Subscribe
Issue No.04 - July/August (2009 vol.15)
pp: 596-604
Gershon Elber , Technion - IIT, Haifa
Tom Grandine , The Boeing Company, Seattle
ABSTRACT
In recent years, several quite successful attempts have been made to solve systems of polynomial constraints, using geometric design tools, exploiting the availability of subdivision-based solvers [7], [11], [12], [15]. This broad range of methods includes both binary domain subdivision as well as the projected polyhedron method of Sherbrooke and Patrikalakis [15]. A prime obstacle in using subdivision solvers is their scalability. When the given constraint is represented as a tensor product of all its independent variables, it grows exponentially in size as a function of the number of variables. In this work, we show that for many applications, especially geometric ones, the exponential complexity of the constraints can be reduced to a polynomial by representing the underlying structure of the problem in the form of expression trees that represent the constraints. We demonstrate the applicability and scalability of this representation and compare its performance to that of tensor product constraint representation through several examples.
INDEX TERMS
Interval arithmetic, multivariate polynomial constraint solver, self-bisectors, contact computation, Hausdorff distance.
CITATION
Gershon Elber, Tom Grandine, "An Efficient Solution to Systems of Multivariate Polynomial Using Expression Trees", IEEE Transactions on Visualization & Computer Graphics, vol.15, no. 4, pp. 596-604, July/August 2009, doi:10.1109/TVCG.2009.42
REFERENCES
 [1] A.V. Aho, R. Sethi, and J.D. Ullman, Compilers: Principles, Techniques, and Tools. Addison-Wesley, 1986. [2] M. Bartoň and B. Jüttler, “Computing Roots of Polynomials by Quadratic Clipping,” Computer Aided Geometric Design, vol. 24, no. 3, pp. 125-141, 2007. [3] M. Bartoň, B. Jüttler, and B. Moore, “Polynomial Solvers with Superquadratic Convergence,” Proc. 10th SIAM Conf. Geometric Design & Computing, 2007. [4] E. Cohen, R.F. Riesenfeld, and G. Elber, Geometric Modeling with Splines. A.K. Peters, 2001. [5] G. Elber and T. Grandine, “Hausdorff and Minimal Distances between Parametric Freeforms in ${\hbox{\rlap{I}\kern 2.0pt{\hbox{R}}}}^2$ and ${\hbox{\rlap{I}\kern 2.0pt{\hbox{R}}}}^3$ ,” Proc. Int'l Conf. Advances in Geometric Modeling and Processing, pp. 191-204, 2008. [6] G. Elber, T. Grandine, and M.S. Kim, “Surface Self Intersection Computation via Algebraic Decomposition,” Computer Aided Design, to be published. [7] G. Elber and M.-S. Kim, “Geometric Constraint Solver Using Multivariate Rational Spline Functions,” Proc. Symp. Solid Modeling and Applications '01, pp. 1-10, 2001. [8] L.H.D. Figueiredo and J. Stolfi, “Affine Arithmetic: Concepts and Applications,” Numerical Algorithms, vol. 37, pp. 147-158, 2004. [9] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, 1979. [10] I. Hanniel and G. Elber, “Subdivision Termination Criteria in Subdivision Multivariate Solvers,” Computer Aided Design, vol. 39, pp. 369-378, 2007. [11] J. Lane and R. Riesenfeld, “Bounds on a Polynomial,” BIT, vol. 21, pp. 112-117, 1981. [12] B. Mourrain and J.P. Pavone, “Subdivision Methods for Solving Polynomial Equations,” Technical Report RR-5658, INRIA Sophia-Antipolis, http://hal.inria.fr/inria-00070350en/, 2006. [13] T. Nishita, T.W. Sederberg, and M. Kakimoto, “Ray Tracing Trimmed Rational Surface Patches,” Proc. ACM SIGGRAPH '90, pp. 337-345, 1990. [14] M. Reuter, T.S. Mikkelsen, E.C. Sherbrooke, T. Maekawa, and N.M. Patrikalakis, “Solving Nonlinear Polynomial Systems in the Barycentric Bernstein Basis,” The Visual Computer, vol. 24, no. 3, pp. 187-200, 2008. [15] E.C. Sherbrooke and N.M. Patrikalakis, “Computation of the Solutions of Nonlinear Polynomial Systems,” Computer Aided Geometric Design, vol. 10, no. 5, pp. 279-405, 1993. [16] C.K. Yap, “Robust Geometric Computation,” Handbook of Discrete and Computational Geometry, J.E. Goodman and J. O'Rourke, eds., second ed., chapter 41, pp. 927-952, Chapman & Hall/CRC, 2004.
20 ms
(Ver 2.0)

Marketing Automation Platform