自由曲线曲面的任意次非均匀细分

提出一种有效的建模自由曲线曲面的非均匀细分算法。首先在节点插入技术基础上推导出任意次自由曲线的非均匀细分规则,然后把它推广到张量积曲面得到任意次自由曲面的非均匀细分规则,最后对奇异点附近曲面采用类Doo-Sabin和Catmull-Clark的细分规则,从而使该算法可以实现建模任意次具有任意拓扑基网格的非均匀细分曲面。此外,该方法也实现了对传统细分格式的统一,例如,当次数为2并采用均匀节点矢量便转化为Doo-Sabin细分,当次数为3并采用均匀节点矢量便转化为Catmull-Clark细分。     

细分曲面拟合的局部渐进插值方法

逼近型细分方法生成的细分曲面其品质要优于插值型细分方法生成的细分曲面.然而,逼近型细分方法生成的细分曲面不能插值于初始控制网格顶点.为使逼近型细分曲面具有插值能力,一般通过求解全局线性方程组,使其插值于网格顶点.当网格顶点较多时,求解线性方程组的计算量很大,因此,难以处理稠密网格.与此不同,在不直接求解线性方程组的情况下,渐进插值方法通过迭代调整控制网格顶点,最终达到插值的效果.渐进插值方法可以处理稠密的任意拓扑网格,生成插值于初始网格顶点的光滑细分曲面.并且经证明,逼近型细分曲面渐进插值具有局部性质,也就是迭代调整初始网格的若干控制顶点,且保持剩余顶点不变,最终生成的极限细分曲面仍插值于初始网格中被调整的那些顶点.这种局部渐进插值性质给形状控制带来了更多的灵活性,并且使得自适应拟合成为可能.实验结果验证了局部渐进插值的形状控制以及自适应拟合能力.     

形状可调的Loop 细分曲面渐进插值方法

针对Loop 细分无法调整形状与不能插值的问题,提出了一种形状可调的Loop 细分 曲面渐进插值方法。首先给出了一个既能对细分网格顶点统一调整又便于引入权因子实现细分曲 面形状可调的等价Loop 细分模板。其次,通过渐进迭代调整初始控制网格顶点生成新网格,运 用本文的两步Loop 细分方法对新网格进行细分,得到插值于初始控制顶点的形状可调的Loop 细分曲面。最后,证明了该方法的收敛性,并给出实例验证了该方法的有效性。     

基于边折叠和质点-弹簧模型的网格简化优化算法

通过边折叠实现网格曲面简化,提出了保持曲面特征的边折叠基本规则,引入边折叠顺序控制因子λ,给出了折叠点坐标获取方法,简化过程中网格边长度趋于均匀．在曲面简化基础上,利用质点-弹簧模型优化网格形状．将网格顶点邻域参数化到二维域上,在质点-弹簧模型中引入约束弹簧,约束调整网格顶点,并逆映射到三维原始曲面上,局部优化网格顶点的相邻网格;调整曲面上所有网格顶点,在全局上优化网格形状．在曲面简化优化过程中,建立原始模型曲面和简化优化后曲面之间的双向映射关系;曲面的网格顶点始终在原始模型表面上滑动,并以双向Hausdorff距离衡量、控制曲面间的形状误差．应用实例表明：文中算法稳定、高效,适合于任意复杂的二维流形网格．     

隐式曲面的快速适应性多边形化算法

通过将隐式曲面多边形化过程分为“构造”和“适应性采样”两个阶段，实现了隐式曲面多边形逼近网格的适应性构造．通过基于空间延展的Marching Cubes方法得到隐式曲面较为粗糙的均匀多边形化逼近，根据曲面上的局部曲率分布，运用适应性细分规则对粗糙网格进行细分迭代，并利用梯度下降法将细分出的新顶点定位到隐式曲面上；最终得到的多边形网格是适应性的单纯复形网格，其在保持规定逼近精度的前提下，减少了冗余三角形的产生，网格质量有明显改善．该算法可用于隐式曲面的交互式可视化过程．     

