NURBS曲面C1连续参数优化算法

为了在NURBS曲面上生成满足C1连续的近似弧长参数化等参线,将NURBS曲面4条边界的积分能量函数作为目标函数,提出一种基于分段三次重新参数化的曲面参数优化算法.首先推导出NURBS曲面4条边界的参数表达式;然后使用Hermite基函数变换NURBS曲面u和v方向参数,重新计算参数化等参线,导出变换后的4条边界的积分能量函数;最后通过数值优化算法计算出曲面的最优参数表示.在MFC和OpenGL环境下实现了多个NURBS曲面的等参线分布和纹理映射实验,结果表明该算法是有效的.     

等距曲面的NURBS放样插值方法

本文给出了等距曲面的一种NURBS放样插值生成方法，该方法主要是在原始NURBS曲面上取得一个能较好反映曲面特征的型值点阵，再交这个型值点阵按某种算法矢方向外推，从而得到原始曲面的等距曲面上的型值点阵，然后，再用NURBS放样插值曲面来逼近等距曲面，本文给出的算法几何意义明显，易于编程实现，且得到的等距曲面其u向和v向参数曲线仍是NURBS曲线，且具有C^2连续性，最后，给出了一个实例。     

NURBS曲面生成算法及仿真研究

讨论了一种生成NURBS曲面的算法,用C语言实现了该算法,并利用MATLAB进行仿真对该算法进行验证。在算法中讨论了曲面及其等距面生成方法以及曲面生成技术中相关的一些技术,如曲线段间参数过渡、曲面生成模式、曲面生成的实时性、改变曲面的形状等。仿真结果证明了算法的有效性。     

基于蚁群的数据点NURBS曲面快速拟合研究

研究数据点的NURBS曲面拟合问题,提高拟合速率。针对所要拟合的数据点分布散乱,传统的基于遗传算法多次迭代,造成曲面拟合速率不高的问题。为解决上述问题,提出一种基于蚁群的数据点NURBS曲面拟合算法。通过采用蚁群寻址算法搜索出控制顶点和边界数据点集,计算曲面的权因子后完成NURBS曲面的拟合,并使用蚁群算法对拟合曲面进行优化,避免了传统方法多代遗传迭代造成的拟合速率不高的问题。实验表明,这种方法能够快速完成散乱数据点的NURBS曲面拟合,并且具有一定的拟合效率,取得了满意的结果。     

插值边界曲线的NURBS 近似极小曲面设计

探索性地设计了一个插值给定边界曲线的NURBS 近似极小曲面算法,弥补了当前NURBS 系统无法有效地设计工程所急需的一般NURBS 极小曲面的缺陷.运用NURBS 曲面的节点插入、Hybrid 多项式逼近等多种技术,将NURBS 曲面转化为相对简单的分片Bézier 曲面求解,并运用各子曲面片的控制顶点优化、整体曲面不断更新的迭代方法,成功地得到高精度的近似分片Bézier 极小曲面.最后,可以按用户的各种要求选择运用相应不同的迭代逼近算法,求取插值给定边界曲线的近似NURBS 极小曲面.     

基于流形的光滑曲面构造

将传统参数曲面的规则定义域推广到一般的二维流形上,并基于定义域微分流形提出了一种新的光滑曲面构造方法.该方法运用最短距离构造基函数,从整体上直接构造出任意拓扑结构的光滑流形曲面,无须分片生成后再进行拼接.而且该算法所生成的流形曲面具有很好的局部性和可控性,NURBS曲面上的很多方法在此曲面上仍然适用.实验结果表明,该构造方法可以构造出非常复杂的曲面.     

基于NURBS曲面拟合的三角网格曲率计算

针对三角网格提出了一种基于NURBS曲面拟合的计算Gauss曲率和平均曲率的算法。首先选取边界检测后的二阶邻点作为局部拟合数据,采用直接投影法实现参数化,由二次NURBS曲面进行最小平方拟合反算控制点矩阵,最后由拟合曲面计算曲率。并从三角网格分辨率和噪声两方面进行了比较,实验结果表明本文算法精度高、较其他算法稳定,因而更具通用性。     

基于NURBS的扭变曲面的计算机建模方法研究

本文首先介绍了扭变曲面CAD在国内外的发展情况和NURBS方法的概念,通过推导曲面初始二维数据向三维数据的转换关系,提出一种将NURBS曲线曲面反求算法用于叶片曲面的构造,完成叶片的计算机建模方法,有效地提高了此类曲面在实际生产中的设计精度.     

