[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Log in

Mechanical theorem proving in projective geometry

  • Published:
Annals of Mathematics and Artificial Intelligence Aims and scope Submit manuscript

Abstract

We present an algorithm that is able to confirm projective incidence statements by carrying out calculations in the ring of all formal determinants (brackets) of a configuration. We will describe an implementation of this prover and present a series of examples treated by the prover, includingPappus' andDesargues' theorems, thesixteen point theorem, Saam's theorem, thebundle condition, theuniqueness of a harmonic point andPascal's theorem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. A. Björner, M. Las Vergnas, B. Sturmfels, N. White and G.M. Ziegler, Oriented Matroids,Encyclopedia of Mathematics and its Applications, Vol. 46 (Cambridge University Press, 1993).

  2. J. Bokowski and J. Richter, On the finding of final polynomials, Eur. J. Comb. 11(1990)21–34.

    Google Scholar 

  3. J. Bokowski and J. Richter-Gebert, Reduction theorems for oriented matroids, Preprint, TH-Darmstadt (1990) 16 p.

  4. J. Bokowski and J. Richter-Gebert, On the classification of non-realizable oriented matroids, Part II: Properties, Preprint, TH-Darmstadt (1990) 9 p.

  5. J. Bokowski, J. Richter and B. Sturmfels, Nonrealizability proofs in computational geometry, Discr. Comp. Geom. 5(1990)333–350.

    Google Scholar 

  6. J. Bokowski and B. Sturmfels, Computational synthetic geometry, Lecture Notes in Math. 1355 (Springer, Heidelberg, 1989).

    Google Scholar 

  7. J. Bokowski and B. Sturmfels, An infinite family of minor-minimal non-realizable 3-chirotopes, Math. Z. 200(1989)583–589.

    Google Scholar 

  8. B. Buchberger, Gröbner bases — an algorithmic method in polynomial ideal theory, in:Multi-dimensional Systems Theory, ed. N.K. Bose (Reidel, 1985) chap. 6.

  9. B. Buchberger, Applications of Gröbner bases in non-linear computational geometry, in:Scientific Software, ed. J.R. Rice, I.A.M. Vol. in Math. and its Appl. 444 (Springer, New York, 1988).

    Google Scholar 

  10. S.C. Chou, Characteristic sets and Gröbner bases in geometry theorem proving,Proc. of Computer-Aided Geometric Reasoning, INRIA, Antibes, France, 1987, pp. 29–56.

    Google Scholar 

  11. S.C. Chou,Mechanical Geometry Theorem Proving (Reidel, Dordrecht, 1988).

    Google Scholar 

  12. C. Collins, Quantifier elimination for real closed fields by cylindrical algebraic decomposition,Proc. 2nd GI Conf. on Automata and Formal Languages, Lecture Notes in Comp. Sci. 33 (Springer, Heidelberg, 1975).

    Google Scholar 

  13. H. Crapo, Invariant-theoretic methods in scene analysis and structural mechanics, J. Symb. Comp. 11(1991)523–548.

    Google Scholar 

  14. H. Crapo and J. Ryan, Spatial realizations of linear scenes, Struct. Topology 13(1986)33–68.

    Google Scholar 

  15. G.M. Crippen and T.F. Havel,Distance Geometry and Molecular Conformation (Research Studies Press, Wiley, New York/Chichester/Toronto/Singapore, 1988).

    Google Scholar 

  16. P. Doubilet, G.C. Rota and J. Stein, On the foundations of combinatorial theory IX: Combinatorial methods in invariant theory, Stud. Appl. Math. 57(1974)185–216.

    Google Scholar 

  17. A. Dreiding, A.W.M. Dress and H. Haegi, Classification of mobile molecules by category theory, in:Symmetries and Properties of Non-Rigid Molecules: A Comprehensive Survey, Stud. Phys. Theor. Chem. 23(1983)39–58.

    Google Scholar 

  18. L.E. Garner,An Outline of Projective Geometry (North-Holland, New York/Oxford, 1981).

    Google Scholar 

  19. E. Goodman and R. Pollack, On the combinatorial classification of non-degenerate configurations in the plane, J. Comb. Theory A29(1980)220–234.

