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三角域上双变量Chebyshev 多项式及其与Bernstein 基的转换

  

  • 出版日期:2013-12-31 发布日期:2015-06-19

Bivariate Chebyshev Polynomials and Transformation of Chebyshev-Bernstein Basis on Triangular Domains

  • Online:2013-12-31 Published:2015-06-19

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摘要： 为了更好的解决三角域上的Bézier 曲面在CAGD 中的最佳一致逼近问题，
构造出了三角域上的双变量Chebyshev 正交多项式，研究了与单变量Chebyshev 多项式相类
似的性质，并且给出了三角域上双变量Chebyshev 基和Bernstein 基的相互转换矩阵。通过
实例比较双变量Chebyshev 多项式与双变量Bernstein 多项式以及双变量Jacobi 多项式的最
小零偏差的大小，阐述了双变量Chebyshev 多项式的最小零偏差性。

关键词: 三角域, Bernstein 基, Chebyshev 多项式

Abstract: For solving least squares approximation problem of Bézier surface effectively and
simply on triangular domains in CAGD, we present a polynomial representation, bivariate
Chebyshev polynomials, adapted to a triangular domain, with properties similar to the univariate
Chebyshev form．We convert and compare this representation to the Bernstein-Bézier and Jacobi
representations．We also give some examples to illustrate that the deviation of the bivariate
Chebyshev polynomials compared with zero is the least than of the bivariate Bernstein
polynomials and bivariate Jacobi polynomials.

Key words: triangular domains, Bernstein basis, Chebyshev polynomial