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Some Geometrical Aspects of Control Points for Toric Patches

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Mathematical Methods for Curves and Surfaces (MMCS 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5862))

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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Author information

Authors and Affiliations

  1. Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin, Madison, WI, 53706, USA

    Gheorghe Craciun

  2. Department of Mathematics and Statistics, Sam Houston State University, Huntsville, TX, 77341, USA

    Luis David García-Puente

  3. Department of Mathematics, Texas A&M University, College Station, TX, 77843, USA

    Frank Sottile

Authors
  1. Gheorghe Craciun
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  2. Luis David García-Puente
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  3. Frank Sottile
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Editor information

Editors and Affiliations

  1. Department of Informatics, University of Oslo, Norway

    Morten Dæhlen  & Tom Lyche  & 

  2. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Norway

    Michael Floater

  3. INSA de Rennes, Rennes, France

    Jean-Louis Merrien

  4. Department of Informatics and Centre of Mathematics for Applications, University of Oslo, Oslo, Norway

    Knut Mørken

  5. Department of Mathematics, Vanderbilt University, 37240, Nashville, TN, USA

    Larry L. Schumaker

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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. https://doi.org/10.1007/978-3-642-11620-9_9

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  • DOI: https://doi.org/10.1007/978-3-642-11620-9_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-11619-3

  • Online ISBN: 978-3-642-11620-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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