Abstract
In this paper a new method is proposed for mechanical geometry theorem proving. It combines vectorial equations solving in Clifford algebra formalism with Wu"s method. The proofs produced have significantly enhanced geometric meaning and fewer nongeometric nondegeneracy conditions.
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Chen, W. H.: Preliminaries of Differential Geometry, Peking Univ. Press, 1990.
Chou, S. C.: Mechanical Geometry Theorem Proving, D. Reidel, Dordrecht, Boston, 1988.
Chou, S. C., Gao, X. S. and Zhang, J. Z.: Machine Proofs in Geometry, World Scientific, Singapore, 1994.
Chou, S. C., Gao, X. S., Yang, L. and Zhang J. Z.: Automated production of readable proofs for theorems in non-Euclidean geometries, in LNAI 1360, Springer, 1997, pp. 171-188.
Chou, S. C., Gao, X. S. and Zhang, J. Z.: Mechanical geometry theorem proving by vector calculation, Proc. ISSAC93, Kiev, ACM Press, 1993, pp. 284-291.
Corrochano, E. B. and Lasenby, J.: Object modeling and motion analysis using Clifford algebra, in R. Mohr and C. Wu (eds.), Proc. Europe-China Workshop on Geometric Modeling and Invariants for Computer Visions, Xi'an, China, 1995, pp. 143-149.
Corrochano, E. B., Buchholz, S. and Sommer, G.: Self-organizing Clifford neural network, in IEEE ICNN'96, Washington D.C., 1996, pp. 120-125.
Crumeyrolle, A.: Orthogonal and Symplectic Clifford Algebras, D. Reidel, Dordrecht, 1990.
Delanghe, R., Sommen, F. and Soucek, V.: Clifford Algebra and Spinor-Valued Functions, D. Reidel, Dordrecht, 1992.
Doran, C., Hestenes, D., Sommen, F. and Acker, N. V.: Lie groups as spin groups, J. Math. Phys. 34(8) (1993), 3642-3669.
Havel, T.: Some examples of the use of distances as coordinates for Euclidean geometry, J. Symbolic Comput. 11 (1991), 579-593.
Havel, T. and Dress, A.: Distance geometry and geometric algebra, Found. Phys. 23 (1992), 1357-1374.
Havel, T.: Geometric algebra and Möbius sphere geometry as a basis for Euclidean invariant theory, in N. L. White (ed.), Invariant Methods in Discrete and Computational Geometry, D. Reidel, Dordrecht, 1995.
Hestenes, D.: Space-Time Algebra, Gordon and Breach, New York, 1966.
Hestenes, D. and Sobczyk, G.: Clifford Algebra to Geometric Calculus, D. Reidel, Dordrecht, Boston, 1984.
Hestenes, D: New Foundations for Classical Mechanics, D. Reidel, Dordrecht, Boston, 1987.
Hestenes, D. and Ziegler, R.: Projective geometry with Clifford algebra, Acta Appl. Math. 23 (1991), 25-63.
Hestenes, D.: The design of linear algebra and geometry, Acta Appl. Math. 23 (1991), 65-93.
Kapur, D.: Using Gröbner bases to reason about geometry problems, J. Symbolic Comput. 2 (1986), 399-408.
Lawson, H. B. and Michelsohn, M. L.: Spin Geometry, Princeton, 1989.
Li, H.: On mechanical theorem proving in differential geometry-local theory of surfaces, Scientia Sinica Series A 40(4) (1997), 350-356.
Li, H. and Cheng, M.: Proving theorems in elementary geometry with Clifford algebraic method, Chinese Math. Progress 26(4) (1997), 357-371.
Li, H. and Cheng, M.: Clifford algebraic reduction method for automated theorem proving in differential geometry, J. Automated Reasoning 21 (1998), 1-21.
Li, H.: Hyperbolic geometry with Clifford algebra, Acta Appl. Math. 48(3) (1997), 317-358.
Mourrain, B. and Stolfi, N.: Computational symbolic geometry, in N. L. White (ed.), Invariant Methods in Discrete and Computational Geometry, D. Reidel, Dordrecht, 1995, pp. 107-139.
Mourrain, B. and Stolfi, N.: Applications of Clifford algebras in robotics, in J.-P. Merlet and B. Ravani (eds.), Computational Kinematics, D. Reidel, Dordrecht, 1995, pp. 141-150.
Stifter, S.: Geometry theorem proving in vector spaces by means of Gröbner bases, Proc. ISSAC93, Kiev, ACM Press, 1993, pp. 301-310.
Sturmfels, B.: Algorithms in Invariant Theory, Springer-Verlag, New York, 1993.
Wang, D. M.: Elimination procedures for mechanical theorem proving in geometry, Ann. of Math. and Artif. Intell. 13 (1995), 1-24.
Wang, D. M.: Clifford algebraic calculus for geometric reasoning, in LNAI 1360, Springer, 1997, pp. 115-140.
Wang, D. M.: A method for proving theorems in differential geometry and mechanics, J. Univ. Computer Sci. 1(9) (1995), 658-673.
White, N. L.: Multilinear Cayley factorization, J. Symbolic Comput. 11 (1991), 421-438.
Whiteley, W.: Invariant computations for analytic projective geometry, J. Symbolic Comput. 11 (1991), 549-578.
Wu, W. T.: Mechanical Theorem Proving in Geometries: Basic Principle (translated from Chinese edition 1984), Springer-Verlag, Wien, 1994.
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Li, H. Vectorial Equations Solving for Mechanical Geometry Theorem Proving. Journal of Automated Reasoning 25, 83–121 (2000). https://doi.org/10.1023/A:1006182023017
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DOI: https://doi.org/10.1023/A:1006182023017