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Granular Geometry

  • Gwendolin WilkeEmail author
Chapter
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 325)

Abstract

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.

Keywords

Geographic Information System Geometric Reasoning Intended Interpretation Classical Geometry Positional Uncertainty 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

Most of the work is part of the author’s PhD thesis and was produced while having been employed as a research assistant at the Research Group Geoinformation at the Vienna University of Technology. It was supported by Scholarships of the Austrian Marshallplan Foundation and the Vienna University of Technology. The author would like to thank Prof. Andrew U. Frank, Dr. John Stell and Prof. Lotfi Zadeh for their continuous support and inspiration.

References

  1. 1.
    Amelunxen, C.: An approach to geocoding based on volunteered spatial data. In: Zipf, A., Behnke, K., Hillen, F., Schäfermeyer, J. (eds.) Geoinformatik 2010. Die Welt im Netz, pp. 7–20 (2010)Google Scholar
  2. 2.
    Bennett, B.: A categorical axiomatisation of region-based geometry. Fundamenta Informaticae 46, 145–158 (2001)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Bennett, B., Cohn, A.G., Torrini, P., Hazarika, S.M.: Describing rigid body motions in a qualitative theory of spatial regions. In: Kautz, H.A., Porter, B. (eds.) Proceedings of AAAI-2000, pp. 503–509 (2000)Google Scholar
  4. 4.
    Bennett, B., Cohn, A.G., Torrini, P., Hazarika, S.M.: A foundation for region-based qualitative geometry. In: Horn, W. (ed.) Proceedings of ECAI-2000, pp. 204–208 (2000)Google Scholar
  5. 5.
    Bennett, B., Cohn, A.G., Torrini, P., Hazarika, S.M.: Region-based qualitative geometry. Research report series, University of Leeds, School of Computer Studies (2000)Google Scholar
  6. 6.
    Biacino, L., Gerla, G.: Connection structures: Grzegorczyk’s and Whitehead’s definitions of points. Notre Dame J. Form. Log. 37, 431–439 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Biacino, L., Gerla, G.: Logics with approximate premises. Int. J. Intell. Syst. 13, 1–10 (1998)CrossRefzbMATHGoogle Scholar
  8. 8.
    Buckley, J., Eslami, E.: Fuzzy plane geometry I: points and lines. Fuzzy Sets Syst. 86, 179–187 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Buckley, J., Eslami, E.: Fuzzy plane geometry II: circles and polygons. Fuzzy Sets Styst. 87, 79–85 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Burrough, P.A., Frank, A.U. (eds.): Geographic Objects with Indeterminate Boundaries. GISDATA. Taylor & Francis, London (1996)Google Scholar
  11. 11.
    Buyong, T.B., Frank, A.U., Kuhn, W.: A conceptual model of measurement-based multipurpose cadastral systems. J. Urban Reg. Inf. Syst. Assoc. URISA 3(2), 35–49 (1991)Google Scholar
  12. 12.
    Cheng, S.-C., Mordeson, J.N.: Fuzzy spheres. In: Annual Meeting of the North American Fuzzy information Processing Society (NAFIPS’97). Fuzzy Information Processing Society (1997)Google Scholar
  13. 13.
    Clementini, E.: A model for uncertain lines. J. Vis. Lang. Comput. 16(4), 271–288 (2005)CrossRefGoogle Scholar
  14. 14.
    Dilo, A.: Representation of and reasoning with vagueness in spatial information—a system for handling vague objects. Ph.D. thesis, C.T. de Wit Graduate School for Production Ecology and Resource Conservation (PE&RC) in Wageningen University, The Netherlands (2006)Google Scholar
  15. 15.
    Douglas, D., Peucker, T.: Algorithms for the reduction of the number of points required to represent a digitized line or its caricature. Can. Cartogr. 10, 112–122 (1973)CrossRefGoogle Scholar
  16. 16.
    Dutta, S.: Qualitative Spatial Reasoning: A Semi-quantitative Approach Using Fuzzy Logic, pp. 345–364. Springer, Heidelberg (1990)Google Scholar
  17. 17.
    Gerla, G.: Pointless metric spaces. J. Symb. Log. 55(1), 207–219 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Gerla, G.: Pointless geometries. In: Buekenhout, F. (ed.) Handbook of Incidence Geometry. Buildings and Foundations, Chapter 18, 1st edn., pp. 1015–1031. Elsevier, North-Holland (1995)Google Scholar
