Arithmetization of a Circular Arc

  • Aurélie Richard
  • Guy Wallet
  • Laurent Fuchs
  • Eric Andres
  • Gaëlle Largeteau-Skapin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5810)


In this paper, we present an arithmetization of the Euler’s integration scheme based on infinitely large integers coming from the nonstandard analysis theory. Using the differential equation that defines circles allows us to draw two families of discrete arc circles using three parameters, the radius, the global scale and the drawing scale. These parameters determine the properties of the obtained arc circles. We give criteria to assure the 8-connectivity. A global error estimate for the arithmetization of the Euler’s integration scheme is also given and a first attempt to define the approximation order of an arithmetized integration scheme is provided.


Discrete circle Discrete arc circle Arithmetization Numerical scheme Error order Connectedness 


  1. 1.
    Bresenham, J.: Algorithm for computer control of a digital plotter. ACM trans. Graphics 4, 25–30 (1965)Google Scholar
  2. 2.
    Bresenham, J.: A linear algorithm for incremental digital display of circular arcs. Comm. of ACM 20, 100–106 (1977)CrossRefzbMATHGoogle Scholar
  3. 3.
    Reveillès, J.P.: Mathématiques discrètes et analyse non standard. In: [19], pp. 382–390Google Scholar
  4. 4.
    Reveillès, J.P.: Géométrie discrète, Calcul en nombres entiers et algorithmique. PhD thesis, Université Louis Pasteur, Strasbourg, France (1991)Google Scholar
  5. 5.
    Reveillès, J.P., Richard, D.: Back and forth between continuous and discrete for the working computer scientist. Annals of Mathematics and Artificial Intelligence, Mathematics and Informatic 16, 89–152 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Andres, E.: Discrete circles, rings and spheres. Computer and Graphics 18, 695–706 (1994)CrossRefGoogle Scholar
  7. 7.
    Nelson, E.: Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society 83, 1165–1198 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Robinson, A.: Non-standard analysis. American Elsevier, New York (1974)Google Scholar
  9. 9.
    Harthong, J.: Une théorie du continu. In: Barreau, H., Harthong, J. (eds.) La mathématique non standard, Editions du CNRS, Paris, pp. 307–329 (1989)Google Scholar
  10. 10.
    Diener, M.: Application du calcul de Harthong-Reeb aux routines graphiques. In: [19], pp. 424–435Google Scholar
  11. 11.
    Harthong, J., Reeb, G.: Intuitionnisme 84. In: Barreau, H., Harthong, J. (eds.) La mathématique nonstandard, CNRS, pp. 213–252 (1989)Google Scholar
  12. 12.
    Harthong, J.: Éléments pour une théorie du continu. Astérisque 109/110, 235–244 (1983)MathSciNetzbMATHGoogle Scholar
  13. 13.
    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
  14. 14.
    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 (2009)Google Scholar
  15. 15.
    Holin, H.: Harthong-Reeb analysis and digital circles. The Visual Computer 8, 8–17 (1991)CrossRefGoogle Scholar
  16. 16.
    Holin, H.: Harthong-Reeb circles. Séminaire non standard, Univ. de Paris 7 2, 1–30 (1989)Google Scholar
  17. 17.
    Holin, H.: Some artefacts of integer computer circles. Ann. Math. Artif. Intell. 16, 153–181 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Quarteroni, A., Sacco, R., Saleri, F.: Numerical mathematics. Springer, Heidelberg (2000)zbMATHGoogle Scholar
  19. 19.
    Salanskis, J.M., Sinaceurs, H. (eds.): Le Labyrinthe du Continu. Springer, Heidelberg (1992)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Aurélie Richard
    • 1
  • Guy Wallet
    • 2
  • Laurent Fuchs
    • 1
  • Eric Andres
    • 1
  • Gaëlle Largeteau-Skapin
    • 1
  1. 1.Laboratoire XLIM-SICUniversité de PoitiersFuturoscope Chasseneuil cedexFrance
  2. 2.Laboratoire MIAUniversité de La RochelleLa Rochelle cedexFrance

Personalised recommendations