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An approach to geometric interpolation by Pythagorean-hodograph curves

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Abstract

The problem of geometric interpolation by Pythagorean-hodograph (PH) curves of general degree n is studied independently of the dimension d ≥ 2. In contrast to classical approaches, where special structures that depend on the dimension are considered (complex numbers, quaternions, etc.), the basic algebraic definition of a PH property together with geometric interpolation conditions is used. The analysis of the resulting system of nonlinear equations exploits techniques such as the cylindrical algebraic decomposition and relies heavily on a computer algebra system. The nonlinear equations are written entirely in terms of geometric data parameters and are independent of the dimension. The analysis of the boundary regions, construction of solutions for particular data and homotopy theory are used to establish the existence and (in some cases) the number of admissible solutions. The general approach is applied to the cubic Hermite and Lagrange type of interpolation. Some known results are extended and numerical examples provided.

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Authors and Affiliations

  1. FMF and IMFM, University of Ljubljana, Jadranska 19, Ljubljana, Slovenia

    Gašper Jaklič, Jernej Kozak, Marjeta Krajnc & Emil Žagar

  2. FAMNIT and PINT, University of Primorska, Glagoljaška 8, Koper, Slovenia

    Vito Vitrih

Authors
  1. Gašper Jaklič
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  2. Jernej Kozak
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  3. Marjeta Krajnc
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  4. Vito Vitrih
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  5. Emil Žagar
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Corresponding author

Correspondence to Emil Žagar.

Additional information

Communicated by R. T. Farouki.

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Jaklič, G., Kozak, J., Krajnc, M. et al. An approach to geometric interpolation by Pythagorean-hodograph curves. Adv Comput Math 37, 123–150 (2012). https://doi.org/10.1007/s10444-011-9209-0

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  • DOI: https://doi.org/10.1007/s10444-011-9209-0

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