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Polynomial Accelerated Iterative Approximation for Higher Order and Rational Bézier Curves

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Abstract

Two accelerated iterative methods for curves approximation are presented in this paper. These presented methods are used to reduce the degree of Bézier curves and approximate rational Bézier curves by polynomials. By employing the preconditioned progressive iterative approximation (PPIA), we approximate the points sampled from target curves, and generate polynomial approximations. The equi-parametric and adaptive sampling methods are introduced. Both of them are well performed in degree reduction of Bézier curves and polynomial approximation of rational Bézier curves. Due to the effectiveness of the preconditioned technique, the accuracy and efficiency of approximation are improved significantly. More importantly, we can obtain the approximation within an user-defined error bound. Numerical examples demonstrate the effectiveness of our methods.

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Acknowledgements

This research is supported by National Natural Science Foundation of China (Grant No. 11771453), Natural Science Foundation of Hunan Province (Grant No. 2020JJ5267), Scientific Research Funds of Hunan Provincial Education Department (Grant No. 18C0877 and 19B301) and Open Research Fund Program of Data Recovery Key Laboratory of Sichuan Province (Grant No. DRN2003).

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Authors and Affiliations

  1. School of Mathematics and Finance, Hunan University of Humanities, Science and Technology, Loudi, 417000, People’s Republic of China

    Chengzhi Liu & Lian Yang

  2. Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang, 641100, People’s Republic of China

    Li Zhang

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  1. Chengzhi Liu
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  2. Lian Yang
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  3. Li Zhang
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Correspondence to Li Zhang.

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Liu, C., Yang, L. & Zhang, L. Polynomial Accelerated Iterative Approximation for Higher Order and Rational Bézier Curves. Results Math 76, 138 (2021). https://doi.org/10.1007/s00025-021-01453-y

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