New Results about Digital Intersections

  • Isabelle Sivignon
  • Florent Dupont
  • Jean-Marc Chassery
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2886)


Digital geometry is very different from Euclidean geometry in many ways and the intersection of two digital lines or planes is often used to illustrate those differences. Nevertheless, while digital lines and planes are widely studied in many areas, very few works deal with the intersection of such objects. In this paper, we investigate the geometrical and arithmetical properties of those objects. More precisely, we give some new results about the connectivity, periodicity and minimal parameters of the intersection of two digital lines or planes.


Digital straight lines and planes intersection 


  1. 1.
    Kim, C.E.: Three-dimensional digital planes. IEEE Trans. on Pattern Analysis and Machine Intelligence 6, 639–645 (1984)zbMATHCrossRefGoogle Scholar
  2. 2.
    Réveillès, J.P.: Géométrie discrète, calcul en nombres entiers et algorithmique. PhD thesis, Université Louis Pasteur, Strasbourg, France (1991)Google Scholar
  3. 3.
    Andrès, E., Acharya, R., Sibata, C.: Discrete analytical hyperplanes. Graphical Models and Image Processing 59, 302–309 (1997)CrossRefGoogle Scholar
  4. 4.
    Kim, C.E., Stojmenovic̀, I.: On the recognition of digital planes in three-dimensional space. Pattern Recognition Letters 12, 665–669 (1991)CrossRefGoogle Scholar
  5. 5.
    Debled-Rennesson, I.: Etude et reconnaissance des droites et plans discrets. PhD thesis, Université Louis Pasteur, Strasbourg, France (1995)Google Scholar
  6. 6.
    Debled, I., Reveillès, J.P.: A new approach to digital planes. In: Spie’s Internat. Symposium on Photonics and Industrial Applications - Technical conference vision geometry 3, Boston (1994)Google Scholar
  7. 7.
    Veelaert, P.: Geometric constructions in the digital plane. Journal of Mathematical Imaging and Vision 11, 99–118 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hardy, G.H., Wright, E.M.: An introduction to the Theory of Numbers. Oxford Society (1989)Google Scholar
  9. 9.
    Hayes, B.: On the teeth of wheels. Computing Science, American Scientist 88(4), 296–300 (2000)Google Scholar
  10. 10.
    McIlroy, M.D.: A note on discrete representation of lines. AT&T Technical Journal 64, 481–490 (1985)Google Scholar
  11. 11.
    Dorst, L., Smeulders, A.N.M.: Discrete representation of straight lines. IEEE Trans. on Pattern Analysis and Machine Intelligence 6, 450–463 (1984)zbMATHCrossRefGoogle Scholar
  12. 12.
    Yaacoub, J.: Enveloppes convexes de réseaux et applications au traitement d’images. PhD thesis, Université Louis Pasteur, Strasbourg, France (1997)Google Scholar
  13. 13.
    Dorst, L., Duin, R.P.W.: Spirograph theory: A framework for calculations on digitized straight lines. IEEE Trans. on Pattern Anal. and Mach. Intell. 6(5), 632–639 (1984)zbMATHCrossRefGoogle Scholar
  14. 14.
    Coeurjolly, D., Sivignon, I., Dupont, F., Feschet, F., Chassery, J.M.: Digital plane preimage structure. In: Del Lungo, A., Di Gesù, V., Kuba, A. (eds.) Electronic Notes in Discrete Mathematics, IWCIA 2003, vol. 12. Elsevier Science Publishers, Amsterdam (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Isabelle Sivignon
    • 1
  • Florent Dupont
    • 2
  • Jean-Marc Chassery
    • 1
  1. 1.Laboratoire LISDomaine universitaire GrenobleSt Martin d’Hères CedexFrance
  2. 2.Laboratoire LIRISUniversité Claude Bernard LyonVilleurbanne cedexFrance

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