A Model of Spatial Position Based on Extremal Points

Two sets of fundamental properties which a spatial representation should describe are the position, extent, and orientation of objects, and topological and directional relations between objects. In previous approaches, the formalization of a particular property, such as topological spatial relations, has often been studied as an independent problem. The resulting theories have been elegant and compact, but may be diicult to use because of the assumptions on available knowledge. For example, several formalizations of topologi-cal relations assume knowledge of the boundaries of objects. Yet, knowledge of exact boundaries requires the knowledge of exact shapes, a property which is very diicult to describe symbolically. Our approach was to rst deene an application, a magnetic elds problem solver capable of doing both spatial and dynamic reasoning, and a broad collection of qualitative problems to be solved before embarking on the task of deening a spatial representation. The nature of the solutions to the various problems provided clues on what spatial properties were of fundamental importance, and what information could be abstracted. We describe an adequacy criterion for abstractions of object shapes, and present a description of the spatial position and orientation of objects based on extremal points. We show that topological and directional relations may be readily described in terms of this object representation. Our spatial representation is compatible with existing tools for the qualitative simulation of dynamic systems, which allows us to reason about changes in spatial relations due to the translational and rotational motion of objects.