Pointless Geometries

Gerla, Giangiacomo

doi:10.1016/b978-044488355-1/50020-7

Cited by 35 publications

(16 citation statements)

References 25 publications

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“…The literature on region-based geometries has a long history, starting with N. I. Lobacevskij [38]. A historical overview is given by G. Gerla [18], and recent results are summarized by D. Vakarelov [70]. Many newer approaches build upon A. Tarski's Geometry of Solids [67].…”

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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“…Shapes are represented as occupying regions of space, and other concepts such as points, boundaries etc., result from interactions between these. There is a long philosophical and mathematical background in such ideas as a foundation for geometry including Whitehead [59], Clarke [60] and Gerla [61]. For design computing, the shape grammar formalism of Stiny [20] is based on a similar premise in which shapes are primarily structured according to parts, identified through the querying mechanism of shape rules.…”

Section: Discussionmentioning

confidence: 99%

Shape Interpretation with Design Computing

Jowers

Earl

2014

Design Computing and Cognition '12

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How information is interpreted has significant impact on how it can be used. This is particularly important in design where information from a wide variety of sources is used in a wide variety of contexts and in a wide variety of ways. This paper is concerned with the information that is created, modified and analysed during design processes, specifically with the information that is represented in shapes. It investigates how design computing seeks to support these processes, and the difficulties that arise when it is necessary to consider alternative interpretations of shape. The aim is to establish the problem of shape interpretation as a general challenge for research in design computing, rather than a difficulty that is to be overcome within specific processes. Shape interpretations are common characteristics of several areas of enquiry in design computing. This paper reviews these, brings an integrated perspective and draws conclusions about how this underlying process can be supported.

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“…So the question arises as to whether this is a mere coincidence of terms or something more. We are going to show that deep links exist between o-algebras and the so-called "connection structures" (for which we refer mainly to [1,2,6]).…”

Section: O-algebras and Connection Structuresmentioning

confidence: 99%

Intuitionistic overlap structures

Ciraulo

2013

LLP

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We study some connections between two kinds of overlap relations: that of point-free geometries in the sense of Grzegorczyk, Whitehead and Clarke, and that recently introduced by Sambin within his constructive approach to topology. The main thesis of this paper is that the overlap relation in the latter sense is a necessary tool for a constructive and intuitionistic development of point-free geometry.

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Pointless Geometries

Cited by 35 publications

References 25 publications

Granular Geometry

Granular Geometry

Shape Interpretation with Design Computing

Intuitionistic overlap structures

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