A Systematic Framework for Solving Geometric Constraints Analytically

Journal of Symbolic Computation

Volumn 30, Issue 5, 2000, Pages 493-519

  Durand, Cassiano ^a   Hoffmann, Christoph M ^a  

^a Purdue University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0005284683     PISSN: 07477171     EISSN: None     Source Type: Journal    
DOI: 10.1006/jsco.2000.0392     Document Type: Article

References (80)

1. Aldefeld B. Variation of geometries based on a geometric-reasoning method. Comput. Aided Des. 20:1988;117-126.
2. Allgower E. L., Georg K. Numerical Continuation Methods. An Introduction. 1990;Springer-Verlag, New York.
3. Allgower E. L., Georg K. Continuation and path following. Acta Numerica. 2:1993;1-64.
4. Allgower E. L., Georg K. Numerical path following. Ciarlet P. G., Lions J. L. Techniques of Scientific Computing (Part 2), volume 5 of Handbook of Numerical Analysis. 1997;3-203 North-Holland, Amsterdam.
5. Betke U. Mixed volumes of polytopes. Arch. Math. 58:1992;388-391.
6. Bonnesen T., Fenchel W. Theory of Convex Bodies. 1987;BCS Associates.
7. Borning A. H. The programming language aspects of ThingLab, a constraint oriented simulation laboratory. ACM TOPLAS. 3:1981;353-387.
8. Bouma W., Fudos I., Hoffmann C. M., Cai J., Paige R. A geometric constraint solver. Comput. Aided Des. 27:1995;485-501.
9. B. D. Brüderlin, 1987.
10. B. Buchberger, 1965.
11. Buchberger B. Gröbner bases: An algorithmic method in polynomial ideal theory. Bose N. K. Recent Trends in Multidimensional Systems Theory. 1985;184-232 D. Reidel, Dordrecht.
12. Chou S. C. An introduction to Wu's method for mechanical theorem proving in geometry. J. Autom. Reasoning. 4:1988;237-267.
13. Cox D., Little J., O'Shea D. Ideals, Varieties and Algorithms. 1992;Springer-Verlag, New York.
14. Crippen G. M., Havel T. F. Distance Geometry and Molecular Conformation. 1988;Research Studies Press, Somerset, England.
15. D-Cubed. The Dimensional Constraint Manager. 1994.
16. Dennis J. E., Schnabel R. B. Numerical Methods of Unconstrained Optimization and Nonlinear Equations. 1983;Prentice-Hall, Englewood Cliffs, NJ.
17. C. Durand, 1998.
18. C. Durand, C. M. Hoffmann, 1998, Department of Computer Sciences, Purdue University, West Lafayette, IN, U.S.A.
19. Dyer M., Gritzmann P., Hufnagel A. On the complexity of computing mixed volumes. SIAM J. Comput. 17:1998;356-400.
20. Emiris I. Z., Canny J. F. Efficient incremental algorithm for the sparse resultant and the mixed volume. J. Symb. Comput. 20:1995;117-149.
21. I. Z. Emiris, B. Mourrain, 1996, INRIA.
22. I. Z. Emiris, J. Verschelde, 1997, INRIA.
23. I. Fudos, 1995.
24. I. Fudos, C. M. Hoffmann, 1993, Department of Computer Sciences, Purdue University, West Lafayette, IN, U.S.A.
25. Fudos I., Hoffmann C. M. Constraint-based parametric conics for CAD. Comput. Aided Des. 28:1996;91-100.
26. Gelfand I. M. Discriminants, Resultants, and Multidimensional Determinants. 1994;Birkhauser, Basel.
27. C. M. Henneberg.
28. Hoffmann C. M. Solid and Geometric Modeling. 1989;Morgan Kaufmann, California, U.S.A.
29. Hoffmann C. M., Joan-Arinyo R. Symbolic constraints in constructive geometry. J. Symb. Comput. 23:1997;287-300.
30. Hoffmann C. M., Lomonosov A., Sitharam M. Finding dense subgraphs of constraint graphs. Smolka G. Constraint Programming '97. 1997a;463-478 Springer-Verlag, New York.
31. Hoffmann C. M., Lomonosov A., Sitharam M. Smolka G. Finding solvable subsets of constraint graphs. 1997b;463-477 Springer, New York.
32. 3. Computing in Euclidean Geometry. 1994;World Scientific Publishing, Singapore.
33. C. M. Hoffmann, P. J. Vermeer, Proceedings of the Computational Kinematics Workshop, Nice, France, September 1995, 1995, ACM, New York.
34. B. Huber, 1996.
35. Huber B., Sturmfels B. A polyhedral method for solving sparse polynomial systems. Math. Comput. 64:October 1995;1541-1555.
36. Huber B., Sturmfels B. Bernstein's theorem in affine space. Discrete Comput. Geom. 17:1997;137-141.
37. Joan-Arinyo R., Soto A. A correct rule-based geometric constraint solver. Comput. Graph. 21:1997a;599-609.
38. Joan-Arinyo R., Soto A. A ruler-and-compass geometric constraint solver. Pratt M. J., Sriram R. D., Wozny M. J. Product Modeling for Computer Integrated Design and Manufacture. 1997b;384-393 Chapman and Hall, London.
39. Kalkbrener M. A generalized Euclidean algorithm for computing triangular representations of algebraic varieties. J. Symb. Comput. 15:1993;143-167.
40. Khovanskii A. G. Newton polyhedra and toroidal varieties. Funct. Anal. Appl. 11:1977;289-296.
