Journal of Automated Reasoning

, Volume 52, Issue 1, pp 99–122 | Cite as

The Reachability Problem in Constructive Geometric Constraint Solving Based Dynamic Geometry

  • Marta R. Hidalgo
  • Robert Joan-Arinyo


An important issue in dynamic geometry is the reachability problem that asks whether there is a continuous path that, from a given starting geometric configuration, continuously leads to an ending configuration. In this work we report on a technique to compute a continuous evaluation path, if one exists, that solves the reachability problem for geometric constructions with one variant parameter. The technique is developed in the framework of a constructive geometric constraint-based dynamic geometry system, uses the A ∗  algorithm and minimizes the variant parameter arc length.


Dynamic geometry Constructive geometric constraint solving Reachability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

10817_2013_9280_MOESM1_ESM.ogv (4.1 mb)
(OGV 4.07 MB)


  1. 1.
    Bouma, W., Fudos, I., Hoffmann, C., Cai, J., Paige, R.: Geometric constraint solver. Comput. Aided Des. 27(6), 487–501 (1995)CrossRefMATHGoogle Scholar
  2. 2.
    Brüderlin, B.D.: Rule-based geometric modelling. PhD thesis, Institut für Informatik der ETH Zürich (1988)Google Scholar
  3. 3.
    Bournez, O., Potapov, I. (eds.): Reachability problems. In: LNCS, Theoretical Notes in Computer Science, vol. 5797. Palaiseau, France (2009)Google Scholar
  4. 4.
    Denner-Broser, B.: On the decidability of tracing problems in dynamic geometry. In: Hong, H., Wang, D. (eds.) LNAI 3763, pp. 111–129. Springer (2006)Google Scholar
  5. 5.
    Denner-Broser, B.: Tracing problems in dynamic geometry. PhD thesis, Institut für Informatik, Freie Universität Berlin (2008)Google Scholar
  6. 6.
    Denner-Broser, B.: An algorithm for the tracing problem using interval analysis. In: SAC’08, pp. 1832–1837. Fortaleza, Ceará, Brazil (2008)Google Scholar
  7. 7.
    DynBCN: A constructive geometric constraint-based dynamic geometry system. Accessed 27 Nov 2012
  8. 8.
    Foley, J., van Dam, A., Feiner, S., Hughes, J.: Computer graphics. In: Principles and Practice, 2nd edn. Addison-Wesley, Reading (1996)Google Scholar
  9. 9.
    Freixas, M., Joan-Arinyo, R., Soto-Riera, A.: A constraint-based dynamic geometry system. Comput. Aided Des. 42(2), 151–161 (2010)CrossRefGoogle Scholar
  10. 10.
    Fudos, I., Hoffmann, C.M.: Correctness proof of a geometric constraint solver. Int. J. Comput. Geom. Appl. 6(4), 405-420 (1996)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Golledge, R., Klatzy, R., Loomis, J., Speigle, J., Tietz, J.: A geographical information system for a GPS based personal guidance system. Int. J. Geogr. Inf. Sci. 12(7), 727–749 (1998)CrossRefGoogle Scholar
  12. 12.
    Halabi, S.: Internet Routing Architectures, 2nd edn. Cisco Press (2000)Google Scholar
  13. 13.
    Hidalgo, M., Joan-Arinyo, R.: Computing parameter ranges in constructive geometric constraint solving: implementation and correctness proof. Comput. Aided Des. 44(7), 709–720 (2012)CrossRefGoogle Scholar
  14. 14.
    Hoffmann, C., Joan-Arinyo, R.: A brief on constraint solving. Comput. Aided Des. Appl. 2(5), 655–663 (2005)Google Scholar
  15. 15.
