Operating on Spatial Relations

examples and a formal definition are given later. As might be anticipated, the notion ofspatial relations is fundamental to the understanding ofgeometry, now regarded not as the study of 'geometrical objects' but of the relations between those objects. The point was made forcibly by the German mathematician David Hilbert, who allegedly said that the words 'point', 'line' and 'plane' could be replaced by 't~ble', 'chair' and 'beer-mug' without changing geometry In the least (Bourbaki, 1968, p 317). Although examples ofvisually plausible spatial relations may be produced easily, there are a number of problems that arise when one considers more carefully the possible kinds of spatial properties. These properties may be numerical, for example, scalars (the distance between two objects), or vectors (the distance between two objects and direction the one defines with respect to the other), or they may be predicates asserting something geometrical or topological (the closure of a curve). It would be desirable to classify the spatial relations of vision within the framework of one or more formal mathematical structures (see e.g..Bourbaki, 1968, Chapter 4), or possibly some fuzzy version of these structures (Kaufmann, 1975). But which structures are the most appropriate for vision, how they should be combined with each other, and how they should be modified to reflect the limitations of the visual apparatus are largely unknown. In addition to these foundational matters, there are u~certainties concerning the status of some commonly cited examples of spatial relations. Consider the set given by Barlow et ale (1972), here slightly modified for later reference: D. H. Foster