to apply GIS in real situations. As curriculum design for GIS courses is normally carried out by a very small group of individuals, curricula generally reflect the specifics of the experience or disciplinary environment of these individuals. For this European project, a widely acceptable course curriculum was needed. This required the cooperation of experts from different disciplines, across many application areas and from different countries. A Delphi survey method combined with a meeting of a small number of GIS education specialists was used to achieve the necessary balance in the curriculum. The survey was used to determine the general content of the course and allowed the varied opinions of the group of European GIS experts to be merged towards a consensus. It resulted in the development of a list of important topics that need to be taught. Following completion of the survey, the GIS education experts met to review this list and discuss the concerns raised in the survey. These discussions led to the development of a Course Blueprint which describes the organisation of the course into 18 instructional units and outlines the objectives and contents that will be achieved in each of them.

This article considers the philosophical and experiential foundations of human perception of geographical phenomena and their abstraction and coding in geographical information systems. It examines the role of culture and language in describing geographical reality and explores the ways geographical data models reflect how people view the world. Differences between those who see the world as made of exact entities and smooth continuous surfaces, and those who prefer to view reality as dynamic and complex are explored in terms of five aspects of spatial data, namely (i) objects versus fields, (ii) single scale versus multiple scales, (iii) Boolean versus multi-valued logic, (iv) static versus dynamic descriptions, and (v) determinism versus uncertainty. These five aspects are further divided into nine factors of geographical data which indicate the differences in the way people perceive spatial data. Eight typical GIS applications and four generic methods of handling spatial data are examined in terms of these nine factors to define a GIS hyperspace. The locations of the typical applications and the generic methods in this hyperspace show why no single generic approach to spatial data handing is sufficient for all possible applications. The analysis reinforces the authors' contention that spatial data analysis tools need to be chosen and developed to match the way users perceive their domains: these tools should not impose alien thought modes on users just because they are impressively high tech. The implications of this conclusion for choosing or developing spatial information systems, for data standardization and generalisation, and for the further development of GIS as a discipline in its own right are presented as topics for further discussion.