Skip to main content
SpringerLink
Log in

Configuration Space of Geometric Objects

  • Published:
Cybernetics and Systems Analysis Aims and scope

Abstract

The concept of configuration space of geometric objects is introduced. Its generalized variables are metric parameters of the spatial form and parameters of the location of objects. The properties of configuration spaces of complex geometric objects are considered. The structures of configuration spaces for various classes of geometric object placement problems, including packing and covering problems, are analyzed. The concept of Φ-function of geometric objects with variable metrical parameters is generalized.

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. B. Grunbaum, “Configurations of points and lines” in: Graduate Studies in Mathematics, American Mathematical Society, Vol. 103, Providence, Rhode Island (2009).

  2. T. Pisanski and B. Servatius, “Configurations from a graphical viewpoint,” in: Combinatorial Configurations, Birkhauser, Boston (2013), pp. 157–191.

    MATH  Google Scholar 

  3. C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs, CRC Press (2010).

  4. H. Gropp, “Configurations between geometry and combinatorics,” Discrete Applied Mathematics, Vol. 138, No. 1, 79–88 (2004).

    Article  MathSciNet  Google Scholar 

  5. C. Berge, Principes de combinatoire, Dunod, Paris (1968).

    MATH  Google Scholar 

  6. H. J. Ryser, “Combinatorial configurations,” SIAM J. on Applied Mathematics, Vol. 17, No. 3, 593–602 (1969).

    Article  MathSciNet  Google Scholar 

  7. V. I. Arnold, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1989).

    Book  Google Scholar 

  8. S. A. Solla, G. B. Sorkin, and S. R. White, “Configuration space analysis for optimization problems,” in: E. Bienenstock et al. (eds.), Disordered Systems and Biological Organization, Springer, Berlin–Heidelberg (1986), pp. 283–293.

    Chapter  Google Scholar 

  9. E. Fadell and L. Neuwirth, “Configuration space,” Math. Scand., Vol. 10, 111–118 (1962).

    Article  MathSciNet  Google Scholar 

  10. C. Westerland, “Configuration spaces in geometry and topology,” Australian Mathematical Society Gazette, Vol. 38, No. 5, 279–283 (2011).

    MathSciNet  MATH  Google Scholar 

  11. E. R. Fadell and S. Y. Husseini, Geometry and Topology of Configuration Spaces, Springer Monographs in Mathematics (2001).

  12. F. R. Cohen and S. Gitler, “On loop spaces of configuration spaces,” Trans. Amer. Math. Soc., Vol. 354, No. 5, 1705–1748 (2002).

    Article  MathSciNet  Google Scholar 

  13. Y. G. Stoyan, “Mathematical methods for geometric design,” in: Advances in CAD/CAM, Proc. of PROLAMAT82, May 1982, Leningrad, USSR, North–Holland, Amsterdam (2003).

  14. Yu. G. Stoyan and S. V. Yakovlev, Mathematical Models and Optimization Methods in Geometric Design [in Russian], Naukova Dumka, Kyiv (1986).

    Google Scholar 

  15. Y. Stoyan, M. Gil, J. Terno, T. Romanova, and G. Schithauer, “Φ-function for complex 2D objects,” 4OR — Quarterly J. of the Belgian, French and Italian Operations Research Societies, Vol. 2, No. 1, 69–84 (2004).

    MathSciNet  Google Scholar 

  16. Yu. Stoyan, G. Scheithauer, and T. Romanova, “Mathematical modeling of interaction of primary geometric 3D objects,” Cybern. Syst. Analysis, Vol. 41, No. 3, 332–342 (2005).

    Article  MathSciNet  Google Scholar 

  17. Yu. Stoyan and T. Romanova, “Mathematical models of placement optimization: Two- and three-dimensional problems and applications,” in: G. Fasano and J. Pintér (eds.), Modeling and Optimization in Space Engineering, Vol. 73, Springer, New York (2013), pp. 363–388.

    Chapter  Google Scholar 

  18. J. Bennell, G. Scheithauer, Y. G. Stoyan, and T. Romanova, “Tools of mathematical modelling of arbitrary object packing problems,” J. Annals of Operations Research, Springer Netherlands Publ., Vol. 179, No. 1, 343–368 (2010).

    Article  Google Scholar 

  19. N. Chernov, Y. Stoyan, and T. Romanova, “Mathematical model and efficient algorithms for object packing problem,” Computational Geometry: Theory and Applications, Vol. 43, No. 5, 535–553 (2010).

    Article  MathSciNet  Google Scholar 

  20. Yu. Stoyan, T. Romanova, A. Pankratov, and A. Chugay, “Optimized object packings using quasi-phi-functions,” in: G. Fasano and J. D. Pintér (eds.), Optimized Packings with Applications, Vol. 105, Springer, New York (2015), pp. 265–293.

