Tolerance geometry

Abstract

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. 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 geometric reasoning, but must be taken into account when formalizing Euclid's first postulate for extended primitives. One important consequence is the positional tolerance of the incidence relation (on-relation). As another consequence, equality of geometric primitives becomes a matter of degree. To account for this fact, we propose a formalization of Euclid's first postulate in Lukasiewicz t-norm fuzzy logic. A model of the proposed formalization is given in the projective plane with elliptic metric. This is not a restriction, since the elliptic metric is locally Euclidean. We introduce graduated geometric reasoning with Rational Pavelka Logic as a means of approximating and propagating positional tolerance through the steps of a geometric construction process. Since the axioms (postulates) of geometry built upon one another, the proposed formalization of Euclid's first postulate provides one building block of a geometric calculus that accounts for positional tolerance in a consistent way.The novel contribution of the paper is to define geometric reasoning with extended primitives as a calculus that propagates positional tolerance. Also new is the axiomatic approach to positional uncertainty and the associated consistency issue.