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Automated Geometry Diagram Construction and Engineering Geometry

  • Xiao-Shan Gao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1669)

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.

Keywords

Geometric Constraint Dynamic Geometry Regular Pentagon Linkage Design Geometry Theorem 
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. 1.
    B. Aldefeld, Variation of Geometries Based on a Geometric-Reasoning Method, Computer Aided Design, 20(3), 117–126, 1988. 234MATHCrossRefGoogle Scholar
  2. 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, 233Google Scholar
  3. 3.
    A. Borning, The Programming Language Aspect of ThingLab, ACM Tras. on Programming Language and Systems, 3(4), 353–387, 1981. 232, 233CrossRefGoogle Scholar
  4. 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.MATHCrossRefGoogle Scholar
  5. 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, 233Google Scholar
  6. 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.CrossRefGoogle Scholar
  7. 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, 241Google Scholar
  8. 8.
    B. Char et al., Maple V, Springer-Verlag, Berlin, 1992. 245MATHGoogle Scholar
  9. 9.
    S. C. Chou, Mechanical Geometry Theorem Proving, D. Reidel Publishing Company, Dordrecht, Netherlands, 1988. 237, 246MATHGoogle Scholar
  10. 10.
    S. C. Chou, A Method for Mechanical Deriving of Formulas in Elementary Geometry, J. of Automated Reasoning, 3, 291–299, 1987. 248MATHCrossRefGoogle Scholar
  11. 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, 248CrossRefGoogle Scholar
  12. 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. 244Google Scholar
  13. 13.
    S. C. Chou, X. S. Gao and J. Z. Zhang, Machine Proofs in Geometry, World Scientific, Singapore, 1994. 237MATHGoogle Scholar
  14. 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, 236Google Scholar
  15. 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, 244Google Scholar
  16. 16.
    J. Chuan, Geometric Constructions with the Computer, in Proc. ATCM’95, pp. 329–338, Springer-Verlag, 1995.Google Scholar
  17. 17.
    K. H. Elster (ed.), Modern Mathematical Methods of Optimization, Akademie Verlag, 1993. 239Google Scholar
  18. 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. 252CrossRefGoogle Scholar
  19. 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.CrossRefGoogle Scholar
  20. 20.
    Gabri Geometry II, Texas Instruments, Dallas, Texas, 1994. 246Google Scholar
  21. 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, 249CrossRefGoogle Scholar
  22. 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, 246CrossRefGoogle Scholar
  23. 23.
    X. S. Gao and S. C. Chou, Implicitization of Rational Parametric Equations, Journal of Symbolic Computation, 14, 459–470, 1992. 247MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    X. S. Gao, J. Z. Zhang and S. C. Chou, Geometry Expert, Nine Chapters Pub., 1998, Taiwan (in Chinese). 246Google Scholar
  25. 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, 247Google Scholar
  26. 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, 252Google Scholar
  27. 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, 253Google Scholar
  28. 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. 239Google Scholar
  29. 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. 232Google Scholar
  30. 30.
    J. Hopcroft and R. Tarjan, Dividing A Graph into Triconnected Components, SIAM J. Computing, 2(3), 135–157, 1973. 237CrossRefMathSciNetGoogle Scholar
  31. 31.
    R. Horaud, B. Conio and O. Leboulleux, An Analytic Solution for the Perspective 4-Point Problem, CVGIP, 47, 33–44, 1989. 252Google Scholar
  32. 32.
    A. Heydon and G. Nelson, The Juno-2 Constraint-Based Drawing Editor, SRC Research Report 131a, 1994. 232, 233Google Scholar
  33. 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, 234Google Scholar
  34. 34.
    C. Hoffmann and I. Fudos, Constraint-based Parametric Conics for CAD, Geometric Aided Design, 28(2), 91–100, 1996.Google Scholar
  35. 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. 247Google Scholar
  36. 36.
    N. Jakiw, Geometer’s Sketchpad, User Guide and Reference Manual, Key Curriculum Press, Berkeley, USA, 1994. 246Google Scholar
  37. 37.
    N. Jacobson, Basic Algebra, Vol. 1, Freeman, San Francisco, 1985. 245MATHGoogle Scholar
  38. 38.
    D. Kapur, Geometry Theorem Proving Using Hilbert’s Nullstellensatz, in Proc. SYMSAC’86, Waterloo, pp. 202–208, ACM Press, 1986. 244Google Scholar
  39. 39.
