A New Interpolation Method by NTP Curves and Surfaces

In this paper, we research on constructing an explicit parametric curve to be taken as the limitation curve of Progressive Iteration Approximation (PIA) which can interpolate some scattered data points by using normalized totally positive (NTP) basis. We prove that when the data points and corresponding parameter list are given, the limitation curve has the unique explicit expression. There is a similar conclusion also for surface case. By specially choosing two kinds of NTP bases, Said-Bézier type generalized Ball (SBGB) basis and DP basis, based on the formula for the inverse matrix of Vandermonde matrix, we deduce an explicit matrix solutions of the limitation curve and surface corresponding to these NTP bases. Our results avoid the tedious calculation of the inverse matrix and hence will gain extensive application in reverse engineering. Keywords-Normalized Totally Positive basis; Said-Bézier type generalized Ball basis; DP basis; Progressive Iteration Approximation; Interpolation