Advertisement

Towards a Geometric-Object-Oriented Language

  • Tielin Liang
  • Dongming Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3763)

Abstract

This paper proposes a geometric-object-oriented language for symbolic geometric computation, reasoning, and visualization. In this language, geometric objects are constructed with indefinite parametric data. Modifications and basic operations on these objects are enabled. Degeneracy and uncertainty are handled effectively by means of imposing conditions and assumptions and geometric statements are formulated by declaring relations among different objects. A system implemented on the basis of this language will allow the user to perform geometric computation and reasoning rigorously and to prove geometric theorems and generate geometric diagrams and interactive documents automatically. We present the overall design of the language, explain the capabilities, features, main components of the proposed system, provide specifications for some of its functors, report our experiments with a preliminary implementation of the system, and discuss some encountered difficulties and research problems.

Keywords

Geometric Meaning Geometric Constraint Geometric Object Algebraic Expression Geometric Relation 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brown, C.: W., Hong, H.: QEPCAD — Quantifier elimination by partial cylindrical algebraic decomposition (2004), http://www.cs.usna.edu/~qepcad/B/QEPCAD.html
  2. 2.
    Chou, S.-C., Gao, X.-S., Liu, Z., Wang, D.-K., Wang, D.: Geometric theorem provers and algebraic equation solvers. In: Gao, X.-S., Wang, D. (eds.) Mathematics Mechanization and Applications, pp. 491–505. Academic Press, London (2000)CrossRefGoogle Scholar
  3. 3.
    Hilbert, D.: Grundlagen der Geometrie. Teubner, Stuttgart (1899)Google Scholar
  4. 4.
    Jaffar, J., Maher, M.J.: J.: Constraint logic programming: A survey. J. Logic Program. 19/20, 503–581 (1994)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Kapur, D.: Using Gröbner bases to reason about geometry problems. J. Symb. Comput. 2, 399–408 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kutzler, B.: Algebraic Approaches to Automated Geometry Theorem Proving. Ph.D. thesis, RISC-Linz, Johannes Kepler University, Austria (1988)Google Scholar
  7. 7.
    Kutzler, B., Stifter, S.: On the application of Buchberger’s algorithm to automated geometry theorem proving. J. Symb. Comput. 2, 389–397 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Wang, D.: Elimination procedures for mechanical theorem proving in geometry. Ann. Math. Artif. Intell. 13, 1–24 (1995)zbMATHCrossRefGoogle Scholar
  9. 9.
    Wang, D.: GEOTHER 1.1: Handling and proving geometric theorems automatically. In: Winkler, F. (ed.) ADG 2002. LNCS (LNAI), vol. 2930, pp. 194–215. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  10. 10.
    Wu, W.-t.: Mechanical Theorem Proving in Geometries: Basic Principles. Springer, Wien New York (1994) (translated from the Chinese by X. Jin and D. Wang)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Tielin Liang
    • 1
  • Dongming Wang
    • 2
    • 3
  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