Design Of Minimal Mesh Surface Surrounded Given Spacial Closed Broken Line

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 Bézier 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 design of minimal mesh surface surrounded given spacial closed broken line.And then we introduce the general idea and the specific Dirichlet approach of the plateau-Bézier problem.We also retrospect the modeling and property of the cubicα-B-spline curve that interpolates given data.Afterwards,we expand a series of in-depth discussions and researches on the minimal mesh surface modeling surrounded given spacial closed broken line,including its general statement,its theoretical mechanism,also bring forward the idea of using the Dirichlet solution of the Plateau-B-spline problem.Then,we get the design of the minimal mesh surface modeling from the view of geometry approach.Finally,we validate this new kind of theory and algorithm by giving a large variety of examples of the double three degree B-spline surfaces.The most important contributions and innovations of this thesis can be summed up as follows:1.It is the first time that the minimal mesh surfaces modeling problem has been proposed and approached successfully in CAGD,which will have a profound influence on some fields of engineering,such as construction and mechanism.2.We extend Dirichlet energy function method into the field of double three degree B-spline surfaces.We use this idea as the foundation to solve the minimal mesh surface surrounded given soacial broken line through Dirichlet energy function method.We induce the area difference of the triangle surface patch and the interpolated triangle plane patch in relation to the parameter field and the control points in detail,and prove when the area of the parameter field approaches zero, with proper blending factorα,the area difference also approaches zero.Then we can conclude that when the number of points on surface approaches infinite, the sum area of the triangle mesh approaches the area of the minimal surface plateau-B-spline.3.We make a study on the specific algorithms of an extension of Dirichlet energy function method to the field of discrete minimal surface.The basic thought is to use the way of generalized inverse matrix to solve the overdetermined linear equations,and the way to normalize non-uniformα-B-spline curve with blending factorα.Then we introduce them into the design of the minimal mesh surface,and transform the problem of continuous Dirichlet energy function into the problem of minimizing a discrete objective function.The theoretical conclusion and numerical experiment demonstrate that the problem of minimal discrete surface has been successfully solved in this paper.