The Least Square Progressive Iterative Approximation Property Of Low Degree Non-uniform Triangular Bézier Surfaces

Progressive-iterative approximation(PIA)is an intuitive and effective method for data fitting.Classical PIA method requires that the number of control points is equal to the number of the data points.It is not suitable for fitting mass data.In order to improve the classical PIA method,the algorithm for fitting data points with triangular surfaces based on PIA method is studied,especially for the low-degree case usually used in practice.Least-squares fitting is one of the most commonly used methods in mathematical tools.Data in 3-D corresponds to the 3-D mesh,corresponding to the triangular surfaces.Therefore,it is very necessary to analyze the LSPIA properties for triangular B-B surfaces and rational triangular B-B surfaces.It has been proved that the quadratic,cubic and quartic non-uniform triangular Bézier surfaces have the property of progressive-iterative approximation for least square fitting(LSPIA).And the limit of the sequence of triangular Bézier surfaces obtained by iteration is just the least square fitting of the data points.Meanwhile,a method is provided to show how to choose the value of the weight so that the iteration has the fastest convergence speed.Some numerical examples are presented to validate the effectiveness of the LSPIA method.At the same time,we also prove that the quadratic,cubic and quartic non-uniform rational triangular Bézier surfaces have the property of progressive-iterative approximation for least square fitting(LSPIA).And the limit of the sequence of rational triangular Bézier surfaces obtained by iteration is just the least square fitting of the data points.Then,an example is presented to verify the validity of the LSPIA method for non-uniform rational triangular Bézier surfaces.