【作者】 郝永霞；

【导师】 王仁宏； 李崇君；

【作者基本信息】 大连理工大学 ， 计算数学， 2013， 博士

【摘要】 现代的产品设计和制造过程11]的很多问题都可以转化为曲线曲面造型问题,例如汽车外形设计、飞机机身设计、服装的设计和加工以及建筑物的外形设计等.而曲线曲面设计也是当前计算机辅助几何设计的核心的研究内容.特别是在实际应用中,满足某种性质和功能的曲线曲面的设计更是尤其重要的.本文重点研究了一类在微分几何和计算几何中尤为重要的曲面-极小曲面,同时也给出了一种简单的曲线细分格式,并考虑将极小曲面与细分格式这两种曲面造型方法有效地结合起来.具体工作主要包括以下几个方面：1.多项式是计算几何辅助设计中曲线曲面的一种重要的表示形式,但是在CAGD领域中我们可以见到的参数多项式形式的极小曲面是非常少的,因此寻求参数形式的多项式极小曲面并将其引入到CAGD领域具有重要的意义.利用微分几何中的一个经典结论,我们给出了等温参数多项式极小曲面的一般形式,得到了其系数需要满足的充要条件,分析了它们具有的对称性质并给出了其共轭极小曲面的一般形式,同时也具体构造了6、7、8、9次的参数多项式极小曲面.2.Bezier曲线曲面是计算机图形学和计算机辅助几何设计的一种基本的表示工具,而quasi-Bezier表示是Bezier表示的一种推广,它所张成的函数空问不仅包含了多项式空间也包括了三角函数空间和双曲函数空间.利用Dirichlet能量来代替面积泛函,我们考虑了在quasi-Bezier表示下的Plateau问题,即Plateau-quasi-Bezier司题,这时的边界曲线不仅可以是多项式曲线也可以圆弧曲线和悬链线.同时,我们也讨论了调和与双调和的quasi-Bezier曲面.3.Plateau司题一直是极小曲面领域的一个重要问题.而在Plateau司题中,要求给定的曲线是曲面的整个边界,但在实际应用中我们获取的边界信息可能只是部分边界曲线.基于这一点,我们考虑了quasi-Plateau问题：在所有以给定的曲线为边界且定义在矩形域的参数曲面中,找一个而积最小的曲面,这里的‘给定的曲线’可以是曲面的整个边界亦可是部分边界.利用Dirichlet泛函来代替面积能量，并利用Ritz-Galerkin方法,这个问题转化为了一个简单的稀疏线性方程组的求解问题,并最终根据不同的边界条件归结为了四个算法.具体实例表明了我们的算法是简单有效的.4.在许多的曲面设计问题中，比如汽车车身、飞机外壳以及机器零部件设计,都要求构造的曲面在给定的边界处满足一定的连续性条件.鉴于此,我们来考虑C1和C2的quasi-Plateau问题,即要求极小曲而不仅以给定的曲线为边界曲线,同时在这些曲线处也要满足C1或C2的连续性.同样地利用Dirichlet能量和Ritz-Galerkin方法，我们得到了对应于不同边界条件的六种算法来求得该问题在不同层的逼近解.具体实例表明了这些算法是简单有效的.5.细分格式也是CAGD领域中曲线曲面的一种重要的表示方法.我们提出了一种形式简单的二进制六点的曲线细分格式,这种细分格式不仅形式简单,而且同时具有很多良好的几何性质,如多项式再生性、保凸性以及高阶连续性.6.基于细分格式与极小曲面在曲面造型中的重要性,我们考虑将两者结合起来,利用Dirichlet能量,通过极小曲面的思想来优化细分曲面的初始网格使得构造的细分极限曲面是满足给定的初始边界控制顶点的拟极小曲面.得到了一般细分格式下的内部控制顶点需要满足的条件,同时通过Loop细分格式作用于几个具体的实例说明了我们的方法.更多还原

【Abstract】 Many problems in the modern product design and manufacturing process can be translated into the problems of curve and surface modeling, such as auto appearance de-sign、design of airplane body、design and process of cloth、and the appearance design of buildings, and so on. The design of curve and surface is also the core content of Computer Aided Geometric Design. Especially in the real applications, the design of curve and surface meeting certain properties and functions is of particular importance. In this paper, our re-search focuses on the minimal surface, a kind of surface important in differential geometry and computational geometry, we also present a simple curve subdivision scheme, and con-sider to combine the research on minimal surface and subdivision scheme. The main work includes the following several aspects:1. Polynomial is an important representation for curve and surface in computational ge-ometry design. However, we can only find very few parametric polynomial minimal surface in CAGD, therefore searching for the parametric polynomial minimal surface and introducing them into CAGD are of great significance. Using a classic result in differential geometry, we present the general formula of parametric polynomial minimal surface, obtain the sufficient and necessary conditions for the corresponding coefficients, analyze their symmetric prop-erties, give the conjugate minimal surfaces, and illustrate concrete examples for polynomial minimal surface of degree6、7、8、9.2. Bezier curve and Bezier surface are basic representation tools in Computer Graph-ics and Computer Aided Geometric Design, while quasi-Bezier is a generalization of Bezier and the space it spans not only includes polynomial space also trigonometric space and hyperbolic space. Using Dirichlet energy, we consider the Plateau problem under the rep-resentation of quasi-Bezier, i.e., Plateau-quasi-Bezier problem. In this case, the boundary curve can not only be polynomial curve also circular curve and catenary. Moreover, we also discuss the harmonic and biharmonic quasi-Bezier surface.3. Plateau problem is always an important problem for minimal surface. In this problem, the given curve is required to be the whole bound of the solution surface, however in applications the boundary information we get maybe only part boundary curve. Based on this, we consider the quasi-Plateau problem:find the surface of minimal area among all the surfaces bounded by the given curve and defined on the rectangular domain, the ’given curve’here can be the whole boundary or part boundary. Replacing the area functional with Dirichlet energy, and using the Ritz-Galerkin method, the quasi-Plateau problem changes into solving a simple sparse linear equations, and finally is summarized as four algorithms according to different boundary conditions. Examples demonstrate that our algorithms are simple and effective.4. Many surface design problems, such as the design of car body、airplane hull and the machine parts, require the surface to satisfy certain continuity conditions at the given bound. Therefore, we consider the C1and C2quasi-Plateau problem, i.e., the surface of minimal area not only is bounded by the given curve, but satisfies the C1or C2continuity. Still using the Dirichlet energy and Ritz-Galerkin method, we obtain six algorithms corresponding to different boundary conditions to get the approximation surface in different levels. Examples demonstrate that these algorithms are simple and effective.5. Subdivision scheme is also an important design method for curve and surface in CAGD. We present a simple binary six point curve subdivision method, this scheme not only has simple form, but also has many good geometric properties at the same time, such as polynomial reproduction%convexity preserving and high continuity.6. Based on the importance of subdivision scheme and minimal surface, we consider to combine the both aspects to construct the quasi-minimal subdivision surface satisfying the given initial boundary control points. Using Dirichlet energy, we obtain the results on inner control points for general subdivision scheme, and illustrate our results by Loop subdivision scheme with several examples.更多还原