Ω-Arithmetization of Ellipses

  • Agathe Chollet
  • Guy Wallet
  • Eric Andres
  • Laurent Fuchs
  • Gaëlle Largeteau-Skapin
  • Aurélie Richard
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6026)


Multi-resolution analysis and numerical precision problems are very important subjects in fields like image analysis or geometrical modeling. In the continuation of our previous works, we propose to apply the method of Ω-arithmetization to ellipses. We obtain a discrete multi-resolution representation of arcs of ellipses. The corresponding algorithms are completely constructive and thus, can be exactly translated into functional computer programs. Moreover, we give a global condition for the connectivity of the discrete curves generated by the method at every scale.


discrete geometry multi-resolution analysis nonstandard analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andres, E.: Discrete circles, rings and spheres. Computer and Graphics 18(5), 695–706 (1994)CrossRefGoogle Scholar
  2. 2.
    Bresenham, J.E.: A linear algorithm for incremental digital display of circular arcs. Comm. of ACM 20(2), 100–106 (1977)zbMATHCrossRefGoogle Scholar
  3. 3.
    Bridges, D.S.: Constructive mathematics: A foundation for computable analysis. Theor. Comput. Sci. 219(1-2), 95–109 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chollet, A., Wallet, G., Fuchs, L., Largeteau-Skapin, G., Andres, E.: Insight in discrete geometry and computational content of a discrete model of the continuum. Pattern Recognition 42(10), 2220–2228 (2009)zbMATHCrossRefGoogle Scholar
  5. 5.
    Chollet, A., Wallet, G., Fuchs, L., Largeteau-Skapin, G., Andres, E.: ω-arithmetization: a discrete multi-resolution representation of real functions. In: Wiederhold, P., Barneva, R.P. (eds.) IWCIA 2009. LNCS, vol. 5852, pp. 316–329. Springer, Heidelberg (2009)Google Scholar
  6. 6.
    Diener, M.: Application du calcul de Harthong-Reeb aux routines graphiques. In: Salanskis, J.-M., Sinaceurs, H. (eds.) Le Labyrinthe du Continu, pp. 424–435. Springer, Heidelberg (1992)Google Scholar
  7. 7.
    Fuchs, L., Largeteau-Skapin, G., Wallet, G., Andres, E., Chollet, A.: A first look into a formal and constructive approach for discrete geometry using nonstandard analysis. In: Coeurjolly, D., Sivignon, I., Tougne, L., Dupont, F. (eds.) DGCI 2008. LNCS, vol. 4992, pp. 21–32. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Harthong, J.: Éléments pour une théorie du continu. Astérisque (109/110), 235–244 (1983)Google Scholar
  9. 9.
    Holin, H.: Harthong-Reeb circles. Séminaire Non Standard, Univ. de Paris 7 (89/2), 1–30 (1989)Google Scholar
  10. 10.
    Holin, H.: Harthong-Reeb analysis and digital circles. The Visual Computer 8(1), 8–17 (1991)CrossRefGoogle Scholar
  11. 11.
    INRIA-consortium. Le langage Caml,
  12. 12.
    Laugwitz, D.: Ω-calculus as a generalization of field extension. In: Hurd, A. (ed.) Nonstandard Analysis – Recent Developments. Lecture Notes in Mathematics, vol. 983, pp. 144–155. Springer, Heidelberg (1983)CrossRefGoogle Scholar
  13. 13.
    Schmieden, C., Laugwitz, D.: Eine erweiterung der Infinitesimalrechnung. Mathematische Zeitschrift 69(1), 1–39 (1958)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Martin-Löf, P.: Constructive mathematics and computer programming. In: Logic, Methodology and Philosophy of Science VI, pp. 153–175 (1980)Google Scholar
  15. 15.
    Martin-Löf, P.: Intuitionnistic Type Theory. Bibliopolis, Napoli (1984)Google Scholar
  16. 16.
    Martin-Löf, P.: Mathematics of infinity. In: Gilbert, J.R., Karlsson, R. (eds.) COLOG 1988. LNCS, vol. 417, pp. 146–197. Springer, Heidelberg (1990)Google Scholar
  17. 17.
    Nelson, E.: Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society 83(6), 1165–1198 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Reveillès, J.-P.: Géométrie Discrète, Calcul en Nombres Entiers et Algorithmique. PhD thesis, Université Louis Pasteur, Strasbourg, France (1991)Google Scholar
  19. 19.
    Richard, A., Wallet, G., Fuchs, L., Andres, E., Largeteau-Skapin, G.: Arithmetization of a circular arc. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 350–361. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  20. 20.
    Robinson, A.: Non-standard Analysis, 2nd edn. American Elsevier, New York (1974)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Agathe Chollet
    • 1
  • Guy Wallet
    • 1
  • Eric Andres
    • 2
  • Laurent Fuchs
    • 2
  • Gaëlle Largeteau-Skapin
    • 2
  • Aurélie Richard
    • 2
  1. 1.Laboratoire MIAUniversité de La RochelleLa Rochelle cedexFrance
  2. 2.Laboratoire XLIM SIC, UMR 6172Université de PoitiersFrance

Personalised recommendations