    Chapter  Google Scholar 

  21. Yu. Stoyan, A. Pankratov, and T. Romanova, “Placement problems for irregular objects: Mathematical modeling, optimization and applications,” in: S. Butenko et al. (eds.), Optimization Methods and Applications, Springer, New York (2017), pp. 521–558.

    Chapter  Google Scholar 

  22. V. L. Rvachev, R-Functions Theory and Some of its Applications [in Russian], Naukova Dumka, Kyiv (1982).

    MATH  Google Scholar 

  23. L. Hulianytskyi and I. Riasna, “Formalization and classification of combinatorial optimization problems,” in: S. Butenko et al. (eds.), Optimization Methods and Applications, Springer, New York (2017), pp. 239–250.

    Chapter  Google Scholar 

  24. I. V. Sergienko, L. F. Hulianytskyi, and S. I. Sirenko, “Classification of applied methods of combinatorial optimization,” Cybern. Syst. Analysis, Vol. 45, No. 5, 732–744 (2009).

    Article  MathSciNet  Google Scholar 

  25. A. Bortfeldt and G. Wascher, “Constraints in container loading: A state of the art review,” Europ. J. of Operational Research, Vol. 229, No. 1, 1–20 (2013).

    Article  MathSciNet  Google Scholar 

  26. G. Fasano, “A modeling-based approach for non-standard packing problems,” in: G. Fasano and J. D. Pintér (eds.), Optimized Packings with Applications, Vol. 105, Springer, New York (2015), pp. 67–85.

    Chapter  Google Scholar 

  27. M. Hifi and R. M’Hallah, “A literature review on circle and sphere packing problems: Model and methodologies,” Advances in Optimization Research, Vol. 2009, 1–22 (2009).

    Article  Google Scholar 

  28. E. G. Birgin, J. M. Martinez, F. H. Nishihara, and D. P. Ronconi, “Orthogonal packing of rectangular items within arbitrary convex regions by nonlinear optimization,” Comput. Oper. Res., Vol. 33, 3535–3548 (2006).

    Article  MathSciNet  Google Scholar 

  29. J. Egeblad, B. K. Nielsen, and M. Brazil, “Translational packing of arbitrary polyhedral,” Comp. Geom., Vol. 142, No. 4, 269–288 (2009).

    Article  Google Scholar 

  30. G. A. Fasano, “Global optimization point of view for non-standard packing problems,” J. of Global Optimization, Vol. 155, No. 2, 279–299 (2013).

    Article  MathSciNet  Google Scholar 

  31. Yu. Stoyan, A. Pankratov, and T. Romanova, “Cutting and packing problems for irregular objects with continuous rotations: Mathematical modeling and nonlinear optimization,” J. of Operational Research Society, Vol. 167, No. 5, 786–800 (2016).

    Article  Google Scholar 

  32. A. Drira, H. Pierreval, and S. Hajri-Gabouj, “Facility layout problems: A survey,” Annual Reviews in Control, Vol. 31, No. 2, 255–267 (2007).

    Article  Google Scholar 

  33. G. M. Fadel and M. M. Wiecek, “Packing optimization of free-form objects in engineering design,” in: G. Fasano and J. Pintér (eds.), Optimized Packings with Applications, Vol. 105, Springer, New York (2015), pp. 37–66.

    Chapter  Google Scholar 

  34. Yu. G. Stoyan, V. V. Semkin, and A. M. Chugay, “Optimization of 3D objects layout into a multiply connected domain with account for shortest distances,” Cybern. Syst. Analysis, Vol. 50, No. 3, 374–385 (2014).

    Article  MathSciNet  Google Scholar 

  35. Zhi-Guo Sun and Hong-Fei Teng, “Optimal layout design of a satellite module,” Engineering Optimization, Vol. 35, No. 5, 513–529 (2003).

    Article  Google Scholar 

  36. Yu. Stoyan, T. Romanova, A. Pankratov, A. Kovalenko, and P. Stetsyuk, “Balance layout problems: Mathematical modeling and nonlinear optimization,” in: G. Fasano and J. Pintér (eds.), Space Engineering. Modeling and Optimization with Case Studies, Vol. 114, Springer, New York (2016), pp. 369–400.

    MATH  Google Scholar 

  37. Yi-Chun Xu, Ren-Bin Xiao, and M. Amos, “A novel genetic algorithm for the layout optimization problem,” in: 2007 IEEE Congr. on Evolutionary Computation, CEC 2007 (2007), pp. 3938–3942.

  38. Yu. G. Stoyan, V. Z. Sokolovskii, and S. V. Yakovlev, “Method of balancing rotating discretely distributed masses,” Energomashinostroenie, No. 2, 4–5 (1982).

  39. Y. G. Stoyan, S. V. Yakovlev, and O. V. Parshin, “Quadratic optimization on combinatorial sets in R n ,” Cybern. Syst. Analysis, Vol. 27, No. 4, 561–567 (1991).

    Article  Google Scholar 

  40. A. Bortfeldt and G. Wascher, “Constraints in container loading: A state of the art review,” Europ. J. of Operational Research, Vol. 229, No. 1, 1–20 (2013).

