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Complex Bézier Curves and the Geometry of Polynomials

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Curves and Surfaces (Curves and Surfaces 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6920))

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Abstract

In this paper, we study the shape of the control polygon of a complex Bézier curve over a complex interval. We show that the location of the complex roots of the polynomial dictates geometrical constraints on the shape of the control polygon. Along the work, new proofs and generalizations of the Walsh coincidence Theorem and the Grace Theorem are given. Applications of the geometry of the control polygon of complex polynomials to Bernstein type inequalities are discussed.

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Author information

Authors and Affiliations

  1. The Center of Advanced Medical Engineering and Informatics, Osaka University, Japan

    Rachid Ait-Haddou & Taishin Nomura

  2. Department of Mechanical Science and Bioengineering, Graduate School of Engineering Science, Osaka University, Japan

    Rachid Ait-Haddou & Taishin Nomura

Authors
  1. Rachid Ait-Haddou
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  2. Taishin Nomura
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Editor information

Editors and Affiliations

  1. INRIA Sophia-Antipolis, 2004 Route des Lucioles, BP 93, 06902, Sophia Antipolis, France

    Jean-Daniel Boissonnat

  2. Laboratoire LJK, University Joseph Fourier, BP 53, 38420, Grenoble Cedex 9, France

    Patrick Chenin

  3. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, Place Jussieu, 75005, Paris, France

    Albert Cohen

  4. Laboratoire de Mathématiques, INSA Rouen, Avenue de l’Université, 76800, St. Etienne du Rouvray, France

    Christian Gout

  5. University of Oslo, DMA, Blindern, P.O.Box 1053, 0316, Oslo, Norway

    Tom Lyche

  6. Laboratoire LJK, Université Joseph Fourier, BP 53, 38041, Grenoble Cedex 9, France

    Marie-Laurence Mazure

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

    Larry Schumaker

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© 2012 Springer-Verlag Berlin Heidelberg

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Ait-Haddou, R., Nomura, T. (2012). Complex Bézier Curves and the Geometry of Polynomials. In: Boissonnat, JD., et al. Curves and Surfaces. Curves and Surfaces 2010. Lecture Notes in Computer Science, vol 6920. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27413-8_3

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  • DOI: https://doi.org/10.1007/978-3-642-27413-8_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-27412-1

  • Online ISBN: 978-3-642-27413-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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