The Visual Computer

, Volume 24, Issue 3, pp 187–200 | Cite as

Solving nonlinear polynomial systems in the barycentric Bernstein basis

  • Martin Reuter
  • Tarjei S. Mikkelsen
  • Evan C. Sherbrooke
  • Takashi Maekawa
  • Nicholas M. Patrikalakis
Original Article


We present a method for solving arbitrary systems of N nonlinear polynomials in n variables over an n-dimensional simplicial domain based on polynomial representation in the barycentric Bernstein basis and subdivision. The roots are approximated to arbitrary precision by iteratively constructing a series of smaller bounding simplices. We use geometric subdivision to isolate multiple roots within a simplex. An algorithm implementing this method in rounded interval arithmetic is described and analyzed. We find that when the total order of polynomials is close to the maximum order of each variable, an iteration of this solver algorithm is asymptotically more efficient than the corresponding step in a similar algorithm which relies on polynomial representation in the tensor product Bernstein basis. We also discuss various implementation issues and identify topics for further study.


CAD CAGD CAM Geometric modeling Solid modeling Intersections Distance computation Engineering design 


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  1. 1.
    Abrams, S.L., Cho, W., Hu, C.Y., Maekawa, T., Patrikalakis, N.M., Sherbrooke, E.C., Ye, X.: Efficient and reliable methods for rounded-interval arithmetic. Comput. Aided Des. 30(8), 657–665 (1998)zbMATHCrossRefGoogle Scholar
  2. 2.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. MIT Press, Cambridge, MA (1990)zbMATHGoogle Scholar
  3. 3.
    Elber, G., Kim, M.S.: Geometric constraint solver using multivariate rational spline functions. In: SMA ’01: Proceedings of the Sixth ACM Symposium on Solid Modeling and Applications, pp. 1–10. ACM Press, New York, NY, USA (2001) (DOI Scholar
  4. 4.
    Farin, G.: Triangular Bernstein–Bézier patches. Comput. Aided Geom. Des. 3(2), 83–128 (1986)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Farin, G.: Curves and Surfaces for Computer-Aided Geometric Design: A Practical Guide, 4th edn. Academic Press, San Diego (1997)zbMATHGoogle Scholar
  6. 6.
    Farin, G.E.: Geometric Modeling: Algorithms and New Trends. SIAM, Philadelphia (1987)zbMATHGoogle Scholar
  7. 7.
    Goldman, R.N.: Subdivision algorithms for Bézier triangles. Comput. Aided Des. 15(3), 159–166 (1983)CrossRefGoogle Scholar
  8. 8.
    Goldman, R.N.: Blossoming and knot insertion algorithms for B-spline curves. Comput. Aided Geom. Des. 7, 69–81 (1990)zbMATHCrossRefGoogle Scholar
  9. 9.
    Goldman, R.N., Lyche, T. (eds.): Knot Insertion and Deletion Algorithms for B-Spline Curves and Surfaces. SIAM, Philadelphia (1993)Google Scholar
  10. 10.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science, 2nd edn. Addison–Wesley, Boston (1994)zbMATHGoogle Scholar
  11. 11.
    Hanniel, I., Elber, G.: Subdivision termination criteria in subdivision multivariate solvers using dual hyperplanes representations. Comput. Aided Des. 39(5), 369–378 (2007) (DOI Scholar
  12. 12.
    Hoschek, J., Lasser, D.: Fundamentals of Computer Aided Geometric Design. A.K. Peters, Wellesley, MA (1993) (Translated by L.L. Schumaker)Google Scholar
  13. 13.
    Hu, C.Y., Maekawa, T., Patrikalakis, N.M., Ye, X.: Robust interval algorithm for surface intersections. Comput. Aided Des. 29(9), 617–627 (1997)zbMATHCrossRefGoogle Scholar
  14. 14.
    Lane, J.M., Riesenfeld, R.F.: A theoretical development for the computer generation and display of piecewise polynomial surfaces. IEEE Trans. Pattern Anal. Mach. Intell. 2, 35–46 (1980)zbMATHCrossRefGoogle Scholar
  15. 15.
    Mourrain, B., Pavone, J.P.: Subdivision Methods for Solving Polynomial Equations. Tech. Rep. 5658, INRIA Sophia-Antipolis (2005)Google Scholar
  16. 16.
    Sablonnière, P.: Spline and Bézier polygons associated with a polynomial spline curve. Comput. Aided Des. 10(4), 257–261 (1978)CrossRefGoogle Scholar
  17. 17.
    Sederberg, T.W.: Algorithm for algebraic curve intersection. Comput. Aided Des. 21(9), 547–554 (1989)zbMATHCrossRefGoogle Scholar
  18. 18.
    Sherbrooke, E.C., Patrikalakis, N.M.: Computation of the solutions of nonlinear polynomial systems. Comput. Aided Geom. Des. 10(5), 379–405 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Waggenspack, W.N., Anderson, C.D.: Converting standard bivariate polynomials to Bernstein form over arbitrary triangular regions. Comput. Aided Des. 18(10), 529–532 (1986)CrossRefGoogle Scholar
  20. 20.
    Wilkinson, J.H.: Rounding Errors in Algebraic Processs. Prentice-Hall, Englewood Cliffs (1963)Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Martin Reuter
    • 1
  • Tarjei S. Mikkelsen
    • 1
  • Evan C. Sherbrooke
    • 1
  • Takashi Maekawa
    • 2
  • Nicholas M. Patrikalakis
    • 1
  1. 1.Massachusetts Institute of TechnologyCambridgeUSA
  2. 2.Yokohama National UniversityYokohamaJapan

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