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A Surface Modeling Method by Using \(C^2\) Piecewise Rational Spline Interpolation

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Abstract

A piecewise bivariate rational spline interpolation is presented only based on the values of the interpolated function. The interpolation has the following advantages: it is \(C^2\) continuous in the whole interpolating region; the interpolation function has a explicit rational mathematical representation, and can be represented by the basis functions; more important, since there are three free parameters \(\alpha _{i,j}\), \(\alpha _{i,j+1}\) and \(\beta _{i,j}\) in this interpolant, the shape of the interpolating surfaces can be modified by selecting suitable parameters for the unchanged interpolating data. Also, the values of the interpolation function are bounded no matter what the parameters might be, and the approximation expressions of the interpolant are derived.

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Acknowledgments

This research was supported by the National Nature Science Foundation of China (No.61070096) and the Natural Science Foundation of Shandong Province (No.ZR2012FL05).

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Qinghua Sun, Fangxun Bao & Qi Duan

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  1. Qinghua Sun
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  2. Fangxun Bao
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  3. Qi Duan
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Correspondence to Fangxun Bao.

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Sun, Q., Bao, F. & Duan, Q. A Surface Modeling Method by Using \(C^2\) Piecewise Rational Spline Interpolation. J Math Imaging Vis 53, 12–20 (2015). https://doi.org/10.1007/s10851-014-0543-y

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  • DOI: https://doi.org/10.1007/s10851-014-0543-y

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