Researches On Some Problems To Curves And Surfaces In Local Progressive Iterative Approximation

Curves and surfaces are main topics for research in the field of Computer AidedGeometric Design (CAGD for short). Among these curves and surfaces, the researchesabout Bézier curves and surfaces with shape parameter are very mature. Shapeparameters play adjustment roles for the shapes of curves and surfaces. The main idea isBézier curves and surfaces can be adjusted by changing the shape parameters withoutmoving their initial control points or control meshes. Local progressive iterativeapproximation (LPIA for short) method is a new kind of fitting and approximationtechnique in CAGD and Reverse Engineering. It also has some advantages inprocessing scattered data. The main idea is we can get a sequence of curves or surfacesby only adjusting a set of the initial control points iteratively, the limit curve and surfacewill interpolate the initial control points. We can use this property on normalized totallypositive (NTP for short) bases because they satisfy the LPIA property. In thisdissertation, we have made some researches on these two topics, and provided results asfollows:1. LPIA of Bézier curves with a shape parameter: Given the initial data points, byonly adjusting a set of the data points iteratively, we can get a sequence of curves.With the increasing times of iterations, a sequence of Bézier curves with a shapeparameter can be generated with finer and finer precision for data point fitting. Andthe limit curve can interpolate the initial data points. The convergence rate of LPIAincreases with the increasement of the shape parameter values.2. An iterative method for polynomial approximation of cubic uniform rationalB-spline curves: the idea and algorithm of approximating rational curves withpolynomial forms is generalized to the case of low-degree uniform rationalB-spline curves. We present a new iteration method for approximating cubicuniform rational B-spline curves. The main idea is sampled control points from thegiven rational B-spline curve as new control points, generating an initial B-splineapproximation curve by progressive iterative approximation. The limit curveconverges to the given rational curve.3. Weighted local progressive iterative approximation (WLPIA for short) fortensor-product Bézier surfaces: we first prove that tensor-product Bézier surfacesalso satisfy the LPIA property. In order to accelerate convergence rate, we present a WLPIA method for tensor-product surfaces. We also prove that we can improvethe WLPIA convergence rate by changing the weight.