Some Geometrical Aspects of Control Points for Toric Patches

  • Gheorghe Craciun
  • Luis David García-Puente
  • Frank Sottile
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5862)


We use ideas from algebraic geometry and dynamical systems to explain some ways that control points influence the shape of a Bézier curve or patch. In particular, we establish a generalization of Birch’s Theorem and use it to deduce sufficient conditions on the control points for a patch to be injective. We also explain a way that the control points influence the shape via degenerations to regular control polytopes. The natural objects of this investigation are irrational patches, which are a generalization of Krasauskas’s toric patches, and include Bézier and tensor product patches as important special cases.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dahmen, W.: Convexity and Bernstein-Bézier polynomials. In: Curves and surfaces (Chamonix-Mont-Blanc, 1990), pp. 107–134. Academic Press, Boston (1991)CrossRefGoogle Scholar
  2. 2.
    Craciun, G., Feinberg, M.: Multiple equilibria in complex chemical reaction networks. I. The injectivity property. SIAM J. Appl. Math. 65(5), 1526–1546 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Krasauskas, R.: Toric surface patches. Adv. Comput. Math. 17(1-2), 89–133 (2002); Advances in geometrical algorithms and representationsGoogle Scholar
  4. 4.
    Brown, L.D.: Fundamentals of statistical exponential families with applications in statistical decision theory. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 9. Institute of Mathematical Statistics, Hayward, CA (1986)Google Scholar
  5. 5.
    Darroch, J.N., Ratcliff, D.: Generalized iterative scaling for log-linear models. Ann. Math. Statist. 43, 1470–1480 (1972)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Sottile, F.: Toric ideals, real toric varieties, and the moment map. In: Topics in algebraic geometry and geometric modeling. Contemp. Math., vol. 334, pp. 225–240. Amer. Math. Soc., Providence (2003)CrossRefGoogle Scholar
  7. 7.
    Garcia-Puente, L.D., Sottile, F.: Linear precision for parametric patches. Advances in Computational Mathematics (to appear, 2009)Google Scholar
  8. 8.
    Karčiauskas, K., Krasauskas, R.: Comparison of different multisided patches using algebraic geometry. In: Laurent, P.J., Sablonniere, P., Schumaker, L. (eds.) Curve and Surface Design: Saint-Malo 1999, pp. 163–172. Vanderbilt University Press, Nashville (2000)Google Scholar
  9. 9.
    Cox, D.: What is a toric variety? In: Topics in algebraic geometry and geometric modeling. Contemp. Math., vol. 334, pp. 203–223. Amer. Math. Soc., Providence (2003)CrossRefGoogle Scholar
  10. 10.
    DeRose, T., Goldman, R., Hagen, H., Mann, S.: Functional composition algorithms via blossoming. ACM Trans. on Graphics 12, 113–135 (1993)CrossRefMATHGoogle Scholar
  11. 11.
    Fulton, W.: Introduction to toric varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993); The William H. Roever Lectures in GeometryCrossRefMATHGoogle Scholar
  12. 12.
    Agresti, A.: Categorical Data Analysis. Wiley series in Probability and Mathematical Statistics. Wiley, New York (1990)MATHGoogle Scholar
  13. 13.
    Graf von Bothmer, H.C., Ranestad, K., Sottile, F.: Linear precision for toric surface patches, ArXiv:math/0806.3230 (2007)Google Scholar
  14. 14.
    Pachter, L., Sturmfels, B. (eds.): Algebraic statistics for computational biology. Cambridge University Press, New York (2005)MATHGoogle Scholar
  15. 15.
    Keller, O.: Ganze cremonatransformationen. Monatschr. Math. Phys. 47, 229–306 (1939)MathSciNetGoogle Scholar
  16. 16.
    Pinchuk, S.: A counterexample to the strong real Jacobian conjecture. Math. Z. 217(1), 1–4 (1994)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Dahmen, W., Micchelli, C.A.: Convexity of multivariate Bernstein polynomials and box spline surfaces. Studia Sci. Math. Hungar. 23(1-2), 265–287 (1988)MathSciNetMATHGoogle Scholar
  18. 18.
    Leroy, R.: Certificats de positivité et minimisation polynomiale dans la base de Bernstein multivariée. PhD thesis, Institut de Recherche Mathématique de Rennes (2008)Google Scholar
  19. 19.
    Sturmfels, B.: Gröbner bases and convex polytopes. American Mathematical Society, Providence (1996)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Gheorghe Craciun
    • 1
  • Luis David García-Puente
    • 2
  • Frank Sottile
    • 3
  1. 1.Department of Mathematics and Department of Biomolecular ChemistryUniversity of WisconsinMadisonUSA
  2. 2.Department of Mathematics and StatisticsSam Houston State UniversityHuntsvilleUSA
  3. 3.Department of MathematicsTexas A&M UniversityCollege StationUSA

Personalised recommendations