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Generalized Functionality for Arithmetic Discrete Planes

  • Valerie Berthé
  • Christophe Fiorio
  • Damien Jamet
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3429)

Abstract

The discrete plane β (a,b,c,μ,ω) is the set of integer points (x,y,z)∈ ℤ satisfying 0 ≤ ax+by+cz + μ < ω. In the case ω=max(|a|,|b|,|c|),the discrete plane is said naive and is well-known to be functional on one of the coordinate planes, that is, for any point of P of this coordinate plane, there exists a unique point in the discrete plane obtained by adding to P a third coordinate. Naive planes have been widely studied, see for instance [Rev91, DRR94, DR95, AAS97, VC97, Col02, BB02].

Keywords

digital planes arithmetic planes local configurations functionality of discrete planes 

References

  1. [AAS97]
    Andres, É., Acharya, R., Sibata, C.: The Discrete Analytical Hyperplanes. Graph. Models Image Process 59(5), 302–309 (1997)CrossRefGoogle Scholar
  2. [ABS04]
    Arnoux, P., Berthé, V., Siegel, A.: Two-dimensional Iterated Morphisms and Discrete Planes. Theoret. Comput. Sci. 319, 145–176 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  3. [BB99]
    Brimkov, V.E., Barneva, R.P.: Graceful planes and thin tunnel-free meshes. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 53–64. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  4. [BB02]
    Brimkov, V.E., Barneva, R.P.: Graceful Planes and Lines. Theoret. Comput. Sci. 283, 151–170 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [BV00]
    Berthé, V., Vuillon, L.: Tilings and Rotations on the Torus: A Two-Dimensional Generalization of Sturmian Sequences. Discrete Math. 223, 27–53 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [BV01]
    Berthé, V., Vuillon, L.: Palindromes and Two-Dimensional Sturmian Sequences. J. Autom. Lang. Comb. 6(2), 121–138 (2001)zbMATHMathSciNetGoogle Scholar
  7. [Col02]
    Jacob-Da Col, M.A.: About Local Configurations in Arithmetic Planes. Theoret. Comput. Sci. 283, 183–201 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  8. [DR95]
    Debled-Renesson, I.: Reconnaissance des Droites et Plans Discrets. In: Thèse de doctorat, Université Louis Pasteur, Strasbourg, France (1995)Google Scholar
  9. [DRR94]
    Debled-Renesson, I., Reveillès, J.-P.: A New Approach to Digital Planes. In: Proc. SPIE Vision geometry III, Boston, USA, vol. 2356 (1994)Google Scholar
  10. [FP99]
    Françon, J., Papier, L.: Polyhedrization of the boundary of a voxel object. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 425–434. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  11. [FST96]
    Françon, J., Schramm, J.-M., Tajine, M.: Recognizing Arithmetic Straight Lines and Planes. In: DGCI, 6th International Workshop. LNCS, pp. 141–150. Springer, Heidelberg (1996)Google Scholar
  12. [Gér99]
    Gérard, Y.: Local configurations of digital hyperplanes. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 65–75. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  13. [Rev91]
    Reveillès, J.-P.: Calcul en Nombres Entiers et Algorithmique. In: Thèse d’état, Université Louis Pasteur, Strasbourg, France (1991)Google Scholar
  14. [Rev95]
    Reveillès, J.-P.: Combinatorial Pieces in Digital Lines and Planes. In: Proc. SPIE Vision geometry IV, San Diego, CA, vol. 2573, pp. 23–24 (1995)Google Scholar
  15. [Sch97]
    Schramm, J.-M.: Coplanar Tricubes. In: Ahronovitz, E. (ed.) DGCI 1997. LNCS, vol. 1347, pp. 87–98. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  16. [VC97]
    Vittone, J., Chassery, J.-M.: Coexistence of Tricubes in Digital Naive Plane. In: Ahronovitz, E. (ed.) DGCI 1997. LNCS, vol. 1347, pp. 99–110. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  17. [VC99]
    Vittone, J., Chassery, J.M. (n,m)-cubes and farey nets for naive planes understanding. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 76–87. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  18. [VC00]
    Vittone, J., Chassery, J.-M.: Recognition of digital naive planes and polyhedrization. In: Nyström, I., Sanniti di Baja, G., Borgefors, G. (eds.) DGCI 2000. LNCS, vol. 1953, pp. 296–307. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  19. [Vui99]
    Vuillon, L.: Local Configurations in a Discrete Plane. Bull. Belgian Math. Soc. 6, 625–636 (1999)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Valerie Berthé
    • 1
  • Christophe Fiorio
    • 1
  • Damien Jamet
    • 1
  1. 1.LIRMMUniversité Montpellier IIMontpellierFrance

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