【作者】 张文斌；

【导师】 王晓明；

【作者基本信息】 大连理工大学 ， 机械设计及理论， 2009， 硕士

【摘要】 在计算机辅助几何设计(CAGD)领域中,熟知有两种定义曲面的方法,即参数形式与隐式形式。所谓隐式曲面,是指用实系数三元代数多项式的零点(f(x,y,z)=0)所定义的曲面,故也称其为代数曲面。参数形式的曲面以其构造简单计算容易等特点而流行于世并成为几何设计的主流。与参数形式的曲面相比,隐式形式有如下优点:隐式形式的曲面表示易于判别一个点是否在曲面上,易于表示封闭的形体,在几何操作下运算封闭;任何参数形式或隐式形式的曲面间的几何运算的结果均可表示成隐式形式。基于隐式形式的上述优点,光滑拼接隐式曲线曲面的研究显得具有重大的意义。等值线和等值面作为一类常见的隐式曲线和曲面,其应用价值受到越来越多的关注。等值线和等值面技术在可视化中应用广泛,许多标量场中的可视化问题都归纳为等值线和等值面的抽取和绘制,Marching Cubes方法是目前应用最为广泛的等值面抽取方法之一。本文阐述了MC算法的基本原理,讨论了其优缺点和产生这些优缺点的原因。算法在构造等值面的过程中,太依赖于直观的构造,构建体元状态模型时,对于对称,旋转等情况的处理缺乏全面考虑,忽视了立方体内部可能存在的环状结构和存在的临界点(等值面发生变化的点),直接使用求得的边界等值点,根据基本的体元状态模型,简单连接成三角片构成等值面,导致生成的等值面拓扑结构不一致,不能满足实际应用中的需求。本文针对MC算法的缺点,将研究多元样条函数的光滑余因子方法引入到MC算法中,首先对数据场进行正方形或立方体剖分,得到MC算法需要的正方形单元或立方体体元。接着建立了单元和体元间光滑连接所应满足的协调条件,进而使求隐式曲线曲面光滑拼接等问题转化为求解协调方程的问题。通过构造插值适定结点组,进而完成了多元样条函数的插值问题,求得了各单元和体元内的具体样条函数表达式。继续利用MC方法对给定值的样条函数进行等值线或等值面的抽取,分别实现了正方形单元间二次隐式曲线段的C~1光滑拼接和三次隐式曲线段的C~1光滑拼接,进而实现了立方体体元间二次隐式曲面片的C~1光滑拼接和三次隐式曲面片的C~1光滑拼接,即抽取的等值线和等值面在整个数据场达到C~1连续。更多还原

【Abstract】 In the field of the Computer Aided Geometric Design(CAGD),surfaces can be classified into two categories:parametric surfaces and implicit surfaces.An implicit surface refers to the surface that is defined by the set of solutions of a real coefficient algebraic polynomial equation(f(x,y,z)=0);therefore we also call it the algebraic surface.The parameter surfaces have been at the center of research in geometric design for a long time due to their highly desirable properties such as the simple structure and easy computation. Compares with the parameter surface,the implicit surface has the following advantages: easily to judge a point in the surface,and the closure property under some geometric operations;for a parametric surface and an implicit surface,all results under geometric operations are denoted to implicit surfaces.Based on these advantages of implicit surfaces,it is very significant to study smoothly blending implicit algebraic curves and surfaces.Contour and iso-surface as common implicit algebraic curves and surfaces,its application value received more and more attention.Contour and iso-surface technology is widely used in visualization and many scalar field visualization problems can be solved by isosurfacing and rendering.Marching cubes algorithm is the most widely used iso-surface method till now.This paper introduces the basic principle of marching cubes algorithm,then gives out the definition,the disadvantages and advantages of the algorithms and explains how these problems are brought about.In the process of isosurfacing,the algorithm doesn’t deal with the complexity of iso-surface in the voxel,and takes the triangles as iso-surfaces.However,an iso-surface should change its shape at some critical points and loops.Therefore,the iso-surface algorithm may encounter many problems in topology,precision and efficiency and can not satisfy many applications.In order to overcome the disadvantages of marching cubes algorithm,this paper incorporates spline function of smoothing cofactor into marching cubes algorithm.First,the data field is discretized into square or cube cells,and then the coordination conditions of element or voxel smooth connection are established.As a result,the smooth blending problems of implicit curves and surfaces are transformed into the problem to solve the coordination equations.By constructing a well-posed interpolation node group,the problem of multi-spline interpolation is successfully solved,and the specific expression of spline for unit element and voxel is obtained.And then using the marching cubes algorithm,the contour or iso-surface is extracted for a given value of the spline function.The C~1 smooth blending of quadratic implicit curve segments and the C~1 smooth blending of cubic implicit curve segments is achieved within element respectively,thereby the C~1 smooth blending of quadratic implicit surface patches and the C~1 smooth blending of cubic implicit surface patches is realized within voxel,which means that the contour and iso-surface reach the C~1 smooth continuous in the entire data fields.更多还原