Tolerance geometry: Euclid's first postulate for points and lines with extension

Object representation and reasoning in vector based geographic information systems (GIS) is based on Euclidean geometry. Euclidean geometry is built upon Euclid's first postulate, stating that two points uniquely determine a line. This postulate makes geometric constructions unambiguous and thereby provides the foundation for consistent geometric reasoning. It holds for exact coordinate points and lines, but is violated, if points and lines are allowed to have extension. As an example for a point that has extension consider a point feature that represents the city of Vienna in a small scale GIS map representation. Geometric constructions with such a point feature easily produce inconsistencies in the data [23]. The present paper addresses the issue of consistency by formalizing Euclid's first postulate for geometric primitives that have extension. We identify a list of six consequences from introducing extension: these are `new qualities' that are not present in exact g...

• Type: Journal
• Year: 2010
• Where: GIS
• Authors: Gwen Wilke, Andrew U. Frank
• Keywords: Automated Reasoning, Geometric, Geometric Construction, GIS 2010, Postulate