【作者】 徐岗；

【导师】 汪国昭；

【作者基本信息】 浙江大学 ， 应用数学， 2008， 博士

【摘要】 极小曲面问题的研究是微分几何领域巾最活跃的分支之一。由于极小曲面在工程领域有着广泛应用,因此将极小曲面引入CAGD(计算机辅助几何设计)领域具有重要意义。本文围绕着CAGD中的极小曲面造型这一主题,就以下几个方面给出了研究成果:(1)极小曲面的控制网格表示与构造控制网格是CAD系统中进行交互设计的重要工具。首先构造了空间{sint,cost,sinht,cosht,1,t,t~2,…,t~n}中的拟非均匀B样条基及相应的曲线曲面模型。由于节点情况比较复杂,我们利用行列式巧妙地把所有的节点情况统一到一个表达式之中。接着,基于该空间中的混合曲线曲面模型,给出了广义螺旋面的控制网格的几何构造方法。最后,利用混合曲线曲面模型给出了平面曲率线极小曲面的控制网格表示。这些结果为将这些极小曲面模型引入CAGD造型系统提供了一个有力工具,从而可以利用细分算法生成这些极小曲面,并在CAGD系统中与其他的曲而统一处理。(2)高次参数多项式极小曲面的挖掘与性质参数多项式形式是CAD系统中曲线曲面的标准形式。根据微分几何中的一个经典结论,给出了五次和六次调和参数多项式曲面为极小曲面的充分条件。基于这些条件,构造出了几类新的极小曲面,并深入研究了它们所具有的几何性质,如对称性、自交性、包含直线等。我们还进一步发现了两类新的共轭极小曲面,并实现了共轭极小曲面之间的动态变形。这些新的极小曲面不仅丰富了极小曲面的种类,而且可直接为当前的CAD系统所采用;最后,提出了关于任意次参数多项式极小曲面的存在性及其性质的三个猜想。(3)基于线性PDE的极小曲面逼近造型基于调和曲面与极小曲面的密切关系,首先利用方向导数研究了三角域上的调和B-B曲面的性质,给出了三角域上的B-B曲面为调和曲面的充要条件,并且证明了任何一个三角域上的B-B曲面调和面的控制网格均由它的第一层和第二层控制顶点完全决定。接着,研究了一类更广泛的负高斯曲率曲面的构造方法,得到了与调和曲面情形相类似的结果。最后,为了实现PDE曲面造型技术与CAD造型系统的数据交换,基于约束优化的思想,给出了PDE曲面的Bézier曲面逼近算法,并利用张量积Bézier曲面的细分性质对该算法进行了优化。(4)基于平均曲率平方能量的Plateau-Bézier问题求解Plateau-Bézier问题是Plateau问题在Bézier曲面形式下的推广,它以内部控制顶点为求解目标。我们从平均曲率为零出发,基于平均曲率平方能量来解决Plateau-Bézier问题。主要研究了矩形域上的张量积Bézier曲面和三角域上的B-B曲面两种情形,分别给出了其内部控制顶点所要满足的充要条件,并与基于Dirichlet能量的方法进行了比较,发现两者各有千秋。可以证明,若在给定的边界条件下,其所对应的极小曲面为等温参数多项式极小曲面,则按照本文方法所得到的Bézier曲面便是该等温参数多项式极小曲面。(5)极小曲面造型中的边界优化问题如何选择边界曲线以满足造型要求,是极小曲面造型中的一个重要问题。首先基于拉伸能量、弯曲能量和jerk能量,研究如下问题:给定部分控制顶点,如何构造其余的控制顶点使得所产生的Bézier曲线具有极小能量。推导出了待定控制顶点所要满足的充要条件。通过曲率图和曲率梳,对三种能量极小Bézier曲线进行了比较,并发现了四次能量最小Bézier曲线的共线性质。最后,对三次均匀B样条曲线进行了两个方面的扩展:首先,构造出了五次和六次调配函数,并以它们为基础定义了两种带形状参数的样条曲线,提高了曲线与其控制多边形的逼近程度;其次,构造出了两类三次和四次调配函数,并以此为基础定义了两类带局部形状参数的样条曲线,其优点是在保持曲线连续性不变的同时可以通过改变局部形状参数的取值对曲线进行局部调整。(6)极小曲面在建筑设计中的应用建筑设计是极小曲面的应用大户。首先基于曲面裁剪技术,讨论了本文所提出的几类新的极小曲面和三角域上的调和B-B曲面在张拉膜结构设计中的应用;最后,提出了旋转圆锥网格的概念,给出了它们的简单构造方法,并将其用于玻璃/钢结构设计。更多还原

