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Automated Geometry Diagram Construction and Engineering Geometry

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Book cover Automated Deduction in Geometry (ADG 1998)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 1669))

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Abstract

This paper reviews three main techniques for automated geometry diagram construction: synthetic methods, numerical computation methods, and symbolic computation methods. We also show how to use these techniques in parametric mechanical CAD, linkage design, computer vision, dynamic geometry, and CAI (computer aided instruction). The methods and the applications reviewed in this paper are closely connected and could be appropriately named as engineering geometry.

This work was supported in part by an Outstanding Youth Grant from the Chinese NSF and the National “973” Project.

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References

  1. B. Aldefeld, Variation of Geometries Based on a Geometric-Reasoning Method, Computer Aided Design, 20(3), 117–126, 1988. 234

    Article  MATH  Google Scholar 

  2. F. Arbab and B. Wang, Reasoning About Geometric Constraints, in Intelligent CAD II, H. Yoshikawa and T. Holden (eds.), pp. 93–107, North-Holland, 1990. 232, 233

    Google Scholar 

  3. A. Borning, The Programming Language Aspect of ThingLab, ACM Tras. on Programming Language and Systems, 3(4), 353–387, 1981. 232, 233

    Article  Google Scholar 

  4. W. Bouma, C. M. Hoffmann, I. Fudos, J. Cai and R. Paige, A Geometric Constraint Solver, Computer Aided Design, 27(6), 487–501, 1995.

    Article  MATH  Google Scholar 

  5. B. Brudelin, Constructing Three-Dimensional Geometric Objects Defined by Constraints, in Proc. Workshop on Interactive 3D Graphics, pp. 111–129, ACM Press, 1986. 232, 233

    Google Scholar 

  6. S. A. Buchanan and A. de Pennington, Constraint Definition System: A Computer Algebra Based Approach to Solving Geometric Problems, Computer Aided Design,25(12), 740–750, 1993.

    Article  Google Scholar 

  7. B. Buchberger, Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory, in Recent Trends in Multidimensional Systems Theory, D. Reidel Publ. Comp., 1985. 233, 241

    Google Scholar 

  8. B. Char et al., Maple V, Springer-Verlag, Berlin, 1992. 245

    MATH  Google Scholar 

  9. S. C. Chou, Mechanical Geometry Theorem Proving, D. Reidel Publishing Company, Dordrecht, Netherlands, 1988. 237, 246

    MATH  Google Scholar 

  10. S. C. Chou, A Method for Mechanical Deriving of Formulas in Elementary Geometry, J. of Automated Reasoning, 3, 291–299, 1987. 248

    Article  MATH  Google Scholar 

  11. S. C. Chou and X. S. Gao, Mechanical Formula Derivation in Elementary Geometries, in Proc. ISSAC-90, pp. 265–270, ACM Press, New York, 1990. 244, 248

    Chapter  Google Scholar 

  12. S. C. Chou and X. S. Gao, Ritt-Wu’s Decomposition Algorithm and Geometry Theorem Proving, in Porc. CADE-10, M. E. Stickel (ed.), pp. 207–220, LNCS, Vol. 449, Springer-Verlag, Berlin, 1990. 244

    Google Scholar 

  13. S. C. Chou, X. S. Gao and J. Z. Zhang, Machine Proofs in Geometry, World Scientific, Singapore, 1994. 237

    MATH  Google Scholar 

  14. S. C. Chou, X. S. Gao and J. Z. Zhang, A Fixpoint Approach To Automated Geometry Theorem Proving, WSUCS-95-2, CS Dept, Wichita State University, 1995, To appear in J. of Automated Reasoning. 234, 236

    Google Scholar 

  15. G. E. Collins, Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition, in LNCS vol. 33, pp. 134–183, Springer-Verlag, Berlin, 1975. 233, 241, 244

    Google Scholar 

  16. J. Chuan, Geometric Constructions with the Computer, in Proc. ATCM’95, pp. 329–338, Springer-Verlag, 1995.

    Google Scholar 

  17. K. H. Elster (ed.), Modern Mathematical Methods of Optimization, Akademie Verlag, 1993. 239

    Google Scholar 

  18. M. A. Fishler, and R. C. Bolles, Random Sample Consensus: A Paradigm for Model Fitting with Applications to Image Analysis and Automated Cartomated Cartography, Communications of the ACM, 24(6), 381–395, 1981. 252

    Article  Google Scholar 

  19. I. Fudos and C. M. Hoffmann, A Graph-Constructive Approach to Solving Systems of Geometric Constraints, ACM Transactions on Graphics, 16(2), 179–216, 1997.