41. Khovanskii A. G. Newton polyhedra and the genus of complete intersections. Funktsional'nyi Analiz i Ego Prilozheniya. 12:1978;51-61.
42. Kondo K. Algebraic method for manipulation of dimensional relationships in geometric models. Comput. Aided Des. 24:1992;141-147.
43. Kushnirenko A. G. A Newton polytope and the number of solutions of a system of k equations in k unknowns. Uspekhi Matematicheskikh Nauk. 30:1975;266-267.
44. Lahaye E. Une méthode de resolution d'une categorie d'equations transcendantes. Comptes Rendus Séances Acad. Sci. 198:1934;1840-1842.
45. Laman G. On graphs and the rigidity of plane skeletal structures. J. Eng. Math. 4:1970;331-340.
46. Lamure H., Michelucci D. Solving geometric constraints by homotopy. Third Symposium on Solid Modeling and its Applications, Salt Lake City, UT. 1995;ACM, New York. p. 263-269.
47. Lazard D. A new method for solving algebraic systems of positive dimension. Discrete Appl. Math. 33:1991;147-160.
48. Lazard D. On theories of triangular sets. J. Symb. Comput. 28:1999;105-124.
49. Li T. Y. Numerical solutions of multivariate polynomial systems by homotopy continuation methods. Acta Numerica. 6:1997;399-436.
50. Li T. Y., Sauer T., Yorke J. A. The Cheater's homotopy: an efficient procedure for solving system of polynomial equations. SIAM J. Numer. Anal. 26:1989;1241-1251.
51. Manocha D. Efficient algorithms for multipolynomial resultant. Comput. J. Special issue on Quantifier Elimination. 36:1993;485-496.
52. Manocha D., Canny J. Multipolynomial resultant algorithms. J. Symb. Comput. 15:1993;99-122.
53. Morgan A. P. A homotopy for solving polynomial systems. Appl. Math. Comput. 18:1986a;87-92.
54. Morgan A. P. A transformation to avoid solutions at infinity for polynomial systems. Appl. Math. Comput. 18:1986b;77-86.
55. Morgan A. Solving Polynomial Systems using Continuation for Engineering and Scientific Problems. 1987;Prentice-Hall, New Jersey.
56. Morgan A. P. Polynomial continuation and its relationship to the symbolic reduction of polynomial systems. Donald B. R., Kapur D., Mundy J. L. Symbolic and Numerical Computation for Artificial Inteligence. 1992;Academic Press, San Diego.
57. Morgan A. P., Sommese A. J. Coefficient-parameter polynomial continuation. Appl. Math. Comput. 29:1989;123-160.
58. Nanua P., Waldron K. J., Murthy V. Direct kinematic solution of a Stewart platform. IEEE Trans. Robot. Autom. 6:1990;438-443.
59. Ortega J., Rheinboldt W. Iterative Solution of Nonlinear Equations in Several Variables. 1970;Academic Press, San Diego.
60. Owen J. Algebraic solution for geometry from dimensional constraints. ACM Symposium on the Foundations of Solid Modeling, Austin, TX. 1991;ACM, New York. p. 397-407.
61. Patrikalakis N. M. Surface-to-surface intersections. IEEE Comput. Graph. Appl. 13:1992;89-95.
62. Peitgen H. O., Ritcher P. H. The Beauty of Fractals, Images of Complex Dynamical Systems. 1986;Springer-Verlag, New York.
63. Ritt J. F. Differential Equations from the Algebraic Standpoint. 1932;American Mathematical Society, Providence, R.I., U.S.A.
64. Ritt J. F. Differential Algebra. 1950;American Mathematical Society, Providence, R.I., U.S.A.
65. Rojas J. M. Solving degenerate sparse polynomial systems faster. J. Symb. Comput. 28:1999;155-186.
66. Schneider R. Convex Bodies. 1993;Cambridge University Press, Cambridge.
67. T. W. Sederberg, August 1983.
68. Stoer J., Bulirsch R. Introduction to Numerical Analysis. 1993;Springer-Verlag, New York.
69. B. Sturmfels, Lecture Notes at the AMS Short Course on Applications of Computational Algebraic Geometry, San Diego, January 1997.
70. Verroust A., Schonek F., Roller D. Rule-oriented method for parametrized computer-aided design. Comput. Aided Des. 24:1992;531-540.
71. J. Verschelde, 1996.
72. J. Verschelde.
73. J. Verschelde, 1997, Department of Computer Science, Katholieke Universiteit Leuven.
74. J. Verschelde.
75. Verschelde J., Gatermann K., Cools R. Mixed-volume computation by dynamic lifting applied to polynomial system solving. Discrete Comput. Geom. 16:1996;69-112.
76. Wampler C. W., Morgan A. P., Sommese A. J. Numerical continuation methods for solving systems arising in kinematics. J. Mech. Des. 112:1990;59-68.
77. Wang D. On Wu's Method for Solving Systems of Algebraic Equations. 1991;Johannes Kepler University, Austria.
78. Wang D. Decomposing polynomial systems into simple systems. J. Symb. Comput. 25:1998;295-314.
79. Wu W. Basic principles of mechanical theorem proving in elementary geometries. J. Autom. Reasoning. 2:1986;221-252.
80. Wu W. Mechanical Theorem Proving in Geometries: Basic Principles. 1994;Springer-Verlag, New York.