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Facet Connectedness of Arithmetic Discrete Hyperplanes with Non-Zero Shift

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Part of the Lecture Notes in Computer Science book series (LNIP,volume 11414)

Abstract

We present a criterion for the arithmetic discrete hyperplane to be facet connected when \(\theta \) is the connecting thickness . We encode the shift \(\mu \) in a numeration system associated with the normal vector and we describe an incremental construction of the plane based on this encoding. We deduce a connectedness criterion and we show that when the Fully Subtractive algorithm applied to has a periodic behaviour, the encodings of shifts \(\mu \) for which the plane is connected may be recognised by a finite state automaton.

Keywords

  • Discrete hyperplane
  • Connectedness
  • Connecting thickness
  • Fully subtractive algorithm
  • Numeration system
  • Finite state automaton

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References

  1. Andrès, E., Acharya, R., Sibata, C.: Discrete analytical hyperplanes. CVGIP: Graph. Model Image Process. 59(5), 302–309 (1997)

    Google Scholar 

  2. Avila, A., Delecroix, V.: Some monoids of Pisot matrices, preprint (2015). https://arxiv.org/abs/1506.03692

  3. Brimkov, V.E., Barneva, R.P.: Connectivity of discrete planes. Theor. Comput. Sci. 319(1–3), 203–227 (2004). https://doi.org/10.1016/j.tcs.2004.02.015

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. Domenjoud, E., Jamet, D., Toutant, J.-L.: On the connecting thickness of arithmetical discrete planes. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 362–372. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04397-0_31

    CrossRef  MATH  Google Scholar 

  5. Domenjoud, E., Provençal, X., Vuillon, L.: Facet connectedness of discrete hyperplanes with zero intercept: the general case. In: Barcucci, E., Frosini, A., Rinaldi, S. (eds.) DGCI 2014. LNCS, vol. 8668, pp. 1–12. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-09955-2_1

    CrossRef  MATH  Google Scholar 

  6. Domenjoud, E., Vuillon, L.: Geometric palindromic closures. Uniform Distribution Theory 7(2), 109–140 (2012). https://math.boku.ac.at/udt/vol07/no2/06DomVuillon13-12.pdf

    MathSciNet  MATH  Google Scholar 

  7. Jamet, D., Toutant, J.L.: Minimal arithmetic thickness connecting discrete planes. Discrete Appl. Math. 157(3), 500–509 (2009). https://doi.org/10.1016/j.dam.2008.05.027

    MathSciNet  CrossRef  MATH  Google Scholar 

  8. Kraaikamp, C., Meester, R.: Ergodic properties of a dynamical system arising from percolation theory. Ergodic Theory Dyn. Syst. 15(04), 653–661 (1995). https://doi.org/10.1017/S0143385700008592

    MathSciNet  CrossRef  MATH  Google Scholar 

  9. Réveillès, J.P.: Géométrie discrète, calcul en nombres entiers et algorithmique. Thèse d’état, Université Louis Pasteur, Strasbourg, France (1991)

    Google Scholar 

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Domenjoud, E., Laboureix, B., Vuillon, L. (2019). Facet Connectedness of Arithmetic Discrete Hyperplanes with Non-Zero Shift. In: Couprie, M., Cousty, J., Kenmochi, Y., Mustafa, N. (eds) Discrete Geometry for Computer Imagery. DGCI 2019. Lecture Notes in Computer Science(), vol 11414. Springer, Cham. https://doi.org/10.1007/978-3-030-14085-4_4

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  • DOI: https://doi.org/10.1007/978-3-030-14085-4_4

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-14084-7

  • Online ISBN: 978-3-030-14085-4

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