Mathematical Methods for Curves and Surfaces

Volume 5862 of the series Lecture Notes in Computer Science pp 111-135

Some Geometrical Aspects of Control Points for Toric Patches

  • Gheorghe CraciunAffiliated withDepartment of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin
  • , Luis David García-PuenteAffiliated withDepartment of Mathematics and Statistics, Sam Houston State University
  • , Frank SottileAffiliated withDepartment of Mathematics, Texas A&M University

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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.