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G. Wilke, A. Frank:
"On Equality of Lines with Positional Uncertainty";
Talk: GIScience 2010, Zurich, Switzerland; 2010-09-14 - 2010-09-17.

Geometric reasoning in vector based geographic information systems (GIS) is based on Euclidean geometry. Euclid´s first postulate, saying that the line determined by two points is unique, makes geometric constructions unambiguous, and, e.g., allows for specifying line features and polygons by tupels of points. It relies on the assumption that the location of a point can be unambiguously and accurately described by a coordinate pair. In contrast to this, the geographic location of a point in space can not be exactly determined, but is subject to positional uncertainty. One way to establish a uniqueness property for points and lines with positional uncertainty in analogy to Euclid´s first postulate is to introduce fuzzy equality predicates measuring a degree of equality of points and lines with positional uncertainty,
respectively. This abstract focuses on equality of lines with uncertainty in location, and derives a list of four requirements to such an equality predicate. We focus on the simplest form of positional uncertainty, namely on location constraints: In this case, the true location of a point or a line is delineated by one or several regions in a coordinate space, and no distribution or weighting of the coordinate points within that regions is given. For many practical cases, assuming a single simply connected
and regular constraining region is sufficient. We define extended points (extended lines) to be points (lines) whose exact location is unknown, and whose set of possible locations is specified by a simply connected regular subset of a coordinate space (parameter space). We interpret the incidence relation (on-relation) between extended points and extended lines by the subset relation of the underlying coordinate space.