Some Geometrical Aspects of Control Points for Toric Patches

  • Gheorghe Craciun
  • Luis David García-Puente
  • Frank Sottile
Conference paper

DOI: 10.1007/978-3-642-11620-9_9

Part of the Lecture Notes in Computer Science book series (LNCS, volume 5862)
Cite this paper as:
Craciun G., García-Puente L.D., Sottile F. (2010) Some Geometrical Aspects of Control Points for Toric Patches. In: Dæhlen M., Floater M., Lyche T., Merrien JL., Mørken K., Schumaker L.L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2008. Lecture Notes in Computer Science, vol 5862. Springer, Berlin, Heidelberg

Abstract

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.

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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

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