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Reasoning about Geometric Problems using an Elimination Method

  • Dongming Wang
Part of the Texts and Monographs in Symbolic Computation book series (TEXTSMONOGR)

Abstract

The present work relates to a paper by (1991) in which he explained the reasoning about a set of selected, geometry-related problems by using the algebraic methods of characteristic sets (Ritt 1950; Wu 1984a, b), Gröbner bases (Buchberger 1985) and cylindrical algebraic decomposition (Collins 1975). Its main purpose is to demonstrate how to deal with the same set of geometric problems by using another algebraic method which is based on some elimination procedures proposed by (1993). We use the same formulations of the problems (with slight modifications when necessary) and the same set of illustrative examples given previously (Wang 1991). It is shown that for most of the examples our new method takes less computing time than the methods of characteristic sets and Gröbner bases do.

Keywords

Algebraic Variety Polynomial System Geometric Problem Triangular Form Subsidiary Condition 
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.

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References

  1. Arnon, D. S. (1986): Geometric reasoning with logic and algebra. Artif. Intell. 37: 37–60.MathSciNetCrossRefGoogle Scholar
  2. Arnon, D. S., Sederberg, T. W. (1984): Implicit equation for a parametric surface by Gröbner bases. In: Golden, V. E. (ed.): Proceedings of the 1984 MACSYMA User’s Conference. General Electric, Schenectady, NY, pp. 431–436.Google Scholar
  3. Buchberger, B. (1985): Gröbner bases: an algorithmic method in polynomial ideal theory. In: Bose, N. K. (ed.): Multidimensional systems theory. Reidel, Dordrecht, pp. 184–232.CrossRefGoogle Scholar
  4. Buchberger, B. (1987): Applications of Gröbner bases in non-linear computational geometry. In: Rice, J. R. (ed.): Mathematical aspects of scientific software. Springer, Berlin Heidelberg New York Tokyo, pp. 59–87 (The IMA volumes in mathematics and its applications, vol. 14).Google Scholar
  5. Buchberger, B., Collins, G. E., Kutzler, B. (1988): Algebraic methods for geometric reasoning. Annu. Rev. Comput. Sci. 3: 85–119.MathSciNetCrossRefGoogle Scholar
  6. Chou, S. C. (1987): A method for the mechanical derivation of formulas in elementary geometry. J. Autom. Reason. 3: 291–299.MATHCrossRefGoogle Scholar
  7. Chou, S. C. (1988): Mechanical geometry theorem proving. Reidel, Dordrecht.MATHGoogle Scholar
  8. Chou, S. C., Gao, X. S. (1990): Mechanical formula derivation in elementary geometries. In: Proceedings ISS AC’ 90, Tokyo, August 20–24, 1990, pp. 265–270.Google Scholar
  9. Collins, G. E. (1975): Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.): Automata theory and formal languages. Springer, Berlin Heidelberg New York, pp. 134–165 (Lecture notes in computer science, vol. 33).Google Scholar
  10. Davenport, J. M. (1986): A “Piano movers” problem. SIGSAM Bull. 20: 15–17.MATHCrossRefGoogle Scholar
  11. Gao, X. S., Chou, S. C. (1991): Computations with parametric equations. In: Proceedings ISSAC’ 91, Bonn, July 15–17, 1991, pp. 122–127.Google Scholar
  12. Hodge, W. V. D., Pedoe, D. (1947/1952): Methods of algebraic geometry, vols. I, II. Cambridge University Press, Cambridge.Google Scholar
  13. Hoffmann, C. (1989): Geometric and solid modeling. Morgan Kaufmann, Los Altos.Google Scholar
  14. Kalkbrener, M. (1991): Three contributions to elimination theory. Ph.D. thesis, University of Linz, Linz, Austria.Google Scholar
  15. Kapur, D., Mundy, J. L. (eds.) (1989): Geometric reasoning. MIT Press, Cambridge, MA.Google Scholar
  16. Li, Z. M. (1989): Automatic implicitization of parametric objects. Math. Mech. Res. Preprints 4: 54–62.Google Scholar
  17. Neff, C. A. (1989): Decomposing algebraic sets using Gröbner bases. Comput. Aided Geom. Des. 6: 249–263.MathSciNetMATHCrossRefGoogle Scholar
  18. Nutbourne, A. W., Martin, R. R. (1988): Differential geometry applied to curve and surface design, vol. 1, foundations. Ellis Horwood, Chichester.Google Scholar
  19. Paul, R. P. (1981): Robot manipulators: mathematics, programming and control. MIT Press, Cambridge, MA.Google Scholar
  20. Pfalzgraf, J., Stokkermans, K., Wang, D. M. (1990): The robotics benchmark. In: Proceedings 12-Month MEDLAR Workshop (Weinberg Castle, Austria, November 4–7, 1990), Department of Computing, Imperial College, University of London.Google Scholar
  21. Ritt, J. F. (1950): Differential algebra. American Mathematical Society, New York.MATHGoogle Scholar