    Google Scholar 

  20. P.M. Gruber and C.G. Lekkerkerker,Geometry of Numbers (North-Holland, Amsterdam/New York/Oxford/Tokyo, 1987).

    Google Scholar 

  21. A. Heyting, Axiomatic projective geometry,Bibliotheca Mathematica V (North-Holland, Amsterdam, 1963).

    Google Scholar 

  22. B. Kutzler, Algebraic approaches to automated geometry theorem proving, Ph.D. Dissertation, Johanned Kepler Universität, Linz (1988).

    Google Scholar 

  23. B. Kutzler and S. Stifter, On the application of Buchberger's algorithm to automated geometry theorem proving, J. Symb. Comp. 2(1986)289–297.

    Google Scholar 

  24. R. Lauffer, Die nichtkonstruierbare Konfiguration (103), Math. Nachr. 11(1954)303–304.

    Google Scholar 

  25. J. Richter-Gebert, On the realizability problem of combinatorial geometries — decision methods, Dissertation, Technische Hochschule Darmstadt (1992).

  26. R.F. Ritt,Differential Equations from an Algebraic Standpoint, Vol. 14 (AMS Colloq. Publ., New York, 1938).

    Google Scholar 

  27. R.F. Ritt,Differential Algebra (AMS Colloq. Publ., New York, 1950).

    Google Scholar 

  28. A. Saam, Ein neuer Schließungssatz für projektive Ebenen, J. Geom. 29(1987)36–42.

    Google Scholar 

  29. A. Saam, Schließungssätze als Eigenschaften von Projektivitäten, J. Geom. 32(1988)86–130.

    Google Scholar 

  30. B. Sturmfels, Aspects of computational synthetic geometry; I. Algorithmic coordinatization of matroids,Proc. of Computer-Aided Geometric Reasoning, INRIA, Antibes, 1987, pp. 57–86.

    Google Scholar 

  31. B. Sturmfels, Computational algebraic geometry of projective configurations, J. Symb. Comp. 11(1991)595–618.

    Google Scholar 

  32. B. Sturmfels, Computing final polynomials and final syzygies using Buchberger's Gröbner bases method, Result. Math. 15(1989)351–360.

    Google Scholar 

  33. B. Sturmfels and N. White, Gröbner bases and invariant theory, Adv. Math. 76(1989)245–259.

    Google Scholar 

  34. B. Sturmfels and W. Whiteley, On the synthetic factorization of homogeneous invariants, J. Symb. Comp. 11(1991)439–454.

    Google Scholar 

  35. A. Tarski,A Decision Method for Elementary Algebra and Geometry, 2nd revised ed. (University of California Press, 1951).

  36. N. White, The bracket ring of a combinatorial geometry. I, Trans. Am. Math. Soc. 202(1975)79–103.

    Google Scholar 

  37. N. White, Multilinear Cayley factorization, J. Symb. Comp. 11(1991)421–438.

    Google Scholar 

  38. N. White and W. Whiteley, The algebraic geometry of stresses in frameworks, SIAM J. Algebr. Discr. Meth. 4(1983)481–511.

    Google Scholar 

  39. W. Whiteley, Applications of the geometry of ridid structures,Proc. of Computer-Aided Geometric Reasoning, INRIA, Antibes, 1987, pp. 219–254.

    Google Scholar 

  40. W.T. Wu, On the decision problem and the mechanization of theorem-proving in elementary geometry, Contemp. Math. 29(1984)213–234.

    Google Scholar 

  41. W.T. Wu, Basic principles of mechanical theorem proving in elementary geometries, J. Syst. Sci. Math. Sci. 4(1984)207–235.

    Google Scholar 

  42. W.T. Wu, Some recent advances in mechanical theorem-proving of geometries, Contemp. Math. 29(1984)235–241.

    Google Scholar 

  43. A. Young, On quantitative substitutionals analysis (3rd paper), Proc. London Math. Soc. Ser. 2, 28(1928)255–292.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Richter-Gebert, J. Mechanical theorem proving in projective geometry. Ann Math Artif Intell 13, 139–172 (1995). https://doi.org/10.1007/BF01531327

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01531327

Keywords

Navigation