G. Wilke:

"Propagating Worst Case Error in Tolerance Geometry";

Talk: Berkeley Initiative in Soft Computing, Berkeley, California, USA (invited); 2010-11-05.

In geographic information systems (GIS), vector representations of user data sets are usually generated by a process of generalization, overlay and merging of data sets that stem from different data sources. As a consequence of this process, geometric points and lines do not always have unambiguous coordinate representations, but are subject to positional tolerance. Geometric constructions with such geometric primitives easily produce inconsistencies in derived geometric data. Results of spatial analysis that use such derived data can be insignificant, misleading, or even wrong. The talk addresses the issue of geometric reasoning with geometric points and lines that have tolerance in position. The issue of consistency is addressed by adopting an axiomatic formalization of geometry. We focus on Euclid´s first postulate, which is a core axiom of Euclidean and projective geometry, stating that any two distinct geometric points uniquely determine a geometric line. For exact points and lines, the postulate ensures the existence of straight line connections. It makes geometric constructions unambiguous and thereby provides the foundation for consistent geometric reasoning. We show that Euclid's First Postulate is violated if points and lines are subject to positional tolerance. The resulting inconsisteny cannot be resolved in a binary axiomatic formalization. As a solution to the problem we propose to extend the usual binary formalization of Euclid's First Postulate to a graduated formalization, using the language of Lukasiewicz t-norm fuzzy logic. Here, the primitive spatial relations of binary equality and binary incidence are replaced by fuzzy equality and fuzzy incidence relations, i.e. by relations that are allowed to have a "truth degree". We give a model of the fuzzified version of Euclid's First Postulate, where the truth degree of a geometric relation can be interpreted as a measure of the worst case error that results from positional tolerance. We use the deduction apparatus of Rational Pavelka Logic to propagate this measure of worst case positional tolerance through the steps of geometric constructions. As an example we derive a uniqueness degree for straight line connections from the equality degree of two given points with positinal tolerance, and their incidence degrees with a given line with positional tolerance.

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