快速收敛的四边形网格三分细分模式

提出了四边形网格的三分细分模式.对于正则和非正则四边形网格,分别采用不同的细分模板获得新的细分顶点.从双三次B样条中推导出正则四边形网格的三分细分模板,极限曲面C²连续;对细分矩阵进行傅里叶变换,推导出非正则四边形网格的三分细分模板,极限曲面C¹连续.提出的三分细分模式可以解决任意拓扑四边形网格的曲面细分问题.与其他细分模式相比,具有收敛速度快、适用范围广等优点.最后给出了四边形网格细分的实例.     

基于蝶形细分自适应算法的三维地形仿真

基于格网法提出了蝶形细分自适应算法进行三维地形模拟,以原网格顶点的法向量为约束条件,通过对初始三角形控制网格进行多阶曲线迭代插值的非静态细分,实现几何造型.插值点的计算依据网格的局部几何特征,根据三角形网格上顶点的平坦度进行有选择性的自适应细分,同时对细分过程中产生的曲面裂缝加以弥补.地形仿真实例显示新的自适应细分方法可以很好地继承原始网格的形状特征,在曲面的光滑度和真实性上更加完善,加快了图形处理的速度.     

渐进插值的LOOP曲面细分*

目前很多细分方法都存在不能用同一种方法处理封闭网格和开放网格的问题。对此,一种新的基于插值技术的LOOP曲面细分方法,其主要思想就是给定一个初始三角网格M,反复生成新的顶点,新顶点是通过其相邻顶点的约束求解得到的,从而构造一个新的控制网格M,在取极限的情况下,可以证明插值过程是收敛的;因为生成新顶点使用的是与其相连顶点的约束求解得到的,本质上是一种局部方法,所以,该方法很容易定义。它在本地方法和全局方法中都有优势,能处理任意顶点数量和任意拓扑结构的网格,从而产生一个光滑的曲面并忠实于给定曲面的形状,其控制     

基于顶点法向量约束的Catmull-Clark细分插值方法

提出一种基于顶点法向量约束实现插值的两步Catmull-Clark细分方法.第一步,通过改造型Catmull-Clark细分生成新网格.第二步,通过顶点法向量约束对新网格进行调整.两步细分分别运用渐进迭代方法和拉格朗日乘子法,使得极限曲面插值于初始控制顶点和法向量.实验结果证明了该方法可同时实现插值初始控制顶点和法向量,极限曲面具有较好的造型效果.     

基于高光线模型消除三角形网格局部不规则区域

高光线是检测自由曲面质量的有效工具.它提供一种直观且便利的手段,在交互设计中提高自由曲面质量.文中提出了在任意三角形网格曲面上生成高光线模型的一种方法.基于该高光线模型,文中给出了一种消除三角形网格上的局部不规则区域的方法.该方法通过求解一个目标函数,并迭代地移动网格顶点位置,来获得修改后的新网格.利用该方法能够同时优化三角形网格表面形状以及网格上的高光线形状.该方法直观易用,适合于三角形网格的局部形状优化.     

Interpolation by geometric algorithm

We present a novel geometric algorithm to construct a smooth surface that interpolates a triangular or a quadrilateral mesh of arbitrary topological type formed by n vertices. Although our method can be applied to B-spline surfaces and subdivision surfaces of all kinds, we illustrate our algorithm focusing on Loop subdivision surfaces as most of the meshes are in triangular form. We start our algorithm by assuming that the given triangular mesh is a control net of a Loop subdivision surface. The control points are iteratively updated globally by a simple local point-surface distance computation and an offsetting procedure without solving a linear system. The complexity of our algorithm is O(mn) where n is the number of vertices and m is the number of iterations. The number of iterations m depends on the fineness of the mesh and accuracy required.     

