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G. Wilke:
"Towards an Axiomatic Approach to Geometric Reasoning with Extended Geographic Objects";
2010; 35 pages.

The report presents a conceptual framework for formal geometric reasoning with points that have size and lines that have width. The research is motivated by the rise of ubiquitous computing and the increasing use of location based services in every day life: As a consequence, it becomes increasingly important to represent and query textual descriptions of spatial configurations in geographic information systems (GIS). Since textual descriptions usually refer to extended locations, e.g. buildings, there is a need to integrate spatial reasoning mechanisms that can handle input with spatial extension in a consistent way. Heuristic solutions can handle specific tasks, but are not generic enough to provide a consistent algebra of operations.
he paper shows that the formalism of Cartesian geometry can not be generalized to handle points and lines with extension in a way that is flexible enough for the purposes of geographic information processing. As a solution we propose to fuzzify the underlying axiomatic system of Euclidean geometry, and thereby achieve more flexibility in the resulting formalism. Fuzzy Łukasiewicz logic with evaluated syntax provides an appropriate tool: it allows for fuzzifying the geometric primitives, as well as the geometric reasoning mechanism itself.
Euclidean geometry consists of five groups of axioms. To illustrate the conceptual framework, the paper discusses the axiom group one of incidence axioms. We choose the axiomatization as given by David Hilbert (1962). The definition of a valid model of the fuzzified version of the incidence axioms that, at the same time, satisfies the needs of geographic information processing, is work in progress. Intermediary results are discussed. The novel contribution of the presented work is the axiomatic approach to geometric reasoning with points that have size and lines that have width.