    Article  Google Scholar 

  20. Gabri Geometry II, Texas Instruments, Dallas, Texas, 1994. 246

    Google Scholar 

  21. X. S. Gao and S. C. Chou, Solving Geometric Constraint Systems, I. A Global Propagation Approach, Computer Aideded Design, 30(1), 47–54, 1998. 232, 233, 234, 236, 249

    Article  Google Scholar 

  22. X. S. Gao and S. C. Chou, Solving Geometric Constraint Systems, II. A Symbolic Computational Approach, Computer Aided Design, 30(2), 115–122, 1998. 233, 241, 243, 244, 246

    Article  Google Scholar 

  23. X. S. Gao and S. C. Chou, Implicitization of Rational Parametric Equations, Journal of Symbolic Computation, 14, 459–470, 1992. 247

    Article  MATH  MathSciNet  Google Scholar 

  24. X. S. Gao, J. Z. Zhang and S. C. Chou, Geometry Expert, Nine Chapters Pub., 1998, Taiwan (in Chinese). 246

    Google Scholar 

  25. X. S. Gao, C. C. Zhu and Y. Huang, Building Dynamic Mathematical Models with Geometry Expert, I. Geometric Transformations, Functions and Plane Curves, in Proc. of ATCM’98, W. C. Yang (ed.), pp. 216–224, Springer-Verlag, 1998. 246, 247

    Google Scholar 

  26. X. S. Gao, C. C. Zhu and Y. Huang, Building Dynamic Mathematical Models with Geometry Expert, II. Linkages, in Proc. of ASCM’98, Z. B. Li (ed.), pp. 15–22, LanZhou Univ. Press, 1998. 246, 250, 252

    Google Scholar 

  27. X. S. Gao and H. F. Cheng, On the Solution Classification of the “P3P” Problem, in Proc. of ASCM’98, Z. B. Li (ed.), pp. 185–200, LanZhou Univ. Press, 1998. 246, 252, 253

    Google Scholar 

  28. J. X. Ge, S. C. Chou and X. S. Gao, Geometric Constraint Satisfaction Using Optimization Methods, WSUCS-98-1, CS Dept, Wichita State University, 1998, submitted to CAD. 239

    Google Scholar 

  29. H. Gelernter, Realization of a Geometry-Theorem Proving Machine, in Computers and Thought, E. A. Feigenbaum and J. Feldman (eds.), pp. 134–152, Mcgraw Hill. 232

    Google Scholar 

  30. J. Hopcroft and R. Tarjan, Dividing A Graph into Triconnected Components, SIAM J. Computing, 2(3), 135–157, 1973. 237

    Article  MathSciNet  Google Scholar 

  31. R. Horaud, B. Conio and O. Leboulleux, An Analytic Solution for the Perspective 4-Point Problem, CVGIP, 47, 33–44, 1989. 252

    Google Scholar 

  32. A. Heydon and G. Nelson, The Juno-2 Constraint-Based Drawing Editor, SRC Research Report 131a, 1994. 232, 233

    Google Scholar 

  33. C. Hoffmann, Geometric Constraint Solving in R2 and R3, in Computing in Euclidean Geometry, D. Z. Du and F. Huang (eds.), pp. 266–298, World Scientific, Singapore, 1995. 232, 233, 234

    Google Scholar 

  34. C. Hoffmann and I. Fudos, Constraint-based Parametric Conics for CAD, Geometric Aided Design, 28(2), 91–100, 1996.

    Google Scholar 

  35. Y. Huang and W. D. Wu, Kinematic Solution of a Steawrt Platform, in Proc. IWMM’92, (W. T. Wu and M. D. Cheng Eds.), pp. 181–188, Inter. Academic Publishers, Beijing, 1992. 247

    Google Scholar 

  36. N. Jakiw, Geometer’s Sketchpad, User Guide and Reference Manual, Key Curriculum Press, Berkeley, USA, 1994. 246

    Google Scholar 

  37. N. Jacobson, Basic Algebra, Vol. 1, Freeman, San Francisco, 1985. 245

    MATH  Google Scholar 

  38. D. Kapur, Geometry Theorem Proving Using Hilbert’s Nullstellensatz, in Proc. SYMSAC’86, Waterloo, pp. 202–208, ACM Press, 1986. 244

