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Extremal Scattered Data Interpolation in \(\mathbb {R}^3\) Using Triangular Bézier Surfaces

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Numerical Methods and Applications (NMA 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8962))

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Abstract

We consider the problem of extremal scattered data interpolation in \(\mathbb {R}^3\). Using our previous work on minimum \(L_2\)-norm interpolation curve networks, we construct a bivariate interpolant \(F\) with the following properties:

  1. (i)

    \(F\) is \(G^1\)-continuous,

  2. (ii)

    \(F\) consists of triangular Bézier surfaces,

  3. (iii)

    each Bézier surface satisfies the tetra-harmonic equation \(\varDelta ^4 F=0\). Hence \(F\) is an extremum to the corresponding energy functional.

We also discuss the case of convex scattered data in \(\mathbb {R}^3\).

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Acknowledgments

This work was partially supported by the Bulgarian National Science Fund under Grant No. DFNI-T01/0001.

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Authors and Affiliations

  1. Faculty of Mathematics and Informatics, Sofia University, “St. Kliment Ohridski” Blvd. James Bourchier 5, 1164, Sofia, Bulgaria

    Krassimira Vlachkova

Authors
  1. Krassimira Vlachkova
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Correspondence to Krassimira Vlachkova .

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Editors and Affiliations

  1. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Dimov

  2. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Stefka Fidanova

  3. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Lirkov

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Vlachkova, K. (2015). Extremal Scattered Data Interpolation in \(\mathbb {R}^3\) Using Triangular Bézier Surfaces. In: Dimov, I., Fidanova, S., Lirkov, I. (eds) Numerical Methods and Applications. NMA 2014. Lecture Notes in Computer Science(), vol 8962. Springer, Cham. https://doi.org/10.1007/978-3-319-15585-2_34

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  • DOI: https://doi.org/10.1007/978-3-319-15585-2_34

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-15584-5

  • Online ISBN: 978-3-319-15585-2

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