四边形网格的去边细分方法

提出一种四边形网格细分算法：每细分一次四边形网格，其数目增加为原来的两倍，细分二次结果相当于一次二分细分和一个旋转．该算法采用三次B样条张量积的形式，其生成曲面在规则点具有C^2连续性，在非规则点具有C^1连续性．由于该细分算法对网格几何操作简单，所得网格数据量增长相对缓慢，适合于3D图像重构及网络传输等应用领域。     

三进制四点法的连续性与误差估计

2002年,Hassan提出了一类三进制四点法．对比Dyn的四点法,前者的插值极限曲线可以达到C^2连续．详细研究了三进制四点细分算法生成的极限曲线的连续性,分别给出了极限曲线C^0,C^1,C^2连续的充分条件和必要条件,并给出了极限曲线在顶点以及中点处的一阶导数与二阶导数显式公式．最后根据算法的二次多项式再生性质,得到算法最高具有三次收敛阶．     

三参数binary细分法

通过在曲线细分过程中引入三个参数,给出一种新的细分曲线构造的算法,并利用生成多项式等方法对细分法的一致收敛性、Ck连续性进行了分析.在给定初始控制数据的条件下,可以通过对形状参数的适当选择来实现对细分极限曲线形状的调控.该方法可以生成C4连续的细分曲线,增加了曲线造型的灵活性.数值试验表明这种算法是有效的.     

曲面三参数binary细分法

在经典四点细分法的基础上,通过在曲线细分过程中引入三个参数,给出一种改进的细分曲线构造的算法,利用生成多项式等方法对细分法的一致收敛性、Ck连续性进行了分析。并把该方法扩展到曲面上,进而提出了曲面三参数binary细分法。在给定初始控制数据的条件下,可以通过对形状参数的适当选择来实现对细分极限曲面形状的调控。数值实验表明该算法较容易控制曲面形状,可方便地应用于工程实际,解决曲线、曲面位置调整和控制问题。     

五点二重逼近细分法

提出了一种新的构造曲线的算法——五点二重逼近细分法。利用细分格式 的生成多项式讨论了该细分格式的一致收敛性及Ck 连续性。该细分格式带有一个张力参数 μ, 通过选取不同的μ值,可以分别生成C1~C5 连续的极限曲线。特别是当μ=9/256 时, 细 分格式生成的极限曲线可以达到C7 连续。最后给出了五点二重逼近曲线细分的实例,表明 了这种细分格式是有效的。     

B样条的p-nary细分

有关B样条曲线曲面的binary细分技巧及其应用的研究已经获得了许多成果,建立在B样条binary细分基础上的binary细分法收敛性连续性分析的生成多项式法就是其中之一。该文研究了B样条曲线的p-nary细分问题,给出并证明了B样条基函数的p尺度细分方程中细分系数的计算公式及其性质,讨论了用p-nary细分生成非有理及有理B样条曲线的细分规则。采用该文的方法可方便而快速地在计算机上绘制有理B样条曲线。文章的结果可用于对一般p-nary曲线细分法收敛性及连续性的分析。     

四点多参数Binary细分曲线

在曲线细分过程中引入六个参数,构造出一种新的四点多参数细分Binary曲线算法。对四点多参数Binary细分法的一致收敛性、连续性进行分析,该算法使Dyn四点法以及2到6次均匀B样条细分曲线成为特例。通过对形状参数的适当选择来实现对细分极限曲线形状的调控,增加曲线造型的灵活性,并给出造型实例。     

基于Catmull-Clark细分的新细分算法的研究

提出一种四边形网格细分算法：每细分一次四边形网格,其数目增加为原来的两倍,细分二次结果相当于一次二分细分,采用边数缓慢增长的策略,使生成的曲面光滑连续。该算法生成曲面在规则点具有C2连续性,在非规则点具有C1连续性。该算法对网格几何操作简单,所得网格数据量增长相对缓慢,适合3D图像重构及网络传输等应用领域。由于文中细分算法对初始网格的拓扑变更,因此第一次细分会产生扭曲现象,但后面的细分会逐步光滑。     

基于Catmull-Clark细分的新细分算法的研究

提出一种四边形网格细分算法:每细分一次四边形网格,其数目增加为原来的两倍,细分二次结果相当于一次二分细分,采用边数缓慢增长的策略,使生成的曲面光滑连续.该算法生成曲面在规则点具有C2连续性,在非规则点具有C1连续性.该算法对网格几何操作简单,所得网格数据量增长相对缓慢,适合3D图像重构及网络传输等应用领域.由于文中细分算法对初始网格的拓扑变更,因此第一次细分会产生扭曲现象,但后面的细分会逐步光滑.     

一类由Laurent多项式诱导的带参数二重细分

为了提高细分曲线的设计灵活性,根据Laurent多项式与细分生成多项式的关系,构造一个可以生成一类多参数细分格式的Laurent多项式.该多项式生成的格式包含许多现有的对称细分,可以用来构造非对称细分;针对一种三参数五点细分格式,分析其产生的极限曲线的光滑性和连续性.通过数值实例分析特定情形下参数对极限曲线的影响,并说明非对称细分有时比对称细分逼近效果更好.     

基于任意三角网格的ternary插值细分曲面造型及其分析

针对任意三角网格,提出一种简单有效且局部性更好的带参数的ternary插值曲面细分法,给出并证明了细分法收敛与G1连续的充分条件.在任意给定三角控制网格的条件下,可通过对形状参数的适当选择来实现对插值细分曲面形状的调整.     

