一种修改NURBS曲线形状的新方法

曲线曲面的形状修改是计算机几何造型过程中的重要部分．文章提出了一种修改NURBS曲线的新方法，使得修改后的曲线在多个参数点处满足用户给定的几何约束(如点约束、切矢约束)，首先引入了一些新的概念如：局部曲线、总曲线、多余约束和多余曲线等．对于每个参数点分别计算出一系列满足该点处几何约束的局部曲线，并由此构造了总曲线．接着插值一条满足多余约束的多余曲线．最后运用构造Coons曲面的思想，计算出最终的修改曲线，它等于总曲线减去多余曲线．同时我们发现两种现存的修改NURBS曲线的方法是一样的．实例表明此方法适用于CAD软件系统。     

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 曲面自由型特征处理方法研究

NURBS曲面自由型特征因不具有确定的参数、准确的边界,使其表示、编辑及操作十分困难。基于一种新的小波多分辨率分析技术将NURBS曲面从单一尺度几何表示的空域变换为频域的表示形式,借助于滤波器组将模型的整体结构和局部细节变换为其全局和局部多分辨率自由型特征,实现了对NURBS曲面的自由型特征的编辑、操作和控制。该方法不仅能提高NURBS曲面几何建模的效率,而且可以克服传统NURBS曲面建模的繁琐和费时。     

基于约束优化的Bézier曲面形状修改

Bézier曲线和曲面广泛应用于CAGD(计算机辅助几何设计)和计算机图形学,对Bézier曲线或者曲面的设计和形状修改是一个重要的问题。研究了基于几何约束的Bézier曲面优化问题,对单点和多点约束的问题,提出了一种通过修改控制点的约束优化方法。用这种方法,通过修改原Bézier曲面的控制点来修改曲面的形状并满足给定的约束条件,同时给出了数值实例,其结果表明,用拉格朗日方法能有效地解决Bézier曲面的形状修改问题。     

NURBS的概念与使用

NURBS是英文Non－UniformRationalB－Spline（即非均匀有理B样条）的缩写。NURBS曲线和曲面造型方法是当今CAD／CAM领域研究热点之一。国际标准化组织ISO在其产品数据模型数据交换的国际标准STEP（STandardfortheEx－changeofProductmodeldata）中把NURBS作为几何描述的主要方法，这因为NURBS对标准的解析曲面，如圆锥曲面、二次曲面、旋转曲面，和自由曲面．提供了统一的、精确的数学表达式。通过调整控制顶点和权因子、可以灵活地改变曲面的形状，个别控制顶点和权因子的调整可对曲面做局部修改。越来越多的CAD／CAM…     

NURBS曲面上积分曲率线的B样条表示

对NURBS曲面的曲率线的积分进行了系统的公式推导，并利用NURBS曲面的离散法向量有效地简化了曲面第二基本量的计算，加速了Euler法迭代求解曲率线微分方程的过程；在求得曲率线上的离散点集以后，应用奇异混合插值技术，在可控精度内把曲率线用显式直接表示为位于NURBS曲面上的B样条曲线．文中的思想与算法有助于曲率线技术在计算机辅助几何设计及曲面造型中的使用与推广．     

基于PIA的B-Spline曲面实时交互修改方法

交互修改是几何设计中一种常用的曲线曲面编辑手段,NURBS曲线曲面是CAD系统中曲线曲面的标准表示形式.现有的B-spline曲线曲面交互修改技术往往需要求解一个带约束的能量优化问题,当曲线曲面的控制顶点较多时,这个优化问题的求解过程较慢,难以满足交互操作的实时要求.为此,基于B-spline曲线曲面的局部迭代逼近(P...     

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．     

NURBS曲面形状修改的一种改进算法

提出一种修改NURBS曲面形状的新算法。该算法运用约束优化方法,通过改变控制顶点和权因子修改NURBS曲面的形状。实验证明,该算法可以获得令人满意的效果。     

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.     

基于外部能量约束的曲面形状修改

提出了一个基于外部能是约束的曲面形状修改ＮＵＲＢ曲面的开矿首先给出曲面内部变形能量的近似计算公式,然后将典面形状修改所需满足的约束条件转化为相应的外部能量约束荐项,并附加在曲面的内部能量项之上,最后通过求解一个使面面能量的变化量为最小的无约束优化问题,得到变形后的曲面,使曲面总能量近亿为最小,这种方法以统一的形式处理各种不同类型的约束条件,从而简化了约束条件下曲面形状修改的计算。     

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.     

BBN曲面的形状分析与控制

研究了调节凸Bézier曲面、B-样条曲面及NURBS曲面（BBN曲面）一个控制点以后,曲面形状变化的规律.通过将BBN曲面分解成一张凸曲面与具有特殊形状曲面的叠加,建立了曲面变形前后一些几何量与变形位移量之间的数量关系,得到了凸BBN曲面失去凸性的充分条件和判据.相应的结果可应用于调节与控制BBN曲面形状的算法设计.     