    Article  MathSciNet  Google Scholar 

  41. Yu. G. Stoyan and V. M. Patsuk, “Covering a convex 3D polytope by a minimal number of congruent spheres,” Intern. J. of Computer Mathematics, Vol. 91, No. 9, 2010–2020 (2014).

    Article  MathSciNet  Google Scholar 

  42. S. V. Yakovlev, “On a class of problems on covering of a bounded set,” Acta Mathematica Hungarica, Vol. 53, No. 3, 253–262 (1989).

    Article  MathSciNet  Google Scholar 

  43. S. N. Gerasin, V. V. Shlyakhov, and S. V. Yakovlev, “Set coverings and tolerance relations,” Cybern. Syst. Analysis, Vol. 44, No. 3, 333–340 (2008).

    Article  Google Scholar 

  44. S. B. Shekhovtsov and S. V. Yakovlev, “Formalization and solution of one class of covering problem in design of control and monitoring systems,” Autom. Remote Control, Vol. 50, No. 5, 705–710 (1989).

    MATH  Google Scholar 

  45. E. M. Kiseleva, L. I. Lozovskaya, and E. V. Timoshenko, “Solution of continuous problems of optimal covering with spheres using optimal set-partition theory,” Cybern. Syst. Analysis, Vol. 45, No. 3, 421–437 (2009).

    Article  Google Scholar 

  46. E. M. Kiseleva and L. S. Koriashkina, Models and Methods of the Solution of Continuous Problems of Optimal Partition of Sets: Linear, Nonlinear, and Dynamic Problems [in Russian], Naukova Dumka, Kyiv (2013).

    Google Scholar 

  47. E. M. Kiseleva and L. S. Koriashkina, “Theory of continuous optimal set partitioning problems as a universal mathematical formalism for constructing Voronoi diagrams and their generalizations,” Cybern. Syst. Analysis, Vol. 51, No. 3, 325–335 (2015).

    Article  MathSciNet  Google Scholar 

  48. Yu. G. Stoyan, S. V. Yakovlev, and O. S. Pichugina, Euclidean Combinatorial Configurations [in Russian], Konstanta, Kharkiv (2017).

    Google Scholar 

  49. S. Yakovlev, “Convex extensions in combinatorial optimization and their applications,” in: S. Butenko et al. (eds.), Optimization Methods and Applications, Springer, New York (2017), pp. 567–584.

    Chapter  Google Scholar 

  50. O. S. Pichugina and S. V. Yakovlev, “Continuous representations and functional extensions in combinatorial optimization,” Cybern. Syst. Analysis, Vol. 52, No. 6, 921–930 (2016).

    Article  MathSciNet  Google Scholar 

  51. S. V. Yakovlev, “Bounds on the minimum of convex functions on Euclidean combinatorial sets,” Cybernetics, Vol. 25, No. 3, 385–391 (1989).

    Article  MathSciNet  Google Scholar 

  52. S. V. Yakovlev and I. V. Grebennik, “Localization of solutions of some problems of nonlinear integer optimization,” Cybern. Syst. Analysis, Vol. 29, No. 5, 727–734 (1993).

    Article  Google Scholar 

  53. S. V. Yakovlev and O. A. Valuiskaya, “Optimization of linear functions at the vertices of a permutation polyhedron with additional linear constraints,” Ukrainian Mathematical J., Vol. 53(9), 1535–1545 (2001).

    Article  MathSciNet  Google Scholar 

  54. S. V. Yakovlev and O. S. Pichugina, “Properties of combinatorial optimization problems over polyhedral–spherical sets,” Cybern. Syst. Analysis, Vol. 54, No. 1, 99–109 (2018).

    Article  Google Scholar 

  55. O. Pichugina and S. Yakovlev, “Optimization on polyhedral-spherical sets: Theory and applications,” in: Proc. 2017 IEEE First Ukrain. Conf. on Electrical and Computer Engeneering, UKRCON (2017), pp. 1167–1175.

  56. S. V. Yakovlev, “The method of artificial dilation in problems of optimal packing of geometric objects,” Cybern. Syst. Analysis, Vol. 53, No. 5, 725–731 (2017).

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. A. M. Pidhorny Institute for Mechanical Engineering Problems, National Academy of Sciences of Ukraine, Kharkiv, Ukraine

    Y. G. Stoyan

  2. M. E. Zhukovsky National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine

    S. V. Yakovlev

Authors
  1. Y. G. Stoyan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. V. Yakovlev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Y. G. Stoyan.

Additional information

Translated from Kibernetika i Sistemnyi Analiz, No. 5, September–October, 2018, pp. 38–50.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Stoyan, Y.G., Yakovlev, S.V. Configuration Space of Geometric Objects. Cybern Syst Anal 54, 716–726 (2018). https://doi.org/10.1007/s10559-018-0073-5

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10559-018-0073-5

Keywords

Search

Navigation