Expo-Rational B-Splines in Geometric Modeling - Methods for Computer Aided Geometric Design

Abstract

Computer aided geometric design is the design of geometric shapes, such as curves and surfaces, by using computer technology. The applications range from product design and computer aided manufacturing to virtual worlds and computer games. Some of the most common representations of curves and surfaces for this purpose are polynomial patches of Bézier type and piecewise polynomials on B-spline form. Real world models are sometimes represented by discrete data, scattered points or polygonal meshes. This thesis considers the use of expo-rational B-splines, a blending type spline construction where the coefficients are local functions that are blended together by infinitely smooth basis functions, as an alternative for the representation of discrete and continuous data. Various applied modeling and reconstruction problems are investigated. A generalization of Bernstein factor matrices, suitable for computation of multivariate Bernstein polynomials, is presented. Results regarding the structure of the factors are given, and the factorization is related to mixed directional derivatives of a B-form, and to the de Casteljau algorithm for evaluating curves and surfaces on Bernstein-Bézier form. Knot insertion is a fundamental operation for splines. It is used to express a spline in a more refined or flexible way without changing the shape of the spline. A knot insertion rule for ERBS curves, where new local functions are computed according to the inserted knot, is presented.