Geometry Without Points

Gerla, Giangiacomo; Volpe, R. C.

doi:10.1080/00029890.1985.11971718

Cited by 11 publications

(3 citation statements)

References 2 publications

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“…Many newer approaches build upon A. Tarski's Geometry of Solids [67]. In particular, the work of Gerla and co-workers [6,17,21,22], is relevant for granular geometry, because some of Gerla's results are used as a basis for it; The work of B. Bennett et al [2][3][4][5], relates to the field of GIS.…”

Section: Axiomatic Approaches To Fuzzy Geometrymentioning

confidence: 99%

Granular Geometry

Wilke

2015

Towards the Future of Fuzzy Logic

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Many approaches have been proposed in the fuzzy logic research community to fuzzifying classical geometries. From the field of geographic information science (GIScience) arises the need for yet another approach, where geometric points and lines have granularity: Instead of being "infinitely precise", points and lines can have size. With the introduction of size as an additional parameter, the classical bivalent geometric predicates such as equality, incidence, parallelity or duality become graduated, i.e., fuzzy. The chapter introduces the Granular Geometry Framework (GGF) as an approach to establishing axiomatic theories of geometries that allow for sound, i.e., reliable, geometric reasoning with points and lines that have size. Following Lakoff's and Núñez' cognitive science of mathematics, the proposed framework is built upon the central assumption that classical geometry is an idealized abstraction of geometric relations between granular entities in the real world. In a granular world, an ideal classical geometric statement is sometimes wrong, but can be "more or less true", depending on the relative sizes and distances of the involved granular points and lines. The GGF augments every classical geometric axiom with a degree of similarity to the truth that indicates its reliability in the presence of granularity. The resulting fuzzy set of axioms is called a granular geometry, if all truthlikeness degrees are greater than zero. As a background logic, Łukasiewicz Fuzzy Logic with Evaluated Syntax is used, and its deduction apparatus allows for deducing the reliability of derived statements. The GGF assigns truthlikeness degrees to axioms in order to embed information about the intended granular model of the world in the syntax of the logical theory. As a result, a granular geometry in the sense of the framework is sound by design. The GGF allows for interpreting positional granules by different modalities of uncertainty (e.g. possibilistic or veristic). We elaborate the framework for possibilistic positional granules and exemplify it's application using the equality axioms and Euclid's First Postulate.

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Section: Axiomatic Approaches To Fuzzy Geometrymentioning

confidence: 99%

Granular Geometry

Wilke

2015

Towards the Future of Fuzzy Logic

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“…The geometries thus developed are unsatisfactory in our view since they make a direct use of points (as a different sort) or regard some of the individuals as point-like entities. The last case is exemplified by [Gli69], which uses the notion of (finite) segment and congruence, and [GV85] based on the notions of solid (among which the so-called ε-points), parthood (here called inclusion) and distance. Among the first type of systems instead we cite [Pam04] that uses as primitives points and triangles (or squares) and the relation of point-triangle (or pointsquare), and [Pra83, Pam00] based on points, circles and the relation of point-circle incidence.…”

Section: Literaturementioning

confidence: 99%

Spheres, cubes and simple

Borgo

2013

LLP

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Abstract. In 1929 Tarski showed how to construct points in a region-based first-order logic for space representation. The resulting system, called the geometry of solids, is a cornerstone for region-based geometry and for the comparison of point-based and region-based geometries. We expand this study of the construction of points in region-based systems using different primitives, namely hyper-cubes and regular simplexes, and show that these primitives lead to equivalent systems in dimension n ≥ 2. The result is achieved by adopting a single set of definitions that works for both these classes of figures. The analysis of our logics shows that Tarski's choice to take sphere as the geometrical primitive might be intuitively justified but is not optimal from a technical viewpoint.

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“…As already mentioned, Whitehead and most point-free geometry scholars proposed a different definition of a point based on the notion of an abstraction class. In this paper we prefer the definition of a point as proposed by A. Śniatycki in[14] and adopted in[6,7] 9. As a trivial consequence of the definition of ≡r, if an r-point P = {l1, l2} is an interior point of a region X, then all the r-points equivalent to P are interior in X.…”

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confidence: 99%