Chapter 8: Fuzzy Geometry Based Computations

In this chapter we present the fuzzy logic for formal geometric reasoning with extended objects. Based on the idea that extended objects may be seen as location constraints to coordinate points, the geometric primitives point, line, incidence and equality are interpreted as fuzzy predicates of a first order language. An additional predicate for the “distinctness” of point like objects is also used. Fuzzy Logic [28] is discussed as a reasoning system for geometry of extended objects. A fuzzification of the axioms of incidence geometry based on the proposed fuzzy logic is presented. In addition we discuss a special form of positional uncertainty, namely positional tolerance that arises from geometric constructions with extended primitives. We also address Euclid’s first postulate, which lays the foundation for consistent geometric reasoning in all classical geometries by taken into account extended primitives and gave a fuzzification of Euclid’s first postulate by using of our fuzzy logic. Fuzzy equivalence relation “Extended lines sameness” is introduced. For its approximation we use fuzzy conditional inference, which is based on proposed fuzzy “Degree of indiscernibility “and “Discernibility measure “of extended points…