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Proving geometric theorems using clifford algebra and rewrite rules

  • Stéphane Fèvre
  • Dongming Wang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1421)

Abstract

We consider geometric theorems that can be stated constructively by introducing points, while each newly introduced point may be represented in terms of the previously constructed points using Clifford algebraic operators. To prove a concrete theorem, one first substitutes the expressions of the dependent points into the conclusion Clifford polynomial to obtain an expression that involves only the free points and parameters. A term-rewriting system is developed that can simplify such an expression to 0, and thus prove the theorem. A large class of theorems can be proved effectively in this coordinate-free manner. This paper describes the method in detail and reports on our preliminary experiments.

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References

  1. 1.
    Ablamowicz, R., Lounesto, P., Parra J. M.: Clifford algebras with numeric and symbolic computations. BirkhÄuser, Boston (1996).zbMATHGoogle Scholar
  2. 2.
    Balbiani, P., Fariñas del Cerro, L.: Affine geometry of collinearity and conditional term rewriting. In: Proc. French Spring School Theor. Comput. Sci. (Font-romeu, May 17–21, 1993), LNCS 909, pp. 196–213 (1995).Google Scholar
  3. 3.
    Chou, S.-C: Mechanical geometry theorem proving. Reidel, Dordrecht (1988).zbMATHGoogle Scholar
  4. 4.
    Chou, S.-C., Schelter, W. F.: Proving geometry theorems with rewrite rules. J. Automat. Reason. 2: 253–273 (1986).CrossRefzbMATHGoogle Scholar
  5. 5.
    Chou, S.-C., Gao, X.-S., Zhang, J.-Z.: Automated geometry theorem proving by vector calculation. In: Proc. ISSAC '93 (Kiev, July 6–8, 1993), pp. 284–291.Google Scholar
  6. 6.
    Chou, S.-C., Gao, X.-S., Zhang, J.-Z.: Machine proofs in geometry. World Scientific, Singapore (1994).zbMATHGoogle Scholar
  7. 7.
    Contejean, E., Marché, C., Rabehasaina, L.: Rewrite systems for natural, integral and rational arithmetic. Technical report, L.R.I., Université de Paris-Sud, France (1997).Google Scholar
  8. 8.
    Fèvre, S.: Integration of reasoning and algebraic calculus in geometry. In: Automated deduction in geometry (D. Wang et al., eds.), LNAI 1360, pp. 218–234 (1998).Google Scholar
  9. 9.
    Havel, T. F.: Some examples of the use of distances as coordinates for Euclidean geometry. J. Symb. Comput. 11: 579–593 (1991).zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Hestenes, D., Sobczyk, G.: Clifford algebra to geometric calculus. Reidel, Dordrecht (1984).zbMATHGoogle Scholar
  11. 11.
    Kapur, D.: Using Gröbner bases to reason about geometry problems. J. Symb. Comput. 2: 399–408 (1986).zbMATHMathSciNetGoogle Scholar
  12. 12.
    Kapur, D., Saxena, T., Yang, L.: Algebraic and geometric reasoning using Dixon resultants. In: Proc. ISSAC '94 (Oxford, July 20–22, 1994), pp. 99–107.Google Scholar
  13. 13.
    Kutzler, B., Stifter, S.: On the application of Buchberger's algorithm to automated geometry theorem proving. J. Symb. Comput. 2: 389–397 (1986).MathSciNetzbMATHGoogle Scholar
  14. 14.
    Li, H.: New explorations on mechanical theorem proving of geometries. Ph.D thesis, Beijing University, China (1994).Google Scholar
  15. 15.
    Li, H., Cheng, M.-t.: Proving theorems in elementary geometry with Clifford algebraic method. Chinese Math. Progress 26: 357–371 (1997).MathSciNetzbMATHGoogle Scholar
  16. 16.
    Marché, C.: Normalized rewriting: An alternative to rewriting modulo a set of equations. J. Symb. Comput. 3: 253–288 (1996).zbMATHCrossRefGoogle Scholar
  17. 17.
    Richter-Gebert, J.: Mechanical theorem proving in projective geometry. Ann. Math. Artif. Intell. 13: 139–172 (1995).zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Stifter, S.: Geometry theorem proving in vector spaces by means of Gröbner bases. In: Proc. ISSAC '93 (Kiev, July 6–8, 1993), pp. 301–310.Google Scholar
  19. 19.
    Wang, D.: Elimination procedures for mechanical theorem proving in geometry. Ann. Math. Artif. Intell. 13: 1–24 (1995).CrossRefzbMATHGoogle Scholar
  20. 20.
    Wang, D.: Clifford algebraic calculus for geometric reasoning with application to computer vision. In: Automated deduction in geometry (D. Wang et al., eds.), LNAI 1360, pp. 115–140 (1998).Google Scholar
  21. 21.
    Wu, W.-t.: Basic principles of mechanical theorem proving in elementary geometries. J. Syst. Sci. Math. Sci. 4: 207–235 (1984).Google Scholar
  22. 22.
    Wu, W.-t.: Mechanical theorem proving in geometries: Basic principles. Springer, Wien (1994).zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Stéphane Fèvre
    • 1
  • Dongming Wang
    • 1
  1. 1.LEIBNIZ-IMAGGrenoble CedexFrance

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