Constructing iterative non-uniform B-spline curve and surface to fit data points

In this paper, based on the idea of profit and loss modification, we presentthe iterative non-uniform B-spline curve and surface to settle a key problem in computeraided geometric design and reverse engineering, that is, constructing the curve (surface)fitting (interpolating) a given ordered point set without solving a linear system. We startwith a piece of initial non-uniform B-spline curve (surface) which takes the given point setas its control point set. Then by adjusting its control points gradually with iterative formula,we can get a group of non-uniform B-spline curves (surfaces) with gradually higherprecision. In this paper, using modern matrix theory, we strictly prove that the limit curve(surface) of the iteration interpolates the given point set. The non-uniform B-spline curves(surfaces) generated with the iteration have many advantages, such as satisfying theNURBS standard, having explicit expression, gaining locality, and convexity preserving,etc     

A subdivision scheme for Poisson curves and surfaces

The de Casteljau evaluation algorithm applied to a finite sequence of control points defines a Bézier curve. This evaluation procedure also generates a subdivision algorithm and the limit of the subdivision process is this same Bézier curve. Extending the de Casteljau subdivision algorithm to an infinite sequence of control points defines a new family of curves. Here, limits of this stationary non-uniform subdivision process are shown to be equivalent to curves whose control points are the original data points and whose blending functions are given by the Poisson distribution. Thus this approach generalizes standard subdivision techniques from polynomials to arbitrary analytic functions. Extensions of this new subdivision scheme from curves to tensor product surfaces are also discussed.     

Conjugate-Gradient Progressive-Iterative Approximation for Loop and Catmull-Clark Subdivision Surface Interpolation

Loop and Catmull-Clark are the most famous approximation subdivision schemes, but their limit surfaces do not interpolate the vertices of the given mesh. Progressive-iterative approximation (PIA) is an efficient method for data interpolation and has a wide range of applications in many fields such as subdivision surface fitting, parametric curve and surface fitting among others. However, the convergence rate of classical PIA is slow. In this paper, we present a new and fast PIA format for constructing interpolation subdivision surface that interpolates the vertices of a mesh with arbitrary topology. The proposed method, named Conjugate-Gradient Progressive-Iterative Approximation (CG-PIA), is based on the Conjugate-Gradient Iterative algorithm and the Progressive Iterative Approximation (PIA) algorithm. The method is presented using Loop and Catmull-Clark subdivision surfaces. CG-PIA preserves the features of the classical PIA method, such as the advantages of both the local and global scheme and resemblance with the given mesh. Moreover, CG-PIA has the following features. 1) It has a faster convergence rate compared with the classical PIA and W-PIA. 2) CG-PIA avoids the selection of weights compared with W-PIA. 3) CG-PIA does not need to modify the subdivision schemes compared with other methods with fairness measure. Numerous examples for Loop and Catmull-Clark subdivision surfaces are provided in this paper to demonstrate the efficiency and effectiveness of CG-PIA.     

Non-Uniform Subdivision Surfaces with Sharp Features

Sharp features are important characteristics in surface modelling. However, it is still a significantly difficult task to create complex sharp features for Non-Uniform Rational B-Splines compatible subdivision surfaces. Current non-uniform subdivision methods produce sharp features generally by setting zero knot intervals, and these sharp features may have unpleasant visual effects. In this paper, we construct a non-uniform subdivision scheme to create complex sharp features by extending the eigen-polyhedron technique. The new scheme allows arbitrarily specifying sharp edges in the initial mesh and generates non-uniform cubic B-spline curves to represent the sharp features. Experimental results demonstrate that the present method can generate visually more pleasant sharp features than other existing approaches.     

用C-C细分法和流形方法构造G2连续的自由型曲面

通过改进Cotrina等利用流形方法构造n边曲面片的算法,以C-C细分网格奇异点的5一环作为控制网构造出了带有均匀三次B样条边界的n边曲面片,使得该曲面片和C-C细分曲面G^2拼接．在此基础上,讨论了C-C细分曲面中n边域的构造和填充,从而为基于任意拓扑网格构造低次G^2连续曲面的问题给出了一个有效的解决方案,实现了用流形方法构造的曲面和C-C细分曲面的融合．最后,给出了几个具体算例．