Approximation And Interpolation Of NTP Curves

Normalized totally positive (NTP) parametric curves and surfaces constructed by polynomial NTP basis blending control points are one of basic tools for computer aid-ed geometric design and geometric modeling. This manuscript focuses the main attention on the approximation and interpolation problem of NTP curves and surfaces.1. Constraint approximation of NTP curves. For common NTP curves, Said-Bezier type generalized Ball curves and Delgado-Pena curve, we present an unified algorithm to approximate these curves by lower degree curves. By using the transformation relation be-tween NTP polynomial basis function and univariate Jacobi polynomial and the orthonor-mality of Jacobi polynomials, the approximation problem can be simplified to a least square problem, hence the control points of lower curve can be computed. This algorithm has sev-eral advantages, such as approximation error in the L2norm is minimal; high continuities at the endpoints of the two curves; multi-degree reduction once; explicit approximation expressions; error bound computed in advance and so on. The algorithm is simple and fast and hence will gain extensive application in data communication, data compression of CAD systems.2. Progressive iteration approximation (PIA) for NTP curves and surfaces. In the range of weighted PIA algorithm convergence, we give out the explicit exact PIA solutions for common NTP parametric polynomial curve and surface, and triangle Bezier surface. For the two NTP basis, Said-Bezier type generalized Ball basis and Delgado-Pena basis, the matrix solution of corresponding interpolation curves and surfaces are given based on explicit inverse matrix of Vandermonde matrix. This results avoid the tedious calculation of the inverse matrix and hence will gain extensive application in reverse engineering.3. Accurate computations with NTP-Vandermonde matrices and the application in interpolating and fitting given data points. Given l(l≥n) increasing nodes in the interval (0,1), the collocation matrix for degree n Said-Bezier type generalized Ball basis at these nodes:Said-Bezier-Vandermonde matrix is a strictly totally positive (STP) matrix. For this type NTP-Vandermonde matrix, we present the bidiagonal decomposition formula, and fast algorithm to compute the bidiagonal decomposition matrix. The algorithm has high accuracy and low complexity for many operations. For example, the bidiagonal decomposi-tion can be used to compute the linear system generated by Said-Bezier basis interpolating given data. The complexity can be reduces from (?)(n3) to (?)(n2). By some application such as solving Said-Bezier linear system, computing eigenvalues of Said-Bezier-Vandermonde matrix, and least square fitting data points, we demonstrate the correctness and high accu-racy of the algorithm.4. Chord-length reparametrization. We present a new reparametrization method to make the parameter of parameterized curve as close as possible to chord-length param-eter. The solution of this problem can be analytically reduced to finding the unique real root on (0,1) of a quadratic equation. The exact and explicit solutions of chord-length reparametrization are presented for Bezier curves and rational quadratic Bezier curves, whereas the numerical solutions are given by Composite-Simpson’s integral formula for higher degree rational Bezier curves.