Bivariate S-λ bases and S-λ surface patches

In this paper two kinds of bivariate S-λ basis functions, tensor product S-λ basis functions and triangular S-λ basis functions, are constructed by means of the technique of generating functions and transformation factors. These two kinds of bivariate S-λ basis functions have lots of important properties, such as non-negativity, partition of unity, linear independence and so on. The framework of the tensor product S-λ basis functions provides a unified scheme for dealing with several kinds of tensor product basis functions, such as the tensor product Bernstein basis functions, the tensor product Poisson basis functions and some other new tensor product basis functions. The framework of the triangular S-λ surface basis functions includes the triangular Bernstein basis functions, the rational triangular Bernstein basis functions and some other new triangular basis functions. Moreover, the corresponding two kinds of S-λ surfaces are constructed by means of these two kinds of bivariate basis functions, respectively. These two kinds of S-λ surface patches have the important properties of surface modeling, such as affine invariance, convex hull property and so on. We construct two kinds of bivariate S-λ bases-tensor product S-λ bases and triangular S-λ bases.We show that the frameworks of these two kinds of S-λ bases include the classical Bezier type bases.We construct two kinds of S-λ surface patches by means of the corresponding S-λ bases.We obtain the important properties of these two kinds of bivariate S-λ bases and the corresponding S-λ surface patches.