Coplanar tricubes

  • Jean-Maurice Schramm
Discrete Shapes and Planes
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1347)


Within the framework of the arithmetic discrete geometry introduced by J.P.Reveillés, I. Debled has defined the concept of tricubes and found out that the total number of the tricubes that may appear in a naive plane is fourty.

This study concerns the coexistence of tricubes in a plane. We call complete combination of coplanar tricubes any set of tricubes, such as a naive plane exists, which contains all the tricubes of this set without any other. We present an algorithm which calculates the set of these combinations.

It appears that the number of these combinations is quite small : only 99, although a “combinatory explosion” could have been expected during their calculation. Their list is given in the appendix.

Key words

arithmetic discrete geometry discrete planes Fourier's algorithm 


  1. [Deb:95]
    I. Debied-Rennesson. Etude et reconnaissance des droites et plans discrets. Thèse de doctorat, Université Louis Pasteur, Strasbourg, 1995.Google Scholar
  2. [Deb,Rev:94]
    I. Debled-Rennesson, J.P. Reveillès. A new approach to digital planes. Vision Geometry III, Boston, 1994.Google Scholar
  3. [Fra,Sch,Taj:96]
    J. Frangon, J.M. Schramm, M. Tajine. Recognizing Arithmetic Straight Lines and Planes. 6 th Confèrence on Discrete Geometry for Computer Imagery, Lyon, 1996. Proceedings: Lecture Notes in Computer Science no1176, Springer, 1996.Google Scholar
  4. [Kuh:56]
    H. W. Kuhn. Solvability and consistency for linear equations and inequalities. The Americain Mathematical Monthly, vol. 63, p. 217–232, 1956.Google Scholar
  5. [Rev:91]
    J.P. Reveiflös. Géomhétrie discréte, calcul en nombres entiers et algorithmique. Thèse de doctorat d'état, Université Louis Pasteur, Strasbourg, 1991.Google Scholar
  6. [Rev:95]
    J.P. ReveiMs. Combinatorial pieces in digital lines and planes. Vision Geometry 4, SPIE'95, San Diego, 1995.Google Scholar
  7. [Sch:97]
    J.M.Schramm. Tricubes coplanaires. RR97-12. LSIIT, Université Louis Pasteur, Strasbourg, 1997.Google Scholar
  8. [Sto-Wit:70]
    J. Stoer, C. Witzgall. Convexity and Optimization in Finite Dimensions I. Die Grundlehren der mathernatischen Wissenschaften in Einzeldarstellungen, Band 163, Springer, 1970.Google Scholar
  9. [Yac:97]
    J.Yaacoub. Enveloppes convexes de réseaux et applications au traitement d'images. Thèse de doctorat, Université Louis Pasteur, Strasbourg, 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Jean-Maurice Schramm
    • 1
  1. 1.Laboratoire des Sciences de l'Image, d'Informatique et de TélédétectionStrasbourg cedexFrance

Personalised recommendations