  19. 19.
    Gerla, G.: Fuzzy Logic: Mathematical Tools for Approximate Reasoning. Trends in Logic, vol. 11. Kluwer Academic Publishers, Dordrecht (2001)Google Scholar
  20. 20.
    Gerla, G.: Approximate similarities and Poincaré paradox. Notre Dame J. Form. Log. 49(2), 203–226 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Gerla, G., Miranda, A.: Graded inclusion and point-free geometry. Int. J. Appl. Math. 11(1), 63–81 (2004)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Gerla, G., Volpe, R.: Geometry without points. Am. Math. Mon. 92, 707–711 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Ghilani, C.D., Wolf, P.R.: Adjustment Computation—Spatial Data Analysis, 4th edn. Wiley, Hoboken (2006)CrossRefGoogle Scholar
  24. 24.
    Godo, L., Rodríguez, R.O.: Logical approaches to fuzzy similarity-based reasoning: an overview. In: Riccia, G.D., Dubois, D., Kruse, R., Lenz, H.-J. (eds.) Preferences and Similarities, chapter 2, vol. 504, pp. 75–128. Springer, New York (2008)CrossRefGoogle Scholar
  25. 25.
    Goodchild, M.: Citizens as sensors: the world of volunteered geography. GeoJournal 69(4), 211–221 (2007)CrossRefGoogle Scholar
  26. 26.
    Gupta, K.C., Ray, S.: Fuzzy plane projective geometry. Fuzzy Sets Syst. 54, 191–206 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Hájek, P.: Metamathematics of Fuzzy Logic. Trends in Logic, vol. 4. Kluwer Academic Publishers, Dordrecht (1998)zbMATHGoogle Scholar
  28. 28.
    Haklay, M.: How good is volunteered geographical information? A comparative study of OpenStreetMap and ordnance survey datasets. Environ. Plan. B: Plan. Des. 37(4), 682–703 (2010)CrossRefGoogle Scholar
  29. 29.
    Haklay, M.M., Basiouka, S., Antoniou, V., Ather, A.: How many volunteers does it take to map an area well? The validity of Linus law to volunteered geographic information. Cartogr. J. 47, 315–322 (2010)CrossRefGoogle Scholar
  30. 30.
    Hartshorne, R.: Geometry: Euclid and Beyond. Undergraduate Texts in Mathematics. Springer, New York (2000)CrossRefGoogle Scholar
  31. 31.
    Hobbs, J.R.: Granularity. In: Proceedings of Ninth International Joint Conference on Artificial Intelligence, pp. 432–435. Morgan Kaufmann, Los Angeles (1985)Google Scholar
  32. 32.
    Katz, M.: Inexact geometry. Notre Dame J. Form. Log. 21, 521–535 (1980)CrossRefzbMATHGoogle Scholar
  33. 33.
    Kuijken, L.: Fuzzy projective geometries. In: Mayor, G., Jaume, J.S. (eds.) Proceedings of the 1999 EUSFLAT-ESTYLF Joint Conference, Palma de Mallorca, Spain (1999)Google Scholar
  34. 34.
    Kuijken, L., Maldeghem, H.V.: Fibered geometries. Discret. Math. 255(1–3), 259–274 (2002)CrossRefzbMATHGoogle Scholar
  35. 35.
    Lakoff, G., Núñez, R.E.: Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Basic Books, New York (2000)Google Scholar
  36. 36.
    Leung, Y., Ma, J.-H., Goodchild, M.F.: A general framework for error analysis in measurement-based GIS, parts 1–4. J. Geogr. Syst. 6(4), 323–428 (2004)CrossRefGoogle Scholar
  37. 37.
    Liu, H., Coghill, G.M.: Can we do trigonometry qualitatively? In: Proceedings of the 19th International Workshop on Qualitative Reasoning QR-05 (2005)Google Scholar
  38. 38.
    Lobačevskij, N.I.: Novye načala geometry s polnoj teoriej parallel’nyh (New principles of geometry with complete theory of parallels). Polnoe sobranie socinenij 2 (1835)Google Scholar
  39. 39.
    Menger, K.: Propabilistic geometry. In: PNAS Proceedings of the National Academy of the United States of America, vol. 37, pp. 226–229 (1951)Google Scholar
  40. 40.
    Menger, K.: Selected Papers in Logic and Foundations, Didactics, Economics. Vienna Circle Collection, vol. 10. D. Reidel Publishing Company, Dordrecht (1979)CrossRefzbMATHGoogle Scholar
  41. 41.
    Novák, V., Perfilieva, I., Močkoř, J.: Mathematical Principles of Fuzzy Logic. Kluwer Academic Publishers, Boston (1999)CrossRefzbMATHGoogle Scholar
  42. 42.
    Pavelka, J.: On fuzzy logic I, II, III. Zeitschrift für Mathematik, Logik und Grundlagen der Mathematik 25, 45–52, 119–134, 447–464 (1979)Google Scholar
  43. 43.
