In the 4th century, Pythagorizing mathematicians made a significant advance in the theory of irrational numbers, such as the-square-root-of-n (√n), n being any rational number, when they developed a method for finding progressive approximations to √2 by forming sets of so-called diagonal numbers.
But the quest for the perfect, in progressive approximations, is what separates science from fiction, opinion from evidence and guesswork from clinical medicine.
This paper presents the weighted progressive iteration approximation (WPIA) property for the triangular Bernstein basis over a triangle domain with uniform parameters, which is extended from the PIA property for triangular Bernstein basis proposed by Chen and Wang in [J. Chen, G.J. Wang, Progressive-iterative approximation for triangular Bézier surfaces, Computer-Aided Design 43 (2011) 889 895].
Recently, for the sake of fitting scattered data points, an important method based on the PIA (progressive iterative approximation) property of the univariate NTP (normalized totally positive) bases has been effectively adopted.
The progressive-iterative approximation brings more flexibility for shape controlling in data fitting.
Just by adjusting the control points iteratively, progressive-iterative approximation presents an intuitive and straightforward way to fit data points.
Progressive-iteration approximation (PIA) is a new data fitting technique developed recently for blending curves and surfaces.
Moreover, a special adjustment manner is designed to make the local progressive-iterative approximation format is convergent for a generic blending patch with normalized totally positive basis.
The local progressive-iterative approximation format adjusts only a part of the control points of a blending curve or patch, and the limit curve or patch interpolates the corresponding data points.
