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Issue No.02 - March/April (1994 vol.14)
pp: 46-55
ABSTRACT
<p>Geometric and solid modelling deal with the representation and manipulation of physical objects. Currently most geometric objects are formulated in terms of polynomial equations, thereby reducing many application problems to manipulating polynomial systems. Solving systems of polynomial equations is a fundamental problem in these geometric computations. The author presents an algorithm for solving polynomial equations. The combination of multipolynomial resultants and matrix computations underlies this efficient, robust and accurate algorithm.</p>
CITATION
Dinesh Manocha, "Solving Systems of Polynomial Equations", IEEE Computer Graphics and Applications, vol.14, no. 2, pp. 46-55, March/April 1994, doi:10.1109/38.267470
REFERENCES
1. D. Lavender et al, "Voronoi Diagrams of Set-Theoretic Solid Models,"IEEE CG&A, Vol. 12, No. 5, Sept. 1992, pp. 69-77.
2. J. F. Canny,The Complexity of Robot Motion Planning. Cambridge, MA: MIT Press, 1988.
3. T.W. Sederberg,Implicit and Parametric Curves and Surfaces for Computer-Aided Geometric Design, doctoral dissertation, Purdue Univ., West Lafayette, Ind., 1983.
4. J. Kajiya, "Ray Tracing Parametric Patches,"Computer Graphics, Vol. 16, No. 3, 1982, pp. 245-254.
5. C. M. Hoffmann,Geometric and Solid Modeling: An Introduction. Los Altos, CA: Morgan Kaufmann, 1989.
6. D. Manocha and J.F. Canny, "A New Approach for Surface Intersection,"Int'l J. Computational Geometry and Applications(Special issue on Solid Modeling), Vol. 1, No. 4, 1991, pp. 491-516.
7. C.M. Hoffmann, "A Dimensionality Paradigm for Surface Interrogations,"Computer Aided Geometric Design, Vol. 7, No. 6, 1990, pp. 517-532.
8. J.H. Wilkinson, "The Evaluation of the Zeros of Ill-Conditioned Polynomials, Parts i and ii,"Numerical Mathematics, Vol. 1, 1959, pp. 150-166 and 167-180.
9. A.A.G. Requicha and J.R. Rossignac, "Solid Modeling and Beyond,"IEEE CG&A, Vol. 12, No. 5, Sept. 1992, pp. 31-44.
10. A.P. Morgan, "Polynomial Continuation and Its Relationship to the Symbolic Reduction of Polynomial Systems," inSymbolic and Numerical Computation for Artificial Intelligence, B. Donald et al., eds., Academic Press, New York, 1992, pp. 23-45.
11. T.W. Sederberg and T. Nishita, "Curve Intersection Using Bézier Clipping,"Computer-Aided Design, Vol. 22, No. 9, Nov. 1990, pp. 538-549.
12. D. Manocha and J.F. Canny, "Multipolynomial Resultant Algorithms,"J. Symbolic Computation, Vol. 15, No. 2, 1993, pp. 99-122.
13. F.S. Macaulay, "On Some Formula in Elimination,"Proc. London Mathematical Society, Vol. 1, No. 33, 1902, pp. 3-27.
14. G. Salmon,Lessons Introductory to the Modern Higher Algebra, G.E. Stechert&Co., New York, 1885.
15. B. Sturmfels and A. Zelevinsky, "Multigraded Resultants of Sylvester Type," to appear inJ. Algebra, 1994.
16. J. Canny and I. Emiris, "An Efficient Algorithm for the Sparse Mixed Resultant," inProc. 10th Int'l Symp. Applied Algebra, Algebraic Algorithms, and Error Correction Codes, Springer-Verlag, New York, 1993, pp. 89-104.
17. D. Manocha,Algebraic and Numeric Techniques for Modeling and Robotics, doctoral dissertation, Univ. of California, Berkeley, 1992.
18. C. Bajaj, T. Garrity, and J. Warren, "On the Applications of Multi-Equational Resultants," Tech. Report CSD-TR-826, Dept. of Computer Science, Purdue Univ., 1988.
19. W. Auzinger and H.J. Stetter, "An Elimination Algorithm for the Computation of All Zeros of a System of Multivariate Polynomial Equations," inInt'l Series of Numerical Mathematics, Vol. 86, Birkhauser, Basil, Switzerland, 1986, pp. 11-30.
20. D. Manocha and J. Demmel, "Algorithms for Intersecting Parametric and Algebraic Curves," inGraphics Interface 92, Canadian Information Processing Society, Toronto, 1992, pp. 232-241.
21. B.L. Van Der Waerden,Modern Algebra, 3rd ed., F. Ungar Publishing, New York, 1950.
22. G.H. Golub and C.F. Van Loan,Matrix Computations, Johns Hopkins Press, Baltimore, 1989.
23. Z. Bai and J. Demmel, "Design of a Parallel Nonsymmetric Eigenroutine Toolbox, Part I," Tech. Report CS-718, Univ. of Calif., Berkeley, 1992.
24. E. Anderson et al.,Lapack User's Guide, Soc. for Industrial and Applied Math., Philadelphia, 1992.
25. J.C. Owen, "Algebraic Solution for Geometry from Dimensional Constraints,"Proc. Symp. Solid Modeling Foundations and CAD/CAM Applications, ACM, New York, 1991, pp. 397-407.
26. C. Wampler and A.P. Morgan, "Solving the 6R Inverse Position Problem Using a Generic-Case Solution Methodology,"Mechanisms and Machine Theory, Vol. 26, No. 1, 1991, pp. 91-106.
27. G.W. Stewart, "Simultaneous Iteration for Computing Invariant Subspaces of Nonhermitian Matrices,"Numerische Mathematik, Vol. 25, 1976, pp. 12-36.
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