Matrix-valued 4-point spline and 3-point non-spline interpolatory curve subdivision schemes

The objective of this paper is to study and construct matrix-valued templates for interpolatory curve subdivision. Since our investigation of this problem was motivated by the need of such subdivision stencils as boundary templates for interpolatory surface subdivision, we provide both spline and non-spline templates that are necessarily symmetric, due to the lack of direction-orientation in carrying out surface subdivision in general. For example, the minimum-supported Hermite interpolatory C¹ cubic spline curve subdivision scheme, with the skew-symmetric basis function for interpolating first derivatives, does not meet the symmetry specification. Non-spline C² interpolatory templates constructed in this paper are particularly important, due to their smaller support needed to minimize undesirable surface oscillations, when adopted as boundary templates for interpolatory C² surface subdivision. The curve subdivision templates introduced in this paper are adopted as boundary stencils for interpolatory surface subdivision with matrix-valued templates.     

An Improving Subdivision Scheme for Curve Design

This paper extends the classical 4-point interpolatory subdivision scheme, and brings forward a new 4-point subdivision scheme with Three Parameters for Curve Design. We discuss the influence of three parameters to limit curve and realize the adjustment and control to it by choosing these three parameters appropriately, The sufficient conditions of the uniform convergence and continuity properties of the subdivision scheme are given.     

3点Binary插值细分法的性质以及应用

为了使细分格式具有好的性质：如光滑性、保凸性,提出了3点binary插值细分格式,然后分析了该细分格式的连续性、保凸性以及分形等性质与参数之间的关系,最后给出了该细分格式性质的一些应用。     

四点插值细分算法极限曲线曲面C2连续的充分必要条件

研究了四点插值细分算法的连续性．用若当标准形重新证明了Dyn的一个定理，从而得到了一个极限函数具有二阶导函数的充分必要条件及二阶导函数的解析表达式；并将结果推广到曲面的情形．     

A Method for Constructing Interpolatory Subdivision Schemes and Blending Subdivisions

This paper presents a universal method for constructing interpolatory subdivision schemes from known approximatory subdivisions. The method establishes geometric rules of the associated interpolatory subdivision through addition of further weighted averaging operations to the approximatory subdivision. The paper thus provides a novel approach for designing new interpolatory subdivision schemes. In addition, a family of subdivision surfaces varying from the given approximatory scheme to its associated interpolatory scheme, namely the blending subdivisions, can also be established. Based on the proposed method, variants of several known interpolatory subdivision schemes are constructed. A new interpolatory subdivision scheme is also developed using the same technique. Brief analysis of a family of blending subdivisions associated with the Loop subdivision scheme demonstrates that this particular family of subdivisions are globally C¹ continuous while maintaining bounded curvature for regular meshes. As a further extension of the blending subdivisions, a volume‐preserving subdivision strategy is also proposed in the paper.     

曲线插值的一种具有还圆性的细分方法

传统的线性四点插值细分方法不能表示圆等非多项式曲线,为了解决这种 问题,基于几何特性提出了一种带有一个参数的四点插值型曲线细分方法。细分过程中,过 相邻三插值点作圆,过相邻二插值点的圆弧有两个中点,将其加权平均得到新插值点,文中 给出了插值公式和算法描述。所给方法具有还圆性,可以实现保凸性。实例分析对比了本方 法与多种细分方法的差异,说明本方法是有效的,当参数取值较小时,曲线靠近控制多边形。     

一种n次均匀B样条曲线细分算法

利用 次均匀B样条细分的掩模与Pascal三角形关系,并借助控制多边形在每次加细过程中新旧控制顶点对应的几何位置关系,给出一种新的 次均匀B样条曲线细分算法,基于该算法构造出带有形状参数的局部插值约束的奇次均匀B样条细分曲线。通过理论和算例说明,该算法几何直观性强、新旧点对应明确、应用灵活且能保持良好的参数连续性。     

Interpolatory,solid subdivision of unstructured hexahedral meshes

This paper presents a new, volumetric subdivision scheme for interpolation of arbitrary hexahedral meshes. To date, nearly every existing volumetric subdivision scheme is approximating, i.e., with each application of the subdivision algorithm, the geometry shrinks away from its control mesh. Often, an approximating algorithm is undesirable and inappropriate, producing unsatisfactory results for certain applications in solid modeling and engineering design (e.g., finite element meshing). We address this lack of smooth, interpolatory subdivision algorithms by devising a new scheme founded upon the concept of tri-cubic Lagrange interpolating polynomials. We show that our algorithm is a natural generalization of the butterfly subdivision surface scheme to a tri-variate, volumetric setting.     

Interpolatory and Mixed Loop Schemes

This paper presents a new interpolatory Loop scheme and an unified and mixed interpolatory and approximation subdivision scheme for triangular meshes. The former which is C¹ continuous as same as the modified Butterfly scheme has better effect in some complex models. The latter can be used to solve the “popping effect” problem when switching between meshes at different levels of resolution. The scheme generates surfaces coincident with the Loop subdivision scheme in the limit condition having the coefficient k equal 0. When k equal 1, it will be changed into a new interpolatory subdivision scheme. Eigen‐structure analysis demonstrates that subdivision surfaces generated using the new scheme are C¹ continuous. All these are achieved only by changing the value of a parameter k. The method is a completely simple one without constructing and solving equations. It can achieve local interpolation and solve the “popping effect” problem which are the method's advantages over the modified Butterfly scheme.