    Perkal, J.: On epsilon length. Bulletin de l’Academie Polonaise des Sciences 4, 399–403 (1956)zbMATHMathSciNetGoogle Scholar
  44. 44.
    Perkal, J.: On the length of empirical curves. discussion paper no. 10. Michigan Inter-University Community of Mathematical Geographers, Ann Arbor (1966)Google Scholar
  45. 45.
    Peucker, T.K.: Chapter A theory of the cartographic line. International Yearbook of Cartography, vol. 16, pp. 134–143 (1975)Google Scholar
  46. 46.
    Poincaré, H.: Science and Hypothesis. Walter Scott Publishing, New York (1905). First published by Flammarion, Paris (1902)Google Scholar
  47. 47.
    Poston. T.: Fuzzy Geometry. PhD thesis, University of Warwick (1971)Google Scholar
  48. 48.
    Pullar, D.V.: Consequences of using a tolerance paradigm in spatial overlay. Proc. Auto-Carto 11, 288–296 (1993)Google Scholar
  49. 49.
    Ramer, U.: An iterative procedure for the polygonal approximation of plane curves. Comput. Graph. Image Process. 1(3), 244–256 (1972)CrossRefGoogle Scholar
  50. 50.
    Roberts, F.S.: Representations of indifference relations. Ph.D. thesis, Stanford University (1968)Google Scholar
  51. 51.
    Roberts, F.S.: Tolerance geometry. Notre Dame J. Form. Log. 14(1), 68–76 (1973)CrossRefzbMATHGoogle Scholar
  52. 52.
    Roberts, F.S., Suppes, P.: Some problems in the geometry of visual perception. Synthese 17, 173–201 (1967)CrossRefzbMATHGoogle Scholar
  53. 53.
    Rosenfeld, A.: The diameter of a fuzzy set. Fuzzy Sets Syst. 13(3), 241–246 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  54. 54.
    Rosenfeld, A.: Fuzzy rectangles. Pattern Recognit. Lett. 11(10), 677–679 (1990)CrossRefzbMATHGoogle Scholar
  55. 55.
    Rosenfeld, A.: Fuzzy plane geometry: triangles. Fuzzy Syst. IEEE World Congr. Comput. Intell. 2, 891–893 (1994)Google Scholar
  56. 56.
    Rosenfeld, A., Haber, S.: The perimeter of a fuzzy set. Pattern Recognit. 18(2), 125–130 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  57. 57.
    Ruspini, E.: On the semantics of fuzzy logic. Int. J. Approx. Reason. 5, 45–88 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  58. 58.
    Salesin, D., Stolfi, J., Guibas, L.: Epsilon geometry: building robust algorithms from imprecise computations. In: SCG’89: Proceedings of the Fifth Annual Symposium on Computational Geometry, pp. 208-217. ACM, New York (1989)Google Scholar
  59. 59.
    Schmidtke, H.: A geometry for places: Representing extension and extended objects. In: Kuhn, W., Worboys, M., Timpf, S. (eds.) Spatial Information Theory: Foundations of Geographic Information Science, pp. 221–238. Springer, Berlin (2003)CrossRefGoogle Scholar
  60. 60.
    Shi, W.: A generic statistical approach for modelling error of geometric features in GIS. Int. J. Geogr. Inf. Sci. 12(2), 131–143 (1998)CrossRefGoogle Scholar
  61. 61.
    Shi, W.: Principles of Modeling Uncertainties in Spatial Data and Spatial Analyses. CRC Press Inc., Boca Raton (2009)CrossRefGoogle Scholar
  62. 62.
    Shi, W., Cheung, C.K., Zhu, C.: Modelling error propagation in vector-based buffer analysis. Int. J. Geogr. Inf. Sci. 17(3), 251–271 (2003)CrossRefGoogle Scholar
  63. 63.
    Shi, W., Liu, W.: A stochastic process-based model for the positional error of line segments in gis. Int. J. Geogr. Inf. Sci. 14(1), 51–66 (2000)CrossRefMathSciNetGoogle Scholar
  64. 64.
    Suppes, P., Krantz, D.H., Luce, D.R., Tversky, A.: Foundations of Measurement Volume I–III. Dover Publications, New York (2007). Originally published by Academic Press (1989)Google Scholar
  65. 65.
    Suppes, P., Krantz, D.H., Luce, D.R., Tversky, A.: Foundations of Measurement Volume II: Geometrical, Threshold, and Probabilistic Representations. Dover Publications, New York (2007). Originally published by Academic Press (1989)Google Scholar
  66. 66.
    Tarski, A.: Methodology of deductive sciences. Logics, Semantics, Metamathematics. Oxford University Press, Oxford (1956)Google Scholar
  67. 67.