    Google Scholar 

  39. D. Kapur, T. Saxena and L. Yang, Algebraic and Geometric Reasoning with Dixon Resultants, in Proc. ISSAC’94, Oxford, ACM Press, 1994. 233, 241

    Google Scholar 

  40. A. B. Kempe, On a General Method of Describing Plane Curves of the n-th Degree by Linkwork, Proc. of L. M. S., 213–216, 1876; see also, Messenger of Math., T. VI., 143–144. 247, 250

    Google Scholar 

  41. J. King abd D. Schattschneider, Geometry Turned On, The Mathematical Association of America, 1997. 246

    Google Scholar 

  42. K. Kondo, Algebraic Method for Manipulation of Dimensional Relationships in Geometric Models, Geometric Aided Design, 24(3), 141–147, 1992. 233, 244

    Article  MATH  MathSciNet  Google Scholar 

  43. G. Kramer, Solving Geometric Constraint Systems, MIT Press, 1992. 232, 233, 234

    Google Scholar 

  44. G. Kramer, A Geometric Constraint Engine, Artificial Intelligence, 58, 327–360, 1992.

    Article  MathSciNet  Google Scholar 

  45. H. Lamure and D. Michelucci, Solving Geometric Constraints by Homotopy, IEEE Trans on Visualization and Computer Graphics, 2(1), 28–34, 1996. 239

    Article  Google Scholar 

  46. R. S. Latheam and A. E. Middleditch, Connectivity Analysis: A Tool for Processing Geometric Constraints, Computer Aided Design, 28(11), 917–928, 1994. 233, 234

    Article  Google Scholar 

  47. J. Y. Lee and K. Kim, Geometric Reasoning for Knowledge-Based Parametric Design Using Graph Representation, Computer Aided Design, 28(10), 831–841, 1996. 234

    Article  Google Scholar 

  48. K. Lee and G. Andrews, Inference of the Positions of Components in an Assembly: Part 2, Computer Aided Design, 17(1), 20–24, 1985.

    Article  Google Scholar 

  49. W. Leler, Constraint Programming Languages, Addison Wesley, 1988. 234

    Google Scholar 

  50. R. Light and D. Gossard, Modification of Geometric Models through Variational Geometry, Geometric Aided Design, 14, 208–214, 1982.

    Google Scholar 

  51. V. C. Lin, D. C. Gossard and R. A. Light, Variational Geometry in Computer-Aided Design, Computer Graphics 15(3), 171–177, 1981. 233

    Article  Google Scholar 

  52. G. L. Nemhauser, A. H. G. Rinnooy Kan and M. J. Todd (eds.), Optimization, Elsevier Science Publishers B. V., 1989. 239

    Google Scholar 

  53. J. Owen, Algebraic Solution for Geometry from Dimensional Constraints, in Proc. ACM Symp. Found. of Solid Modeling, ACM Press, pp.397–407, Austin, TX, 1991. 232, 233, 234, 237

    Chapter  Google Scholar 

  54. J. Owen, Constraints of Simple Geometry in Two and Three Dimensions, Inter. J. of Comp. Geometry and Its Applications, 6, 421–434, 1996. 242

    Article  MATH  MathSciNet  Google Scholar 

  55. D. N. Rocheleau and K. Lee, System for Interactive Assembly Modeling, Computer Aided Design, 19 (1), 65–72, 1987.

    Article  Google Scholar 

  56. D. Serrano and D. Gossard, Constraint Management in MCAE, in Artificial Intelligence in Engineering: Design, J. Gero (ed.), pp. 93–110, Elsevier, Amsterdam, 1988.

