Design Of Rational Bézier Minimal Surface: An Extension Of Dirichlet Approach

To begin with, we briefly retrospect the birth and the evolution of the minimal surface problem (Plateau Problem), and we comprehensively introduce some important methods, useful significances, and essential limitations of recent researches on Bezier minimal surface modeling and B-spline minimal surface modeling in the field of CAGD. Furthermore, we point out that it is necessary and reasonable for us to study rational Bezier minimal surface modeling. And then we introduce the general idea and the specific approach of the finite element method. We also retrospect some relevant basic conceptions and important theorems. Afterwards, we expand a series of in-depth discussions and researches on the rational Bezier minimal surface modeling, including its general statement, its theoretical mechanism, and its algorithmic design by using finite element tool. Finally, we validate this new kind of theory and algorithm by giving a large variety of examples of the double two degree and double three degree rational Bezier surfaces. The most important contributions and innovations of this thesis can be summed up as follows:1. It is the first time that the rational Bezier surfaces modeling problem has been proposed and approached successfully, which make it possible for us to research on minimal surface modeling of general conicoid. Hence, it will promote the field of CAD to use the NURBS system to draw and compute minimal surface, which will have a profound influence on some fields of engineering, such as construction and mechanism.2. We make a study on whether it is feasible and potential for us to extend Dirichlet energy function method into the field of rational Bezier surface. We use the idea of isothermal parameter representation to prove a convergence theorem, which serves as an important theoretical base of studying the problem of rational Bezier surface. This theorem shows that according to the given boundary condition, when the degree of the approximate surfaces is high enough, the surface solved by using the method of Dirichlet is nearly the true rational Bezier surface we want to obtain.3. We also make a study on the specific algorithms of an extension of Dirichlet energyfunction method to the field of rational Bezier surface. Because of the complex form of the rational Bezier surface, we adopt a strategy completely different from the Bezier surface. First, we extend the general finite element method from the function surface to the parameter function. Second, we bring this kind of method into the design of rational Bezier minimal surface. Third, we transform the problem of Dirichlet energy function into the problem of minimizing a discrete objective function. Finally, the theoretical conclusion and numerical experiment demonstrate that the problem of rational Bezier minimal surface can finally be transformed into an optimal problem on how to minimize a nonlinear function boasted of a limited number of variables under a linear condition, which has been successfully solved in this paper.