| To begin with, we briefly retrospect the birth and the evolution of the minimal surface problem (Plateau Problem) with prescribed border, and in addition we comprehensively introduce some important methods, for example, Dirichlet functional method and discrete Mask method and so on, of recent researches on Bezier minimal surface modeling in the field of CAGD. These methods are based on the following conclusion that for isothermal regular parametric surface, it is a minimal surface if and only if this is a harmonic surface. Although these studies have some good results, Bezier surface can not be accurately but approximatively expressed for quadric in addition to paraboloid. It is well-known that rational Bezier surface can be accurately expressed for not only paraboloid but also for quadric. Furthermore, we point out that it is necessary and reasonable for us to study Plateau-Rational-Bezier surface modeling surrounded by the closed curve problem. Based on the idea of "dividing and conquering", we apply the method of finite difference method to study the problem of rational Bezier minimal surface modeling. The most important contributions and innovations of this thesis can be summed up as follows:1. We successfully attempt to solve the rational Bezier surfaces modeling problem, which make it possible for us to study arbitrary harmonic rational Bezier surfaces modeling. 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 extend the geometric design on the surface of Bezier harmonic to the rational form. Owing to the complicated form of rational Bezier surfaces, we don't directly apply the solution to harmonic Bezier surface modeling, but we can adopt the dividing and conquering approach. We transform the problem of rational Bezier surfaces modeling into solving linear equations. This method can be applied to arbitrary order of the rational Bezier surfaces modeling problem, which is the same with low-level case by solving a linear equations.3. We implement the algorithm of solving the harmonic rational Bezier surface for given control borders and verify this algorithm by several examples in order to explain that a class of rational Bezier harmonic surface,obtained by finite difference method, is the closest to the rational Bezier minimal surface. On the other hand, in order to objectively analysis and compare the surfaces, we give their absolute mean curvature maps. |
