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.
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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).
J. Bokowski and J. Richter, On the finding of final polynomials, Eur. J. Comb. 11(1990)21–34.
J. Bokowski and J. Richter-Gebert, Reduction theorems for oriented matroids, Preprint, TH-Darmstadt (1990) 16 p.
J. Bokowski and J. Richter-Gebert, On the classification of non-realizable oriented matroids, Part II: Properties, Preprint, TH-Darmstadt (1990) 9 p.
J. Bokowski, J. Richter and B. Sturmfels, Nonrealizability proofs in computational geometry, Discr. Comp. Geom. 5(1990)333–350.
J. Bokowski and B. Sturmfels, Computational synthetic geometry, Lecture Notes in Math. 1355 (Springer, Heidelberg, 1989).
J. Bokowski and B. Sturmfels, An infinite family of minor-minimal non-realizable 3-chirotopes, Math. Z. 200(1989)583–589.
B. Buchberger, Gröbner bases — an algorithmic method in polynomial ideal theory, in:Multi-dimensional Systems Theory, ed. N.K. Bose (Reidel, 1985) chap. 6.
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).
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.
S.C. Chou,Mechanical Geometry Theorem Proving (Reidel, Dordrecht, 1988).
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).
H. Crapo, Invariant-theoretic methods in scene analysis and structural mechanics, J. Symb. Comp. 11(1991)523–548.
H. Crapo and J. Ryan, Spatial realizations of linear scenes, Struct. Topology 13(1986)33–68.
G.M. Crippen and T.F. Havel,Distance Geometry and Molecular Conformation (Research Studies Press, Wiley, New York/Chichester/Toronto/Singapore, 1988).
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.
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.
L.E. Garner,An Outline of Projective Geometry (North-Holland, New York/Oxford, 1981).
E. Goodman and R. Pollack, On the combinatorial classification of non-degenerate configurations in the plane, J. Comb. Theory A29(1980)220–234.
P.M. Gruber and C.G. Lekkerkerker,Geometry of Numbers (North-Holland, Amsterdam/New York/Oxford/Tokyo, 1987).
A. Heyting, Axiomatic projective geometry,Bibliotheca Mathematica V (North-Holland, Amsterdam, 1963).
B. Kutzler, Algebraic approaches to automated geometry theorem proving, Ph.D. Dissertation, Johanned Kepler Universität, Linz (1988).
B. Kutzler and S. Stifter, On the application of Buchberger's algorithm to automated geometry theorem proving, J. Symb. Comp. 2(1986)289–297.
R. Lauffer, Die nichtkonstruierbare Konfiguration (103), Math. Nachr. 11(1954)303–304.
J. Richter-Gebert, On the realizability problem of combinatorial geometries — decision methods, Dissertation, Technische Hochschule Darmstadt (1992).
R.F. Ritt,Differential Equations from an Algebraic Standpoint, Vol. 14 (AMS Colloq. Publ., New York, 1938).
R.F. Ritt,Differential Algebra (AMS Colloq. Publ., New York, 1950).
A. Saam, Ein neuer Schließungssatz für projektive Ebenen, J. Geom. 29(1987)36–42.
A. Saam, Schließungssätze als Eigenschaften von Projektivitäten, J. Geom. 32(1988)86–130.
B. Sturmfels, Aspects of computational synthetic geometry; I. Algorithmic coordinatization of matroids,Proc. of Computer-Aided Geometric Reasoning, INRIA, Antibes, 1987, pp. 57–86.
B. Sturmfels, Computational algebraic geometry of projective configurations, J. Symb. Comp. 11(1991)595–618.
B. Sturmfels, Computing final polynomials and final syzygies using Buchberger's Gröbner bases method, Result. Math. 15(1989)351–360.
B. Sturmfels and N. White, Gröbner bases and invariant theory, Adv. Math. 76(1989)245–259.
B. Sturmfels and W. Whiteley, On the synthetic factorization of homogeneous invariants, J. Symb. Comp. 11(1991)439–454.
A. Tarski,A Decision Method for Elementary Algebra and Geometry, 2nd revised ed. (University of California Press, 1951).
N. White, The bracket ring of a combinatorial geometry. I, Trans. Am. Math. Soc. 202(1975)79–103.
N. White, Multilinear Cayley factorization, J. Symb. Comp. 11(1991)421–438.
N. White and W. Whiteley, The algebraic geometry of stresses in frameworks, SIAM J. Algebr. Discr. Meth. 4(1983)481–511.
W. Whiteley, Applications of the geometry of ridid structures,Proc. of Computer-Aided Geometric Reasoning, INRIA, Antibes, 1987, pp. 219–254.
W.T. Wu, On the decision problem and the mechanization of theorem-proving in elementary geometry, Contemp. Math. 29(1984)213–234.
W.T. Wu, Basic principles of mechanical theorem proving in elementary geometries, J. Syst. Sci. Math. Sci. 4(1984)207–235.
W.T. Wu, Some recent advances in mechanical theorem-proving of geometries, Contemp. Math. 29(1984)235–241.
A. Young, On quantitative substitutionals analysis (3rd paper), Proc. London Math. Soc. Ser. 2, 28(1928)255–292.
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Richter-Gebert, J. Mechanical theorem proving in projective geometry. Ann Math Artif Intell 13, 139–172 (1995). https://doi.org/10.1007/BF01531327
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DOI: https://doi.org/10.1007/BF01531327