On the Use of Conformal Geometric Algebra in Geometric Constraint Solving


To model a geometrical part in Computer Aided Design systems, declarative modeling is a well-adapted solution to declare and specify geometric objects and constraints. In this chapter, we are interested in the representation of geometric objects and constraints using a new language of description and representation, Geometric Algebra (GA). GA is used here in association with the conformal model of Euclidean geometry (CGA) which requires two extra dimensions comparing to the usual vector space model. Topologically and Technologically Related Surfaces (TTRS) Theory is introduced here as a unified framework for geometric objects representation and geometric constraints solving. Based on TTRS, this chapter shows the capability of the CGA to represent geometric objects and geometric constraints through symbolic geometric constraints solving and algebraic classification.


Geometric Constraint Geometric Object Rigid Body Motion Rigid Motion Geometric Algebra 
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  1. 1.
    Ait-Aoudia, S., Bahriz, M., Salhi, L.: 2D geometric constraint solving: an overview. In: Proceedings of 2nd International Conference in Visualisation (VIZ), Barcelona (Spain), July 15–17, 2009, pp. 201–206. IEEE Comput. Soc., Los Alamitos (2009) CrossRefGoogle Scholar
  2. 2.
    Bettig, B., Hoffmann, C.M.: Geometric constraint solving in parametric computer-aided design. doi: 10.1115/1.3593408
  3. 3.
    Chou, S.-C.: Mechanical Geometry Theorem Proving. Springer, Berlin (1988) MATHGoogle Scholar
  4. 4.
    Chiabert, P., Orlando, M.: About a cat model consistent with iso/tc 213 last issues. Achievements in Mechanical and Materials Engineering Conference. J. Mater. Process. Technol. 157–158, 61–66 (2004) CrossRefGoogle Scholar
  5. 5.
    Chiabert, P., Lombardi, F., Vaccarino, F.: Analysis of kinematic methods for invariants based classification in the ISO/TC213 framework. In: Proceedings of the 10th CIRP International Seminar on Computer-Aided Tolerancing, Erlangen (Germany), March 21–23, 2007 Google Scholar
  6. 6.
    Clément, A., Rivière, A., Temmerman, M.: Cotation tridimensionnelle des systèmes mécaniques – Théorie et pratique. PYC, Ivry-sur-Seine (1994) Google Scholar
  7. 7.
    Clément, A., Rivière, A., Serré, P., Valade, C.: The TTRS: 13 constraints for dimensioning and tolerancing. In: Proceedings of the 5th CIRP International Seminar on Computer-Aided Tolerancing, pp. 28–29 (1997) Google Scholar
  8. 8.
    Gaildrat, V.: Declarative modelling of virtual environments, overview of issues and applications. In: Plemenos, D., Miaoulis, G. (eds.) Proceedings of International Conference on Computer Graphics and Artificial Intelligence (3IA), Athens (Greece), May 30–31, 2007 Google Scholar
  9. 9.
    Hervé, J.-M.: The mathematical group structure of the set of displacements. Mech. Mach. Theory 29(1), 73–81 (1994) CrossRefGoogle Scholar
  10. 10.
    Hervé, J.-M.: The Lie group of rigid body displacements, a fundamental tool for mechanism design. Mech. Mach. Theory 34(5), 719–730 (1999) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Hestenes, D.: New tools for computational geometry and rejuvenation of screw theory. In: Bayro-Corrochano, E., Scheuermann, G. (eds.) Geometric Algebra Computing, pp. 3–33. Springer, London (2010) CrossRefGoogle Scholar
  12. 12.
    Hoffmann, C.M., Joan-Arinyo, R.: A brief on constraint solving. Comput-Aided Des. Appl. 2(5), 655–663 (2005) Google Scholar
  13. 13.
    Joan-Arinyo, R.: Basics on geometric constraint solving. In: Proceedings of 13th Encuentros de Geometrfa Computacional (EGC09), Zaragoza (Spain), June 29–July 1, 2009 Google Scholar
  14. 14.
    Li, H.: Invariant Algebras and Geometric Reasoning. World Scientific, Singapore (2008) MATHCrossRefGoogle Scholar
  15. 15.
    Luo, Z., Dai, J.S.: Mathematical methodologies in computational kinematics. In: 14th Biennial Mechanisms Conference, Chong Qing (China), 2004 Google Scholar
  16. 16.
    Selig, J.M., Bayro-Corrochano, E.: Rigid body dynamics using Clifford algebra. Adv. Appl. Clifford Algebras 20, 141–154 (2010) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Selig, J.M.: Clifford algebra of points, lines and planes. Robotica 18(5), 545–556 (2000) CrossRefGoogle Scholar
  18. 18.
    Serré, P., Moinet, M., Clément, A.: Declaration and specification of a geometrical part in the language of geometric algebra. In: Advanced Mathematical and Computational Tools in Metrology and Testing VIII. Series on Advances in Mathematical for Applied Sciences, vol. 78, pp. 298–308 (2009) CrossRefGoogle Scholar
  19. 19.
    Srinivasan, V.: A geometrical product specification language based on a classification of symmetry groups. Comput. Aided Des. 31(11), 659–668 (1999) MATHCrossRefGoogle Scholar
  20. 20.
    van der Meiden, H.A., Bronsvoort, W.F.: A constructive approach to calculate parameter ranges for systems of geometric constraints. Comput. Aided Des. 38(4), 275–283 (2006) CrossRefGoogle Scholar
  21. 21.
    Zaragoza, J., Ramos, F., Orozco, H.R., Gaildrat, V.: Creation of virtual environments through knowledge-aid declarative modeling. In: LAPTEC, pp. 114–132 (2007) Google Scholar

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© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.LISMMAInstitut Supérieur de Mécanique de ParisParisFrance
  2. 2.LURPAÉcole Normale Supérieure de CachanCachanFrance
  3. 3.Robotics and Machine Dynamics LaboratoryBeijing University of TechnologyBeijingP.R. China

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