Orthogonal Polynomials And In Bernstein Form On Triangular Domains

As we know, the orthogonal polynomials were widely applied in mathematics,physics, and other branches of science and technology. In recent years, orthogonalpolynomials have made rapid progress under the leadership of mathematicians likeRichard Askey and George Andrews etc. For solving least squares approximationproblem of Bézier surface effectively and simply in CAGD and CAM, someorthogonal polynomials were constructed, such as univariate Legendre polynomial,univariate Jacobi polynomial, and univariate Chebyshev polynomial, and alsobivariate Legendre polynomial, bivariate Jacobi polynomial on triangular domains.Then, the matrices of transformation of the orthogonal polynomials into Bernsteinbasis form were derived to make up the deficiencies of the Bernstein polynomials.In this thesis, it mainly introduces the univariate and bivariate orthogonalpolynomials, and their applications in CAGD. So, we present bivariate Chebyshevpolynomials on a triangular domain, whose properties are similar to the univariateChebyshev polynomial. We convert and compare this representation to theBernstein-Bézier and Jacobi representations. We also give some examples toillustrate that the deviation of the bivariate Chebyshev polynomials compared withzero is the least than of the bivariate Bernstein polynomials and bivariate Jacobipolynomials. The specific arrangements were organized as follows:In chapter1, it briefly introduces the developing of the orthogonalpolynomials, including their results, application in CAGD, and the main work to bedone in the paper;In chapter2, the expressions and properties of the orthogonal polynomialswere briefly introduced. Also, the conversion matrices were given and theirapplication in CAGD;In chapter3, it introduces the expression and properties of bivariateorthogonal polynomials on triangular domain. Then, the conversion matrix is givenand their application in CAGD;In chapter4, the expression of the bivariate Chebyshev orthogonal polynomialon triangular domain is derived, and it was given out their properties. Also, thematrices of transformation of the bivariate Bernstein basis form into the Chebyshevbasis of the same degree are derived;In chapter5, it concludes the thesis, and some valuable suggestions were made to continue to explore and further research issues.