    D. Kapur, T. Saxena and L. Yang, Algebraic and Geometric Reasoning with Dixon Resultants, in Proc. ISSAC’94, Oxford, ACM Press, 1994. 233, 241Google Scholar
  40. 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, 250Google Scholar
  41. 41.
    J. King abd D. Schattschneider, Geometry Turned On, The Mathematical Association of America, 1997. 246Google Scholar
  42. 42.
    K. Kondo, Algebraic Method for Manipulation of Dimensional Relationships in Geometric Models, Geometric Aided Design, 24(3), 141–147, 1992. 233, 244MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    G. Kramer, Solving Geometric Constraint Systems, MIT Press, 1992. 232, 233, 234Google Scholar
  44. 44.
    G. Kramer, A Geometric Constraint Engine, Artificial Intelligence, 58, 327–360, 1992.CrossRefMathSciNetGoogle Scholar
  45. 45.
    H. Lamure and D. Michelucci, Solving Geometric Constraints by Homotopy, IEEE Trans on Visualization and Computer Graphics, 2(1), 28–34, 1996. 239CrossRefGoogle Scholar
  46. 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, 234CrossRefGoogle Scholar
  47. 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. 234CrossRefGoogle Scholar
  48. 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.CrossRefGoogle Scholar
  49. 49.
    W. Leler, Constraint Programming Languages, Addison Wesley, 1988. 234Google Scholar
  50. 50.
    R. Light and D. Gossard, Modification of Geometric Models through Variational Geometry, Geometric Aided Design, 14, 208–214, 1982.Google Scholar
  51. 51.
    V. C. Lin, D. C. Gossard and R. A. Light, Variational Geometry in Computer-Aided Design, Computer Graphics 15(3), 171–177, 1981. 233CrossRefGoogle Scholar
  52. 52.
    G. L. Nemhauser, A. H. G. Rinnooy Kan and M. J. Todd (eds.), Optimization, Elsevier Science Publishers B. V., 1989. 239Google Scholar
  53. 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, 237CrossRefGoogle Scholar
  54. 54.
    J. Owen, Constraints of Simple Geometry in Two and Three Dimensions, Inter. J. of Comp. Geometry and Its Applications, 6, 421–434, 1996. 242MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    D. N. Rocheleau and K. Lee, System for Interactive Assembly Modeling, Computer Aided Design, 19 (1), 65–72, 1987.CrossRefGoogle Scholar
  56. 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. 57.
    G. L. Steele and G. L. Sussman, CONSTRAINTS-A Language for Expressing Almost-Hierarchical Descriptions, Artificial Intelligence, 14, 1–39, 1980. 234CrossRefGoogle Scholar
  58. 58.
    H. Suzuki, H. Ando and F. Kimura, Geometric Constraints and Reasoning for Geometrical CAD Systems, Computer and Graphics, 14(2), 211–224, 1990CrossRefGoogle Scholar
  59. 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, 233Google Scholar
  60. 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, 233Google Scholar
  61. 61.
    A. Tarski, A Decision Method for Elementary Algebra and Geometry, Univ. of California Press, Berkeley, Calif., 1951. 232MATHGoogle Scholar
  62. 62.
    P. Todd, A k-tree Generalization that Characterizes Consistency of Dimensioned Engineering Drawings, SIAM J. of Disc. Math., 2, 255–261, 1989. 233, 237MATHCrossRefMathSciNetGoogle Scholar
  63. 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, 233Google Scholar
  64. 64.
    A. Verroust, F. Schonek and D. Roller, Rule-oriented Method for Parameterized Computer-aided Design. Geometric Aided Design, 24(3), 531–540, 1992. 234MATHCrossRefGoogle Scholar
  65. 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. 248Google Scholar
  66. 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. 246Google Scholar
  67. 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. 248Google Scholar
  68. 68.
    W. T. Wu, Mechanical Theorem Proving in Geometries: Basic Principles, Springer-Verlag, Wien New York, 1994. 232, 233, 237, 241, 243, 244, 253MATHGoogle Scholar
  69. 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. 248MATHGoogle Scholar
  70. 70.
    W. T. Wu, Mathematics Mechanization, Science Press, Beijing, 1999. 248Google Scholar
  71. 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. 247Google Scholar
  72. 72.
    W. T. Wu and D. K. Wang, On the Surface Fitting Problems in CAGD (in Chinese), Mathematics in Practice and Theory, 3, 1994. 247Google Scholar
  73. 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. 247Google Scholar
  74. 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. 75.
    L. Yang, A Simplified Algorithm for Solution Classification of the P3P Problem, preprint, 1998. 252Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Xiao-Shan Gao
    • 1
  1. 1.Academia SinicaInstitute of Systems ScienceBeijingP.R. China

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