三角域上双变量Chebyshev多项式及其与Bernstein基的转换

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

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。

For solving least squares approximation problem of B6zier 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-B6zier 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.