Frontiers of Computer Science in China

, Volume 1, Issue 2, pp 180–190 | Cite as

On the design and implementation of a geometric-object-oriented language

  • Liang Tielin 
  • Wang Dongming 
Research Article


This paper presents the design and implementation of a geometric-object-oriented language Gool for constructing, representing, manipulating, and visualizing symbolic geometric objects and relations and performing symbolic geometric computation and formal reasoning. The language uses case distinction to formalize symbolic geometric objects and relations, reducing the problem of dealing with uncertainty and degeneracy to that of handling geometric constraints. We describe the capabilities, features, and main components of Gool, propose several techniques for geometric constraint handling, and discuss some of the implementation issues.


Automated reasoning geometric constraint geometry software object-oriented language symbolic computation uncertainty and degeneracy 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Hilbert D. Grundlagen der Geometrie. Teubner, Stuttgart, 1899Google Scholar
  2. 2.
    Wu W-t. Mechanical Theorem Proving in Geometries: Basic Principles (translated from the Chinese version by Jin X, Wang D). Springer-Verlag, Wien New York, 1994Google Scholar
  3. 3.
    Chou S-C, Gao X-S, Liu Z, et al. Geometric theorem provers and algebraic equation solvers. In: Gao X-S, Wang D, eds. Mathematics Mechanization and Applications, Academic Press, London, 2000, 491–505Google Scholar
  4. 4.
    Liang T, Wang D. Towards a geometric-object-oriented language. In: Hong H, Wang D, eds. Automated Deduction in Geometry, LNAI 3763, Springer-Verlag, Berlin Heidelberg, 2006, 130–155CrossRefGoogle Scholar
  5. 5.
    Joan-Arinyo R, Hoffmann C M. A brief on constraint solving., 2005
  6. 6.
    Wang D. GEOTHER 1.1: Handling and proving geometric theorems automatically. In: Winkler F, ed. Automated Deduction in Geometry, LNAI 2930, Springer-Verlag, Berlin Heidelberg, 2004, 194–215Google Scholar
  7. 7.
    Kapur D. Using Gröbner bases to reason about geometry problems. J. Symb. Comput., 1986, 2: 399–408zbMATHMathSciNetGoogle Scholar
  8. 8.
    Kutzler B, Stifter S. On the application of Buchberger’s algorithm to automated geometry theorem proving. J. Symb. Comput., 1986, 2: 389–397zbMATHMathSciNetGoogle Scholar
  9. 9.
    Wang D. Elimination procedures for mechanical theorem proving in geometry. Ann. Math. Artif. Intell., 1995, 13: 1–24zbMATHCrossRefGoogle Scholar
  10. 10.
    Chou S-C, Gao X-S, Zhang J-Z. Machine Proofs in Geometry. World Scientific, Singapore, 1994zbMATHGoogle Scholar
  11. 11.
    Wang D. Elimination Practice: Software Tools and Applications. Imperial College Press, London, 2004zbMATHGoogle Scholar
  12. 12.
    Collins G E, Hong H. Partial cylindrical algebraic decomposition for quantifier elimination. J. Symb. Comput., 1991, 12: 299–328zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Yang L, Hou X-R, Xia B. A complete algorithm for automated discovering of a class of inequality-type theorems. Sci. China (Ser. F), 2001, 44: 33–49zbMATHMathSciNetGoogle Scholar
  14. 14.
    Dolzmann A, Sturm T. Simplification of quantifier-free formulae over ordered fields. J. Symb. Comput., 1997, 24: 209–231zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Zeng G, Zeng X. An effective decision method for semidefinite polynomials. J. Symb. Comput., 2004, 37: 83–99zbMATHCrossRefGoogle Scholar
  16. 16.
    Brown C W, Hong H. QEPCAD — Quantifier elimination by partial cylindrical algebraic decomposition., 2004
  17. 17.
    Chen X, Wang D. Towards an electronic geometry textbook. In: Botana F, Roanes-Lozano E, eds. Automated Deduction in Geometry, Universidad de Vigo, Pontevedra, Spain, 2006, 15–25Google Scholar

Copyright information

© Higher Education Press and Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Science and Technology of ChinaHefeiChina
  2. 2.LMIB-School of ScienceBeihang UniversityBeijingChina
  3. 3.Laboratoire d’Informatique de Paris 6Université Pierre et Marie Curie — CNRSParisFrance

Personalised recommendations