【Abstract】 The research on minimal surface problem is one of the most active fields of interest in differential geometry.As minimal surfaces have extensive applications in engineering,it is very meaningful to introduce minimal surface into the field of CAGD(Computer Aided Geometric Design).In this thesis,some creative contributions on the topic of minimal surface modeling are given as follows.(1) Representation and construction of control mesh of minimal surfacesControl mesh is an important tool for interactive design in CAD systems.We first construct the quasi non-uniform B-spline basis and the corresponding curve and surface models in the space spanned by {sin t,cos t,sinh t,cosh t,1,t,t~2,…,t~n}. Since the knot cases are complex,we unify all the knot cases into a formula using determinate technology.Based on the new hybrid curve and surface model,the geometric construction of control mesh of the generalized helicoid is proposed. Finally,we present the control mesh representation of minimal surfaces with planar lines of curvature.These results provide an efficient tool for introducing these minimal surfaces into CAGD modeling systems,then we can produce these minimal surfaces through subdivision algorithm,and do some geometry processing with other surface in a unified way in CAGD systems.(2) Exploration and properties of parametric polynomial minimal surfaces of high degreeParametric polynomial form is the standard form of curves and surfaces in CAD systems.Based on the classical result in differential geometry,we give the sufficient conditions of harmonic parametric polynomial surfaces of degree five and six being minimal surface.Prom these conditions,several kinds of new minimal surfaces are constructed.We study the interesting geometric properties of these new minimal surfaces,such as symmetry,self-intersection and containing straight lines.We also find two new kinds of conjugate minimal surfaces,and implement the dynamic deformation between the conjugate minimal surfaces. These new minimal surfaces not only enrich the category of minimal surfaces,but also can be integrated into the current CAD systems directly.Finally,we propose three conjectures about the existence and properties of parametric polynomial minimal surfaces of arbitrary degree.(3) Approximate modeling of minimal surfaces based on linear PDEBased on the close relationship between harmonic surfaces and minimal surfaces, the properties of the harmonic B-B surface over the triangular domain are first discussed using the direction derivatives.A sufficient and necessary condition of a B-B surface over the triangular domain being a harmonic surface is obtained.We have proved that the control net of an arbitrary harmonic B-B surface over the triangular domain is fully determined by the first and second layers of control points.Then,we study the construction method of a kinds of surfaces with negative Gaussian curvature,and the result is similar with the harmonic case.In order to exchange data between PDE modeling system and CAD systems,we present a novel algorithm for approximating PDE surface by tensor product Bézier surface based on constrained optimization.We also improve this algorithm by the subdivision property of Bézier surface.(4) Solution of Plateau-Bézier problem based on squared mean curvature energyPlateau-Bézier problem is the extension of Plateau problem in Bézier form. It makes inner control points as objective solution.From the condition of mean curvature being zero,we study this problem in the case of the tensor product Bézier surfaces and the B-B surfaces over triangular domain,and give the sufficient and necessary condition that the inner control points satisfy.We also compare this method with the Dirichlet method.We can prove that,if the minimal surface with respect to the given boundary curves is parametric polynomial minimal surface with isothermal parameter,then the surface obtained by squared mean curvature method is just the parametric polynomial minimal surfaces with isothermal parameter.(5) Boundary optimization in minimal surface modelingHow to choose boundary curves to satisfy the user’s requirement,is an important problem in minimal surface modeling.Firstly,based on the stretch energy, strain energy and jerk energy,we study the following problem:given partial control points of a Bézier curve,how to construct other control points such that the energy of the resulting Bézier curve is a minimum among all the energy of all Bézier curves with the same given control points.We derive the necessary and sufficient condition on the unknown control points for Bézier curves to have minimal energy.We compare the three kinds of energy-minimizing Bézier curves via curvature combs and curvature plots,and also present the collinear property of energy-minimizing quartic Bézier curves.Finally,we propose two extensions of the cubic uniform B-spline curves.First,two classes of polynomial blending functions of degree five and six are constructed.Based on the blending functions, two methods of generating piecewise polynomial curves with a shape parameter are given.It improves the approaching degree of the curves to their control polygon. Secondly,two classes of polynomial blending functions of degree three and four are presented.From these blending functions,we propose two kinds of spline curves with local shape parameter.The advantage of this method is that we can manipulate the shape of the curves locally by changing local shape parameter, while the continuity of the spline curve is unchanged.(6) Application of minimal surface in architecture designArchitecture design is the main application field of minimal surface.Based on the surface trimming technology,we first discuss the application of the new minimal surfaces and the harmonic B-B surface over triangular domain proposed in this thesis in the design of membrane structure.Finally,we propose a new concept- conical mesh of revolution,and give the simple construction method of them.We also employ them in the design of glass/steel structure.更多还原