  22. Schwartz, J. T., Yap, C. K. (1987): Advances in robotics, vol. 1, algorithmic and geometric aspects of robotics. Lawrence Erlbaum, Hillsdale.Google Scholar
  23. Schwartz, J. T., Sharir, M., Hopcroft, J. (1987): Planning, geometry, and complexity of robot motion. Ablex, Norwood.Google Scholar
  24. Sederberg, T. W., Anderson, D. C. (1984): Implicit representation of parametric curves and surfaces. Comput. Vision Graphics Image Process. 28: 72–84.MATHCrossRefGoogle Scholar
  25. Seidenberg, A. (1956a): Some remarks on Hubert’s Nullstellensatz. Arch. Math. 7: 235–240.MathSciNetMATHCrossRefGoogle Scholar
  26. Seidenberg, A. (1956b): An elimination theory for differential algebra. Univ. California Publ. Math. (N.S.) 3: 31–66.MathSciNetGoogle Scholar
  27. Seidenberg, A. (1969): On k-constructable sets, k-elementary formulae, and elimination theory. J. Reine Angew. Math. 239/240: 256–267.MathSciNetGoogle Scholar
  28. Tarski, A. (1951): A decision method for elementary algebra and geometry, 2nd ed. University of California Press, Berkeley.MATHGoogle Scholar
  29. Wang, D. M. (1987): Mechanical approach for polynomial set and its related fields. Ph.D. thesis, Academia Sinica, Beijing, People’s Republic of China.Google Scholar
  30. Wang, D. M. (1989a): On the singularities of algebraic hypersurfaces — the discriminant systems. Preprint, School of Mathematical Sciences, Queen Mary College, University of London.Google Scholar
  31. Wang, D. M. (1989b): Characteristic sets and zero structure of polynomial sets. Lecture Notes, RISC Linz.Google Scholar
  32. Wang, D. M. (1990): Some notes on algebraic methods for geometry theorem proving. Preprint, RISC Linz.Google Scholar
  33. Wang, D. M. (1991): Reasoning about geometric problems using algebraic methods. In: Proceedings MEDLAR 24-Month Workshop, Grenoble, December 8–11, 1991. Department of Computing, Imperial College, University of London.Google Scholar
  34. Wang, D. M. (1992a): A method for factorizing multivariate polynomials over successive algebraic extension fields. Preprint, RISC Linz.Google Scholar
  35. Wang, D. M. (1992b): Irreducible decomposition of algebraic varieties via characteristic sets and Gröbner bases. Comput. Aided Geom. Des. 9: 471–484.MATHCrossRefGoogle Scholar
  36. Wang, D. M. (1993): An elimination method for polynomial systems. J. Symb. Comput. 16: 83–114.MATHCrossRefGoogle Scholar
  37. Wang, D. M. (1994): Algebraic factoring and geometry theorem proving. In: Proceedings CADE-12, Nancy, June 28–July 1, 1994. Springer, Berlin Heidelberg New York Tokyo, pp. 386–400 (Lecture notes in artificial intelligence, vol. 814).Google Scholar
  38. Wang, D. M. (1995): Elimination procedures for mechanical theorem proving in geometries. Ann. Math. Artif. Intell. (to appear)Google Scholar
  39. Wu, W.-t. (1978): On the decision problem and the mechanization of theorem-proving in elementary geometry. Sci. Sin. 21: 159–172 [also in Bledsoe, W. W., Loveland, D. W. (eds.): Automated theorem proving: after 25 years. American Mathematical Society, Providence, pp. 213–234 (1984)].MATHGoogle Scholar
  40. Wu, W.-t. (1984a): Basic principles of mechanical theorem proving in elementary geometries. J. Syst. Sci. Math. Sci. 4: 207–235 [also in J. Automat. Reason. 2: 221–252 (1986)].Google Scholar
  41. Wu, W.-t. (1984b): Basic principles of mechanical theorem proving in geometries (part on elementary geometries). Science Press, Beijing (in Chinese); English-language edition, Springer, Wien New York (1994).Google Scholar
  42. Wu, W.-t. (1986a): On zeros of algebraic equations — an application of Ritt principle. Kexue Tongbao 31: 1–5.MATHGoogle Scholar
  43. Wu, W.-t. (1986b): A mechanization method of geometry and its applications — I. distances, areas and volumes. J. Syst. Sci. Math. Sci. 6: 204–216.MATHGoogle Scholar
  44. Wu, W.-t. (1988): A mechanization method of geometry and its applications — III. mechanical proving of polynomial inequalities and equations-solving. Syst. Sci. Math. Sci. 1: 1–17 [also in Math. Mech. Res. Preprints 2: 1–17 (1987)].MATHGoogle Scholar
  45. Wu, W.-t. (1989): A mechanization method of geometry and its applications — VI. solving inverse kinematic equations of PUMA-type robots. Math. Mech. Res. Preprints 4: 49–53.Google Scholar
  46. Wu, W.-t. (1990): On a projection theorem of quasi-varieties in elimination theory. Chin. Ann. Math. 11B/2: 220–226.Google Scholar

Copyright information

© Springer-Verlag Wien 1995

Authors and Affiliations

  • Dongming Wang

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