    Google Scholar 

  57. G. L. Steele and G. L. Sussman, CONSTRAINTS-A Language for Expressing Almost-Hierarchical Descriptions, Artificial Intelligence, 14, 1–39, 1980. 234

    Article  Google Scholar 

  58. H. Suzuki, H. Ando and F. Kimura, Geometric Constraints and Reasoning for Geometrical CAD Systems, Computer and Graphics, 14(2), 211–224, 1990

    Article  Google Scholar 

  59. G. Sunde, Specification of Shape by Dimensions and Other Geometric Constraints, in Geometric Modeling for CAD Applications, M. J. Wozny et al. (eds.), pp. 199–213, North Holland, 1988. 232, 233

    Google Scholar 

  60. I. Sutherland, Sketchpad, A Man-Machine Graphical Communication System, in Proc. of the Spring Joint Comp. Conference, North-Holland, pp. 329–345, 1963. 232, 233

    Google Scholar 

  61. A. Tarski, A Decision Method for Elementary Algebra and Geometry, Univ. of California Press, Berkeley, Calif., 1951. 232

    MATH  Google Scholar 

  62. P. Todd, A k-tree Generalization that Characterizes Consistency of Dimensioned Engineering Drawings, SIAM J. of Disc. Math., 2, 255–261, 1989. 233, 237

    Article  MATH  MathSciNet  Google Scholar 

  63. R. C. Veltkamp, Geometric Constraint Management with Quanta, in Intelligent Computer Aided Design, D. C. Brown et al. (eds.), pp.409–426, North-Holland, 1992. 232, 233

    Google Scholar 

  64. A. Verroust, F. Schonek and D. Roller, Rule-oriented Method for Parameterized Computer-aided Design. Geometric Aided Design, 24(3), 531–540, 1992. 234

    Article  MATH  Google Scholar 

  65. D. M. Wang, Reasoning about Geometric Problems Using an Elimination Method, in Automated Practical Reasoning: Algebraic Approaches, J. Pfalzgraf and D. Wang (eds.), Springer-Verlag, Wien New York, pp. 147–185, 1995. 248

    Google Scholar 

  66. D. M. Wang, GEOTHER: A Geometry Theorem Prover, in Proc. CADE-13, New Brunswick, 1996, pp. 213–239, LNAI, Vol. 1104, Springer-Verlag, Berlin, 1996. 246

    Google Scholar 

  67. D. M. Wang and X. S. Gao, Geometry Theorems Proved Mechanically Using Wu’s Method, Part on Elementary Geometries, MM Research Preprints, No 2, pp. 75–106, 1987, Institute of Systems Science. 248

    Google Scholar 

  68. W. T. Wu, Mechanical Theorem Proving in Geometries: Basic Principles, Springer-Verlag, Wien New York, 1994. 232, 233, 237, 241, 243, 244, 253

    MATH  Google Scholar 

  69. W. T. Wu, A Mechanizations Method of Geometry and its Applications I. Distances, Areas and Volumes, J. Sys. Sci. and Math. Scis., 6, 204–216, 1986. 248

    MATH  Google Scholar 

  70. W. T. Wu, Mathematics Mechanization, Science Press, Beijing, 1999. 248

    Google Scholar 

  71. W. T. Wu, A Mechanization Method of Geometry and Its Applications VI. Solving Inverse Kinematics Equations of PUMA-Type Robotics, pp. 49–53, MM Research Preprints, No 4, 1989, Institute of Systems Science. 247

    Google Scholar 

  72. W. T. Wu and D. K. Wang, On the Surface Fitting Problems in CAGD (in Chinese), Mathematics in Practice and Theory, 3, 1994. 247

    Google Scholar 

  73. L. Yang, H. Fu and Z. Zeng, A Practical Symbolic Algorithm for Inverse Kinematics of 6R Manipulators with Simple Geometry, in Proc. CADE-14, pp. 73–86, Springer-Verlag, Berlin, 1997. 247

    Google Scholar 

  74. L. Yang, J. Z. Zhang and X. R. Hou, Non-Linear Algebraic Equations and Theorem Machine Proof, ShangHai Science and Education Press, ShangHai, 1997(in Chinese).

    Google Scholar 

  75. L. Yang, A Simplified Algorithm for Solution Classification of the P3P Problem, preprint, 1998. 252

    Google Scholar 

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Authors and Affiliations

  1. Academia Sinica, Institute of Systems Science, Beijing, 100080, P.R. China

    Xiao-Shan Gao

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© 1999 Springer-Verlag Berlin Heidelberg

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Gao, XS. (1999). Automated Geometry Diagram Construction and Engineering Geometry. In: Automated Deduction in Geometry. ADG 1998. Lecture Notes in Computer Science(), vol 1669. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-47997-X_12

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  • DOI: https://doi.org/10.1007/3-540-47997-X_12

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  • Print ISBN: 978-3-540-66672-1

  • Online ISBN: 978-3-540-47997-0

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