Skip to main content
SpringerLink
Log in

Characterization and Detection of Toric Loops in n-Dimensional Discrete Toric Spaces

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

Since a toric space is not simply connected, it is possible to find in such spaces some loops which are not homotopic to a point: we call them toric loops. Some applications, such as the study of the relationship between the geometrical characteristics of a material and its physical properties, rely on three-dimensional discrete toric spaces and require detecting objects having a toric loop.

In this work, we study objects embedded in discrete toric spaces, and propose a new definition of loops and equivalence of loops. Moreover, we introduce a characteristic of loops that we call wrapping vector: relying on this notion, we propose a linear time algorithm which detects whether an object has a toric loop or not.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, Amsterdam (1972)

    Google Scholar 

  2. Bertrand, G., Couprie, M., Passat, N.: A note on 3-D simple points and simple-equivalence. Inf. Process. Lett. 13(109), 700–704 (2009)

    Article  MathSciNet  Google Scholar 

  3. Chaussard, J., Bertrand, G., Couprie, M.: Characterizing and detecting toric loops in n-dimensional discrete toric spaces. In: Coeurjolly, D., Sivignon, I., Tougne, L., Dupont, F. (eds.) DGCI. Lecture Notes in Computer Science, vol. 4992. Springer, Berlin (2008)

    Google Scholar 

  4. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley Longman, Boston (1994)

    MATH  Google Scholar 

  5. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  6. Kong, T.Y.: A digital fundamental group. Comput. Graph. 13(2), 159–166 (1989)

    Article  Google Scholar 

  7. Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Vis. Graph. Image Process. 48(3), 357–393 (1989)

    Article  Google Scholar 

  8. Malgouyres, R.: Computing the fundamental group in digital spaces. Int. J. Pattern Recogn. Artif. Intell. 15(7), 1053–1074 (2001)

    Article  Google Scholar 

  9. Maunder, C.: Algebraic Topology. Dover, New York (1996)

    Google Scholar 

  10. Poincaré, H.: Analysis situs. J. Ecole Polytech., Sér. 2 1, 1–121 (1895)

    Google Scholar 

  11. Stillwell, J.: Classical Topology and Combinatorial Group Theory. Springer, Berlin (1980)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université Paris Est, Labinfo-Igm, A2SI-Esiee, 2, boulevard Blaise Pascal, Cité Descartes, BP 99, 93162, Noisy le Grand Cedex, France

    John Chaussard, Gilles Bertrand & Michel Couprie

Authors
  1. John Chaussard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gilles Bertrand
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michel Couprie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to John Chaussard.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chaussard, J., Bertrand, G. & Couprie, M. Characterization and Detection of Toric Loops in n-Dimensional Discrete Toric Spaces. J Math Imaging Vis 36, 111–124 (2010). https://doi.org/10.1007/s10851-009-0175-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10851-009-0175-9

Search

Navigation