    Hoffmann, C., Lomonosov, A., Sitharam, M.: Decompostion plans for geometric constraint systems, part I: performance measurements for CAD. J. Symb. Comput. 31, 367–408 (2001)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Hoffmann, C., Lomonosov, A., Sitharam, M.: Decompostion plans for geometric constraint problems, part II: new algorithms. J. Symb. Comput. 31, 409–427 (2001)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Jerman, C., Trombettoni, G., Neveu, B., Mathis, P.: Decomposition of geometric constraint systems: a survey. Int. J. Comput. Geom. Appl. 16(5–6), 379–414 (2006)CrossRefGoogle Scholar
  18. 18.
    Joan-Arinyo, R., Soto, A.: A correct rule-based geometric constraint solver. Comput. Graph. 21(5), 599–609 (1997)CrossRefGoogle Scholar
  19. 19.
    Joan-Arinyo, R., Soto-Riera, A., Vila-Marta, S., Vilaplana, J.: On the domain of constructive geometric constraint solving techniques. In: Duricovic, R., Czanner, S. (eds.) Spring Conference on Computer Graphics, pp. 49–54. IEEE Computer Society, Los Alamitos, CA, Budmerice, Slovakia (2001)Google Scholar
  20. 20.
    Joan-Arinyo, R., Luzón, M., Soto, A.: Genetic algorithms for root multiselection in constructive geometric constraint solving. Comput. Graph. 27(1), 51–60 (2003)CrossRefGoogle Scholar
  21. 21.
    Joan-Arinyo, R., Luzón, M., Yeguas, E.: Search space pruning to solve the root identification problem in geometric constraint solving. Comput. Aided Des. Appl. 6(1), 15–25 (2009)Google Scholar
  22. 22.
    Kang, M.W., Pha, M., Hwang, D.: Part I. A GIS-based simulation model for positioning & routing unmanned ground vehicles. Tech. rep., Center for Advanced Transportation and Infrastrcuture Engineering, Morgan State University (2010)Google Scholar
  23. 23.
    Kleene, S.C.: Mathematical Logic. Wiley, New York (1967)MATHGoogle Scholar
  24. 24.
    Kortenkamp, U.: Foundations of dynamic geometry. PhD Thesis, Swiss Federal Institute of Technology Zurich (1999)Google Scholar
  25. 25.
    Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4(4), 331–340 (1970)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Lovász, L., Yemini, Y.: On generic rigidity in the plane. SIAM J. Algebr. Discrete Methods 3(1), 91–98 (1982)CrossRefMATHGoogle Scholar
  27. 27.
    Richter-Gebert, J., Kortenkamp, U.H.: Complexity issues in dynamic geometry. In: Proceedings of the Smale Fest 2000, Foundations Computational Mathematics, pp. 1–37. World Scientific, Singapore (2001)Google Scholar
  28. 28.
    Russell, S., Norvig, P.: Artificial Intelligence: A Modern Approach. Prentice Hall, Upper Saddle River (2003)Google Scholar
  29. 29.
    van der Meiden, H., Bronsvoort, W.: An efficient method to determine the intended solution for a system of geometric constraints. Int. J. Comput. Geom. 15(3), 79–98 (2005)Google Scholar
  30. 30.
    van der Meiden, H., Bronsvoort, W.: A constructive approach to calculate parameter ranges for systems of geometric constraints. Comput. Aided Des. 38(4), 275–283 (2006)CrossRefGoogle Scholar
  31. 31.
    Winroth, H.: Dynamic projective geometry. PhD Thesis, Stockholms Universitet (1999)Google Scholar
  32. 32.
    Yang, J., Dymond, P., Jenkin, M.: Exploiting hierarchical probabilistic motion planning for robot reachable workspace estimation. Informatics Control Autom. Robot 85, 229–241 (2011). doi: 10.1007/978-3-642-19730-7_16 CrossRefGoogle Scholar
  33. 33.
    Ying, Z., Iyengar, S.S.: Robot reachability problem: a nonlinear optimization approach. J. Intell. Robot. Syst. 12(1), 87–100 (2011). doi: 10.1007/BF01258308 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Grup d’Informàtica a l’EnginyeriaUniversitat Politècnica de CatalunyaBarcelonaCatalonia

Personalised recommendations