NURBS曲线曲面间最短距离的计算

针对现有的大多数计算几何形状间最短距离的算法都需要进行大量的多边形检测,且有时计算出的最短距离不够精确的问题,提出一种计算NURBS曲线与曲线、曲线与曲面和曲面与曲面间最短距离的算法.首先将2个NURBS形状分解成分段B啨zier表示的2个集合,给出一种计算2个集合的边界包围球的简单快速算法;然后分别在2个集合中选择包含最短距离的B啨zier表示对形成候选集.该算法采用边界包围球和"四点条件"约束提高计算效率,用多维Newton-Raphson迭代计算所有候选对间的局部最短距离,由此求出全局的最短距离.实验结果表明,文中算法具有速度快、精度高和鲁棒性好的特点,可实时计算2个NURBS曲线曲面间的最短距离.     

基于截面数据的C1插值融合曲面重建

为了避免NURBS曲面重建需要进行节点矢量相容的问题，提出了一种双方向融合插值的[C1]参数曲面重建方法，该方法先后分段插值截面上连续的数据点、截面曲线以构造样条曲线和曲面片，并引入融合算法进行曲线、曲面拼接，从而得到光滑的待建曲面。该方法不会产生由节点插入所带来的大量的数据冗余以及复杂的计算过程，同时采用了融合的思想来处理曲线、曲面的拼接，改良了传统参数曲线、曲面拼接方法需要满足边界条件的缺陷。     

Isogeometric shape optimization of photonic crystals via Coons patches

In this paper, we present an approach that extends isogeometric shape optimization from optimization of rectangular-like NURBS patches to the optimization of topologically complex geometries. We have successfully applied this approach in designing photonic crystals where complex geometries have been optimized to maximize the band gaps.Salient features of this approach include the following: (1) multi-patch Coons representation of design geometry. The design geometry is represented as a collection of Coons patches where the four boundaries of each patch are represented as NURBS curves. The use of multiple patches is motivated by the need for representing topologically complex geometries. The Coons patches are used as a design representation so that designers do not need to specify interior control points and they provide a mechanism to compute analytical sensitivities for internal nodes in shape optimization, (2) exact boundary conversion to the analysis geometry with guaranteed mesh injectivity. The analysis geometry is a collection of NURBS patches that are converted from the multi-patch Coons representation with geometric exactness in patch boundaries. The internal NURBS control points are embedded in the parametric domain of the Coons patches with a built-in mesh rectifier to ensure the injectivity of the resulting B-spline geometry, i.e. every point in the physical domain is mapped to one point in the parametric domain, (3) analytical sensitivities. Sensitivities of objective functions and constraints with respect to design variables are derived through nodal sensitivities. The nodal sensitivities for the boundary control points are directly determined by the design parameters and those for internal nodes are obtained via the corresponding Coons patches.     

Coons曲面的覆盖式设计程序

Coons曲面的边界条件中存在数据冗余，使得在造型过程中边界条件之间的相容性不易保证．本文在阐明了边界条件之间的相容性联系的基础上，针对任意阶的Coons曲面，提出了一个以角点信息矩阵为中心的覆盖式设计程序，通过曲面的设计程序保证了相容性．     

A geometric approach to boundary-conformed toolpath generation

Isoparametric or streamline-like toolpath generation is an important method for CNC surface machining. For a trimmed surface, the generation of such a toolpath is not straightforward because by simply following the isoparametric curves of the parametric surface to generate the toolpath, the resulting toolpath may no longer conform to the trimmed boundary of the surface. Various methods have been proposed to solve this problem. Some methods work well for single surface machining, but may not be feasible for multi-patch surface machining. A new geometric approach to the problem of multi-patch machining is proposed in this paper. The new method works by generating bisectors to partition the region into smaller subregions, and generates the toolpath by offsetting the subregion boundary by using a special offset function. As this new method does not rely on processing on the parametric domain of the surface patches, it can be used for both NURBS and subdivision surfaces. Examples are given to demonstrate the capability of the new method, and to show that the new method compares favorably with existing techniques.     

