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G 2 Hermite Interpolation with Curves Represented by Multi-valued Trigonometric Support Functions

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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

It was recently proved in [27] that all rational hypocycloids and epicycloids are Pythagorean hodograph curves, i.e., rational curves with rational offsets. In this paper, we extend the discussion to a more general class of curves represented by trigonometric polynomial support functions. We show that these curves are offsets to translated convolutions of scaled and rotated hypocycloids and epicycloids. Using this result, we formulate a new and very simple G 2 Hermite interpolation algorithm based on solving a small system of linear equations. The efficiency of the designed method is then presented on several examples. In particular, we show how to approximate general trochoids, which, as we prove, are not Pythagorean hodograph curves in general.

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Author information

Authors and Affiliations

  1. Faculty of Applied Sciences, Department of Mathematics, University of West Bohemia, Univerzitní 8, 301 00, Plzeň, Czech Republic

    Bohumír Bastl, Miroslav Lávička & Zbyněk Šír

Authors
  1. Bohumír Bastl
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  2. Miroslav Lávička
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  3. Zbyněk Šír
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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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Bastl, B., Lávička, M., Šír, Z. (2012). G 2 Hermite Interpolation with Curves Represented by Multi-valued Trigonometric Support Functions. 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_9

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  • DOI: https://doi.org/10.1007/978-3-642-27413-8_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-27412-1

  • Online ISBN: 978-3-642-27413-8

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