NURBS曲线曲面的构形原理及其在产品造型中的应用

本文介绍了NURBS曲线、曲面的数学表达,给出了NURBS曲线和曲面的构形原理,在此基础上,推导了过滤曲面,进而获得了产品造型所要求的曲面。实践表明,应用NURBS曲面,使产品造型设计者能随心所欲地实现自己的设计意图。     

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.     

Optimized GPU evaluation of arbitrary degree NURBS curves and surfaces

This paper presents a new unified and optimized method for evaluating and displaying trimmed NURBS surfaces using the Graphics Processing Unit (GPU). Trimmed NURBS surfaces, the de facto standard in commercial mechanical CAD modeling packages, are currently being tessellated into triangles before being sent to the graphics card for display since there is no native hardware support for NURBS. Other GPU-based NURBS evaluation and display methods either approximated the NURBS patches with lower degree patches or relied on specific hard-coded programs for evaluating NURBS surfaces of different degrees. Our method uses a unified GPU fragment program to evaluate the surface point coordinates of any arbitrary degree NURBS patch directly, from the control points and knot vectors stored as textures in graphics memory. This evaluated surface is trimmed during display using a dynamically generated trim-texture calculated via alpha blending. The display also incorporates dynamic Level of Detail (LOD) for real-time interaction at different resolutions of the NURBS surfaces. Different data representations and access patterns are compared for efficiency and the optimized evaluation method is chosen. Our GPU evaluation and rendering speeds are more than 40 times faster than evaluation using the CPU.     

A practical approach to generating steepest ascent tool-paths for three-axis finish milling of compound NURBS surfaces

NURBS surfaces are commonly used in CAD/CAM software systems to represent the complex shapes of mechanical parts. Well-planned tool paths for machining the surfaces can significantly increase cutting efficiency and improve part quality. The steepest ascent tool-path pattern has been proposed for making sculptured surfaces in a 3-axis finish milling operation, and it has been proven that a steepest ascent tool-path is inherently more efficient in removing material to make these surfaces within tolerances than a tool path of any other type. However, the mathematical representation of steepest ascent paths on NURBS surfaces has not been addressed. In our work, simplified formulae of these paths are derived, and a comprehensive, efficient algorithm to plan steepest ascent tool-paths on compound NURBS surfaces is developed. To verify its validity and efficiency, this innovative approach is applied to a complicated compound surface. Furthermore, a comparison between the steepest ascent and CATIA tool-paths on two NURBS surfaces is conducted to demonstrate the advantages of the steepest ascent tool-paths for NURBS surface part production.     

Computational methods for blending surface approximation

Blending surfaces form a smooth transition between two distinct, intersecting surfaces or smoothly join two or more disconnected surfaces and are normally procedural surfaces which are difficult to exchange and to interrogate in a reliable and efficient manner. In this paper, an approximation method for blending surfaces which are curvature continuous to the underlying surfaces with a non-uniform rational B-spline (NURBS) surface is presented. The use of NURBS is important since it facilitates the exchange of geometric information between various computer aided design and manufacturing systems. In the method, linkage curves on the underlying surfaces are approximated to within a specified tolerance and cross-link curves are created using the linkage curves, a directional curve and the parametric partial derivatives of the underlying surfaces. Cross-link curves are lofted to form the blending surface and an adaptive sampling procedure is used to test the blending surface against specified tolerances. Cross-link curves are added, where necessary, and the surface relofted until the continuity conditions are satisfied to within specified tolerances. Examples illustrate the applicability of the method.     

Interactive Rendering of Deforming NURBS Surfaces

Non-uniform rational B-splines (NURBS) has been widely accepted as a standard tool for geometry representation and design. Its rich geometric properties allow it to represent both analytic shapes and free-form curves and surfaces precisely. Moreover, a set of tools is available for shape modification or more implicitly, object deformation. Existing NURBS rendering methods include de Boor algorithm, Oslo algorithm, Shantz’s adaptive forward differencing algorithm and Silbermann’s high speed implementation of NURBS. However, these methods consider only speeding up the rendering process of individual frames. Recently, Kumar et al. proposed an incremental method for rendering NURBS surfaces, but it is still limited to static surfaces. In real-time applications such as virtual reality, interactive display is needed. If a virtual environment contains a lot of deforming objects, these methods cannot provide a good solution. In this paper, we propose an efficient method for interactive rendering of deformable objects by maintaining a polygon model of each deforming NURBS surface and adaptively refining the resolution of the polygon model. We also look at how this method may be applied to multi-resolution modelling.