基于高光线模型修改NURBS曲面不光顺性

高光线模型是一种评估曲面光顺性的强有力工具．提出一种基于高光线模型修改NURBS曲面不光顺性的新方法，将高光线模型对应的非线性方程组转化成线性方程组，从而大大简化了问题的复杂性，修改后的控制点通过解一组线性方程求得．新NURBS曲面的某些点具有指定的法方向．该方法直观、适用于实时交互设计，通过简单的操作就能修改NURBS强曲面，从而使其具有良好的光顺性．还给出了测试文中方法有效性的实例．     

带形状参数的R-Coons曲面

为了解决传统 Coons 曲面不具备形状可调性,不能精确表示椭球面、椭圆锥面、 椭圆柱面等二次曲面的问题,在有理函数空间上构造了一组有理混合基函数,称之为 R-Hermite 基。首先分析了 R-Hermite 基的性质;其次基于 R-Hermite 基,利用张量积方法,构造了一种 新的带两个形状参数的有理 Coons 曲面,并称之为 R-Coons 曲面,R-Coons 曲面不仅具有传统 Coons 曲面的良好性质,还具备形状可调性;最后给出了曲面精确表示椭球面、椭圆锥面、椭 圆柱面的方法,并通过实例说明方法的有效性。     

NURBS曲面的等距曲面算法

本文将ＮＵＲＢＳ曲线的有理ｄｅＢｏｏｒ算法推广到ＮＵＲＢＳ曲面点的计算，由此可以得到ＮＵＲＢＳ曲面上点的单位法矢量，供其应用于ＮＵＲＢＳ曲面等曲面的生成。该算法几何意义明显，算法简洁，易于编程实现。     

面向等几何分析的B 样条参数体生成方法

三变量体参数化是进行三维等几何分析求解的首要步骤,论文提出了基于 模板的由边界曲面生成B 样条参数体的新方法。首先给出了插值边界曲面的Coons 体的参 数表达式,再由该表达式的离散表示,内部控制顶点可表示为边界控制顶点的线性组合。然 后由其离散表示可得到Coons 模板,并将Coons 模板推广到统一形式,为内部控制顶点的 生成提供了更多选择。论文通过热传导问题的实例对由不同的模板所得到的不同体参数化结 果及其对等几何分析结果的影响进行了比较分析。     

Boundary-conformed toolpath generation for trimmed free-form surfaces

In this paper, we adopt a 2D reparameterization procedure to regenerate boundary-conformed toolpaths. Three methods for the 2D reparameterization of trimmed boundaries in parametric space are examined and compared. They are the Coons method, the Laplace method, and a newly developed boundary-blending method. These three methods represent three different approaches to 2D surface parameterization, namely, the algebraic interpolation approach, the partial differential equation approach, and a geometric offsetting approach, respectively. Complete algorithms for surface reparameterization and toolpath generation are developed and implemented. The results showed that the Coons method is relatively simple yet might cause anomalies when the complexities of the boundary are high. The Laplace method is robust but takes relatively more computational time and also has the problem of uneven distribution of iso-parametrics. For the newly developed boundary-blending method, both the computational efficiency and parameterization robustness are quite good, in addition, it alleviates the uneven distribution problem appeared in the Laplace results.     

Image‐based modeling of 3D objects with curved surfaces

This paper addresses an image‐based method for modeling 3D objects with curved surfaces based on the non‐uniform rational B‐splines (NURBS) representation. The user fits the feature curves on a few calibrated images with 2D NURBS curves using the interactive user interface. Then, 3D NURBS curves are constructed by stereo reconstruction of the corresponding feature curves. Using these as building blocks, NURBS surfaces are reconstructed by the known surface building methods including bilinear surfaces, ruled surfaces, generalized cylinders, and surfaces of revolution. In addition to them, we also employ various advanced techniques, including skinned surfaces, swept surfaces, and boundary patches. Based on these surface modeling techniques, it is possible to build various types of 3D shape models with textured curved surfaces without much effort. Copyright © 2007 John Wiley & Sons, Ltd.     

心血管三维表面的NURBS重建方法

针对心血管三维表面重建的问题,采用NURBS(非均匀有理B样条)重构三维血管表面,研究了NURBS曲面的拟合算法和利用"权值"修改NURBS曲面的方法,并运用OpenGL技术实现了三维血管的虚拟显示。利用临床冠状动脉造影图像进行心血管三维重建实验,结果表明该重建方法比传统方法在重建精度和光顺性上都有明显的改善。