In the field of computer aided design

a new data fitting technique

the progressive iterative approximation (PIA)

has been proposed and attracts plenty of attention. By adjusting the control points iteratively

the PIA method provides a straightforward way to generate a sequences of curves/surfaces with better precision for data fitting. The curve (tensor product surface) has the PIA property as long as the bases are normalized completely positive and the corresponding collocation matrix is non-singular. In order to extend the scope of application of the PIA property

our paper focuses on the triangular surface and the non-totally positive collocation matrix. Furthermore

we assume that it may also possess the PIA property. The theory is based on certain conditions

which are essential for a basis to satisfy the PIA property. Given a set of triangular basis functions and its corresponding parametric values

we can obtain the collocation matrix of the triangular basis functions at the parametric values. Then

we get a new matrix

which is the result of the identity matrix subtracting the collocation matrix

and calculate the spectrum radius of the new matrix. If the value of the spectrum radius is less than 1

we call the triangular basis functions over a triangle domain having the PIA property. Given a collocation matrix

which is diagonally dominant or generalized diagonally

dominant and the elements of the matrix are positive number

then the real part of the eigenvalue of the collocation matrix is also a positive number. Our work proves that if the real part of the eigenvalue collocation matrix is a positive number

then the corresponding bases on triangular domain possess the PIA property (we call it as generalized PIA property). In the end

we build the relationship between diagonally dominant or generalized diagonally dominant matrix and the bases possess progressive iterative approximation property. If the collocation matrix is diagonally dominant or generalized diagonally dominant

the corresponding bases on triangular domain possess the PIA property (we call it as generalized PIA property). As we know

the Said-Ball bases have many good properties

such as the shape preserving property and the convex hull property. Since the Said-Ball bases are much better than Bernstein bases in recursive evaluation and degree elevation or reduction

it is worthwhile to develop the Said-Ball bases on triangle domain for free-form surface design. We take the Said-Ball bases as examples and find that the Said-Ball bases on triangular domain have the generalized PIA property. Numerical examples are given to demonstrate the correctness of our theory. The main work of our paper is that we extend the scope of PIA property to the generalized blending basis over a triangle domain. The basic technique is to give a progressive iterative scheme of the triangle Said-Ball surfaces. Giving some scattered data points to form an initial control mesh

the limit surface can interpolate these original points by constructing an iterative sequence of the triangular Said-Ball patches. Moreover

the numerical examples of the triangular Said-Ball basis of low degree with the uniform or non-uniform parameters are given. Furthermore

we can generalize this method to the study of different kinds of generalized Ball bases in triangular domain.