    Tarski, A.: What is elementary geometry? In: Henkin, L., Suppes, P., Tarski, A. (eds.) The Axiomatic Method. With Special Reference to Geometry and Physics. Proceedings of an International Symposium Held at the University of California, Berkeley, Studies in Logic and the Foundations of Mathematics, pp. 16–29. North-Holland, Amsterdam (1958)Google Scholar
  68. 68.
    Topaloglou, T.: First order theories of approximate space. In: AAAI-94 Workshop on Spatial and Temporal Reasoning, pp. 47–53 (1994)Google Scholar
  69. 69.
    Trillas, E., Valverde, L.: An inquiry into indistinguishability operators. In: Skala, H., Termini, S., Trillas, E. (eds.) Aspects of Vagueness. Theory and Decision Library, vol. 39, pp. 231–256. Springer, Netherlands (1984)CrossRefGoogle Scholar
  70. 70.
    Vakarelov, D.: Region-based theory of space: Algebras of regions, representation theory, and logics. In: Gabbay, D., Goncharov, S., Zakharyaschev, M. (eds.) Mathematical Problems from Applied Logic II. Logics for the XXIst Century. International Mathematical Series, vol. 5. Springer, New York (2007)Google Scholar
  71. 71.
    Veelaert, P.: Geometric constructions in the digital plane. J. Math. Imaging Vis. 11, 99–118 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  72. 72.
    Wilke, G.: Approximate geometric reasoning with extended geographic objects. In: Proceedings of the ISPRS COST-Workshop on Quality, Scale and Analysis Aspects of City Models, Lund, Sweden (2009)Google Scholar
  73. 73.
    Wilke, G.: Towards approximate tolerance geometry for GIS—a framework for formalizing sound geometric reasoning under positional tolerance. PhD thesis, Vienna University of Technology (2012)Google Scholar
  74. 74.
    Wilke, G.: Equality in approximate tolerance geometry. In: Kacprzyk, J. (ed.) Advances in Intelligent Systems and Computing. Springer, New York (2014)Google Scholar
  75. 75.
    Wilke, G.: Fuzzy logic with evaluated syntax for sound geometric reasoning with geographic information. In: Proceedings of the 2014 IEEE Conference on Norbert Wiener in the 21st Century, Boston MA, USA, 24–26 June 2014, incorporating the Proceedings of the Annual Conference of the North American Fuzzy Information Processing Society: NAFIPS 2014 (2014)Google Scholar
  76. 76.
    Wilke, G., Frank, A.: On equality of points and lines. In: GIScience 2010, Zürich, Switzerland (2010)Google Scholar
  77. 77.
    Wilke, G., Frank, A.U.: Tolerance geometry: Euclid’s first postulate for points and lines with extension. In: 18th ACM SIGSPATIAL International Symposium on Advances in Geographic Information Systems, ACM-GIS 2010, 3–5 November 2010, San Jose, CA, USA, Proceedings, pp. 162–171(2010)Google Scholar
  78. 78.
    Ying, M.: A logic for approximate reasoning. J. Symb. Log. 59(3), 830–837 (1994)CrossRefzbMATHGoogle Scholar
  79. 79.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  80. 80.
    Zadeh, L.A.: Similarity relations and fuzzy orderings. Inf. Sci. 3, 177–200 (1971)CrossRefzbMATHMathSciNetGoogle Scholar
  81. 81.
    Zadeh, L.A.: Generalized theory of uncertainty (GTU)—principal concepts and ideas. Comput. Stat. Data Anal. 51, 15–46 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  82. 82.
    Zadeh, L.A.: A note on z-numbers. Inf. Sci. 181(14), 2923–2932 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  83. 83.
    Zadeh, L.A.: Toward a restriction-centered theory of truth and meaning (RCT). Inf. Sci. 248, 1–14 (2013)CrossRefMathSciNetGoogle Scholar
  84. 84.
    Zandbergen, P.A.: Geocoding quality and implications for spatial analysis. Geogr. Compass 3(2), 647–680 (2009)CrossRefGoogle Scholar
  85. 85.
    Zeeman, E.: The topology of the brain and visual perception. Topology of 3-Manifolds and Related Topics (Proceedings of the University of Georgia Institute, 1961), pp. 240–256. Prentice-Hall, Upper Saddle River (1962)Google Scholar
  86. 86.
    Zulfiqar, N.: A study of the quality of OpenStreetMap.org maps: a comparison of OSM data and ordnance survey data. Master’s thesis, University College London (2008)Google Scholar

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Authors and Affiliations

  1. 1.University of Applied Sciences and Arts Northwestern Switzerland (FHNW)OltenSwitzerland

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