A Family Of Bivariate Of Rational Bernstein Operators

In1912, S.N.Bernstein presented the Bernstein operators and Bernstein basis functions creatively when he proved Weierstrass approximation theorem. In the100years that followed, Bernstein basis functions are widely used in approximation theory, numerical calculation and computational geometry. After Bezier and de Casteljau independently used them into design of automobile shape and gave the famous B6zier curves. Their theory research and applied re-search attracted tremendous interests and got quikly development. Rational Bernstein operators are forms of generalization of Bernstein operators, and B6zier curves can express cone curves uniformly. Classical rational Bernstein operators inherit the property of Bernstein operators and are widely used in approximation and geometric modeling, but they can not reproduce linear polynomials. For overcoming this shorcoming, Pitual and Sablonniere presented a new family of univariate rational Bernstein operators and analyzed their properties.In this paper, we generalize their method to construct a new family of bivariate rational Bernstein operators over triangles and rectangular fields. The operators are positive and exactly reproduce linear polynomials. Finally, we give the first derivatives and prove the convergence of the presented operators.In the first chapter, we introduce the concept of Bernstein operators an rational Bernstein operators and present some properties.In the second chapter, we construct rational Bernstein operators on over triangles, prove their reproduction of linear polynomials, first derivatives and convergence.In the third chapter, we construct rational Bernstein operators on the rectangular fields, prove their reproduction of linear polynomials, derivatives and convergence.