Abstract
Toric spaces being non-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.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Bear, J.: Dynamics of Fluids in Porous Media. Elsevier, Amsterdam (1972)
Hatcher, A.: Algebraic topology. Cambridge University Press, Cambridge (2002)
Stillwell, J.: Classical Topology and Combinatorial Group Theory. Springer, Heidelberg (1980)
Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley Longman Publishing Co., Inc, Boston, MA, USA (1994)
Kong, T.Y.: A digital fundamental group. Computers & Graphics 13(2), 159–166 (1989)
Chaussard, J., Bertrand, G., Couprie, M.: Characterization and detection of loops in n-dimensional discrete toric spaces. Technical Report IGM2008-02, Université de Marne-la-Vallée (2008)
Poincaré, H.: Analysis situs. Journal de l’Ecole Polytechnique, 2eme série, cahier 1, 1–121 (1895)
Malgouyres, R.: Computing the fundamental group in digital spaces. IJPRAI 15(7) (2001)
Bertrand, G., Couprie, M., Passat, N.: 3-D simple points and simple-equivalence (submitted, 2007)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chaussard, J., Bertrand, G., Couprie, M. (2008). Characterizing and Detecting Toric Loops in n-Dimensional Discrete Toric Spaces. In: Coeurjolly, D., Sivignon, I., Tougne, L., Dupont, F. (eds) Discrete Geometry for Computer Imagery. DGCI 2008. Lecture Notes in Computer Science, vol 4992. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79126-3_13
Download citation
DOI: https://doi.org/10.1007/978-3-540-79126-3_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-79125-6
Online ISBN: 978-3-540-79126-3
eBook Packages: Computer ScienceComputer Science (R0)
