Reliable Computing

, Volume 11, Issue 3, pp 235–251 | Cite as

Tolerances in Geometric Constraint Problems

  • Johannes WallnerEmail author
  • Hans-Peter Schröcker
  • Shi-Min Hu


We study error propagation through implicit geometric problems by linearizing and estimating the linearization error. The method is particularly useful for quadratic constraints, which turns out to be no big restriction for many geometric problems in applications.


Mathematical Modeling Computational Mathematic Industrial Mathematic Error Propagation Geometric Constraint 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Asimov, L. and Roth, B.: The Rigidity of Graphs, Trans. Amer. Math. Soc. 245 (1978), pp. 171–190.Google Scholar
  2. 2.
    Asimov, L. and Roth, B.: The Rigidity of Graphs II, J. Math. Anal. Appl. 68 (1979), pp. 171–190.CrossRefGoogle Scholar
  3. 3.
    Bhatia, R.: Matrix Analysis, Springer, 1997.Google Scholar
  4. 4.
    Bouma, W., Fudos, I., Hoffmann, C., Cai, J., and Paige, R.: Geometric Constraint Solver, Computer-Aided Design 27 (1995), pp. 487–501.CrossRefGoogle Scholar
  5. 5.
    Bruederlin, B.: Using Geometric Rewriting Rules for Solving Geometric Problems Symbolically, Theoret. Comput. Sci. 116 (1993), pp. 291–303.CrossRefGoogle Scholar
  6. 6.
    Cauchy, A. L.: IIe mémoire sur les polygones et les polyèdres, Journal de l’Ecole Polytechnique 19 (1813), pp. 87–98.Google Scholar
  7. 7.
    Conelly, R.: On Generic Global Rigidity, in: Gritzmann, P. and Sturmfels, B.(eds), Applied Geometry and Discrete Mathematics—The Victor Klee Festschrift, American Mathematical Society,Google Scholar
  8. 8.
    Crippen, G. M. and Havel, T. F.: Distance Geometry and Molecular Conformation, Chemometrics Series 15, Research Studies Press Ltd., Chichester, 1988.Google Scholar
  9. 9.
    Fudos, I. and Hoffmann, C.: A Graph-Constructive Approach to Solving Systems of Geometric Constraints, ACMTransactions on Graphics 16 (1997), pp. 179–216.CrossRefGoogle Scholar
  10. 10.
    Gao, X. S. and Chou, S. C.: Solving Geometric Constraint Systems. I.A Global Propagation Approach, Computer-Aided Design 30 (1998), pp. 47–54.CrossRefGoogle Scholar
  11. 11.
    Gao, X. S. and Chou, S. C.: Solving Geometric Constraint Systems. II.A Symbolic Approach and Decision of rc-Constructibility, Computer-Aided Design 30 (1998), pp. 115–122.CrossRefGoogle Scholar
  12. 12.
    Ghosh, P.: A Unified Computational Framework for Minkowski Operations, Computers & Graphics 17 (1993), pp. 357–378.Google Scholar
  13. 13.
    Higham, N. J.: Accuracy and Stability of Numerical Algorithms, Soc. Industrial and Appl. Math, 1996, Section 6.2.Google Scholar
  14. 14.
    Hoffmann, C. M.: Robustness in Geometric Computations, Journal of Computing and InformationScience in Engineering 1 (2001), pp. 143–156.CrossRefGoogle Scholar
  15. 15.
    Hoffmann, C. M. and Vermeer, P.: Geometric Constraint Solving in R2 and R3, in: Du, D.-Z. and Hwang, F. K.(eds.), Computing in Euclidean Geometry, World Scientific, 1995, pp. 266–298.Google Scholar
  16. 16.
    Hu, S.-M. and Wallner, J.: ErrorPropagation through Geometric Transformations, Technical Report 102, Institut four Geometrie, TU Wien, 2003.Google Scholar
  17. 17.
    Kondo, K.: Algebraic Method for Manipulation of Dimensional Relationships in Geometric Models, Computer-Aided Design 24 (1992), pp. 141–147.CrossRefGoogle Scholar
  18. 18.
    Laman, G.: On Graphs and Rigidity of Plane Skeletal Structures, J. Engrg. Math. 4 (1970), pp. 331–340.CrossRefGoogle Scholar
  19. 19.
    Lamure, H. and Michelucci, D.: Solving Geometric Constraints by Homotopy, IEEE Trans. Vis. Comp. Graph. 2 (1996), pp. 28–34.CrossRefGoogle Scholar
  20. 20.
    Lee, J. Y. and Kim, K.: A 2-D Geometric Constraint Solver Using DOF-Based Graph Reduction, Computer-Aided Design 30 (1998), pp. 883–896.CrossRefGoogle Scholar
  21. 21.
    Lee, K.-Y., Kwon, O.-H., Lee, J.-Y., and Kim, T. W.: A Hybrid Approach to Geometric Constraint Solving with Graph Analysis and Reduction, Adv. Eng. Software 34 (2003), pp. 103–113.CrossRefGoogle Scholar
  22. 22.
    Li, Y.-T., Hu, S.-M., and Sun, J.-G.: A Constructive Approach to Solving 3-D Geometric Constraint Systems Using Dependence Analysis, Computer-Aided Design 34 (2002), pp. 97–108CrossRefGoogle Scholar
  23. 23.
    Light, R. A. and Gossard, D. C.: Modification of Geometric Models through Variational Geometry, Computer-Aided Design 14 (1982), pp. 209–214.CrossRefGoogle Scholar
  24. 24.
    Pottmann, H., Odehnal, B., Peternell, M., Wallner, J., and Haddou, R. A.: On Optimal Tolerancing in Computer-Aided Design, in: Martin, R. and Wang, W.(eds), Geometric Modeling and Processing 2000, IEEE Computer Society, Los Alamitos, 2000, pp. 347–363.Google Scholar
  25. 25.
    Pottmann, H. and Wallner, J.: Computational Line Geometry, Springer, 2001.Google Scholar
  26. 26.
    Requicha, A. A. G.: Towards a Theory of Geometric Tolerancing, Internat. J. Robotics Res. 2 (1983), pp. 45–60.Google Scholar
  27. 27.
    Servatius, B. and Whiteley, W.: Constraining Plane Configurations in Computer-Aided Design: Combinatorics of Directions and Lengths, SIAM J. Discret. Math. 12 (1999), pp. 136–153.CrossRefGoogle Scholar
  28. 28.
    Stachel, H.: Higher-Order Flexibility for a Bipartite Planar Framework, in: Kecskeméthy, A., Schneider, M., and Woernle, C.(eds) Advances in Multi-Body Systems and Mechatronics, Inst. f. Mechanik und Getriebelehre der TU Graz, Duisburg, 1999, pp. 345–357.Google Scholar
  29. 29.
    Verroust, A., Schonek, F., and Roller, D.: Rule-Oriented Method for Parametrized ComputerAided Design, Computer-Aided Design 25 (1993), pp. 531–540.Google Scholar
  30. 30.
    Wallner, J., Krasauskas, R., and Pottmann, H.: Error Propagation in Geometric Constructions, Computer-Aided Design 32 (2000), pp. 631–641.CrossRefGoogle Scholar
  31. 31.
    Whitely, W.: Infinitesimal Motions of a Bipartite Framework, Pacific J. Math. 110 (1984), pp. 233–255.Google Scholar
  32. 32.
    Wunderlich, W.: Ebene Kinematik, BI-Hochschultaschenb ücher 447/447a, Bibliograph. Inst., Mannheim, 1970.Google Scholar
  33. 33.
    Wunderlich, W.: Über Ausnahmefachwerke, deren Knoten auf einem Kegelschnitt liegen, Acta Math. 47 (1983), pp. 291–300.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  • Johannes Wallner
    • 1
    Email author
  • Hans-Peter Schröcker
    • 1
  • Shi-Min Hu
    • 2
  1. 1.Institute of Discrete Mathematics and GeometryVienna University of TechnologyWienAustria
  2. 2.Department of Computer Science and TechnologyTsinghua UniversityBeijingChina

Personalised recommendations