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。     

等弧长原则的NURBS曲线离散算法

NURBS曲线广泛应用于工业产品复杂曲线曲面设计中,但在实际应用中常遇到曲线离散的几何处理问题。针对NURBS曲线离散问题,提出了一种按等弧长原则对NURBS曲线进行离散的方法。该方法引入步长函数控制离散曲线段的弧长,采用积分法和迭代法调整步长函数以控制曲线的离散精度,通过误差检验方法校验曲线离散的逼近精度。通过实际算例,验证了NURBS曲线等弧长离散算法的合理性和有效性。     

曲线曲面局部最小二乘渐进迭代逼近

作为一种有效的大数据拟合方法，曲线曲面最小二乘渐进迭代逼近方法(LSPIA)吸引了众多研究者的关注，并获得了广泛的应用。针对LSPIA算法拟合局部数据点效果较差的问题，提出了一种局部的LSPIA算法，称为LOCAL-LSPIA。首先，给定初始曲线(曲面)并从给定的数据点中选择部分数据点；然后在初始曲线(曲面)上选择需要调整的控制点；最后，LOCAL-LSPIA通过迭代调整这一部分控制点来生成一系列局部变化的拟合曲线(曲面),并且保证生成的曲线(曲面)的极限是在仅调整这部分控制点的情况下拟合部分数据点的最小二乘结果。在多个曲线曲面拟合上的实验结果表明，为达到相同的拟合精度，LOCAL-LSPIA算法比LSPIA算法需要的步骤和运算时间更少。因此，LOCAL-LSPIA是有效的，而且在拟合局部数据的情况下比LSPIA算法的收敛速度更快。     

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．     

基于逼近理论的曲面重构方法的研究

针对三维扫描数据点的曲面重构技术在实际系统中的应用，提出了一种NURBS曲面构造方法。该方法根据已知数据点逼近目标曲面。通过实际系统应用验证，该方法是一种行之有效的曲面拟合方法。     

一种基于NURBS理论的曲面逼近优化算法

针对三维扫描数据点的曲面重构技术在实际系统中的应用,提出了一种NURBS曲面构造方法,该方法根据已知数据点逼近目标曲面.通过实际系统应用验证,该方法是一种行之有效的曲面拟合方法.     

基于最佳圆弧样条逼近的快速距离曲面计算

距离曲面是一种常用的隐式曲面,它在几何造型和计算机动画中具有重要的应用价值,但以往往在对距离曲面进行多边形化时速较慢,为了提高点到曲线最近距离计算的效率,提出了一种基于最佳圆弧样条逼近的快速线骨架距离曲面计算方法,该算法对于一条任意的二维NURBS曲线,在用户给定的误差范围内,先用最少量的圆弧样条来逼近给定的曲线,从而把点到NURBS曲线最近距离的计算问题转化为点到圆弧样条最近距离的计算问题,由于在对曲面进行多边形化时,需要大量的点到曲线最近距离的计算,而该处可以将点到圆弧样条最近距离很少的计算量来解析求得,故该算法效率很高,该实验表明,算法简单实用,具有很大的应用价值。     

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.     

实平面奇异代数曲线的全局B样条逼近

提出了一种用k次B样条曲线全局逼近实平面k次代数曲线的算法,每个连通部分用一条B样条曲线逼近.它适合于任意亏格的不可约的实平面代数曲线(包括含奇异点的曲线).这种逼近建立在所提出的代数曲线胀开采样的基础上,这种胀开采样算法从本质上解决了奇异点周围采样难的问题.实验结果表明,该方法的逼近精度高于已有算法.     

一种NURBS曲面拟合精度的量化评测算法

曲面重构是逆向工程中的核心技术之一,由于NURBS曲面在光顺性和局部可编辑等方面所具有的优点,使其成为点云数据自由曲面重构的常见形式。目前对NURBS曲面重构技术的研究上取得了一些成果,但各方法在拟合精度和效果上各有参差,因此有必要对NURBS曲面拟合精度评价算法进行研究。在采用NURBS实现曲面拟合的基础上,对拟合精度的量化指标进行研究,设计了一种基于区域划分的搜索迭代算法,可快速地计算得到原始点云与NURBS曲面的偏差。     

反求工程中的混合切片技术

提出一种基于平面与“点云”、平面与NURBS曲面求交计算的混合切片方法．该方法可以保证切片曲线在点云和曲面的连接处达到G^1连续，在此基础上的重构曲面既能保证与相邻曲面的连续性要求，又能满足对点云的逼近精度要求，对反求建模尤其是过渡特征的重建有着重要意义．文中详细探讨了平面与曲面求交和点云切片两个核心算法，并对基于模型特征的混合切片方案的选择原则以及不同方法进行了论述和比较．最后用实例证明该方法在反求建模中是切实可行的．     

平面NURBS曲线的等距线算法：圆弧法矢近似法

本文根据产生曲线的特征点与它的等距线的特征点的对应关系,给出了一种平面NURBS曲线的等距线表示方法——圆弧法矢近似法。这种方法的特点是:(1)等距线与产生曲线具有统一的NURBS表示;(2)计算简单、几何意义明确、近似精度高。     

自然地形环境下移动机器人的一种路径规划方法

本文给出了一种规划移动机器人在自然地形中运动的新方法，该方法利用ＮＵＲＢＳ曲面模拟自然地形地貌，以ＴｒｉｍｍｅｄＮＵＲＢＳ曲面描述带有障碍物或不可逾越区域的地形，在综合考虑机器人动力学、地形及障碍描述和曲面特性等各方面因素的情形下，运用测地线的概念和计算方法以及Ａ＊搜索算法，获得了在自然地形环境下任意两点间的距离最短路径和时间最优路径，所有的路径均由ＮＵＲＢＳ曲线表示，实验结果表明，该方法在性能与效率上均十分令人满意．     

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.     

Partition of unity parametrics: a framework for meta-modeling

We propose Partition of Unity Parametrics (PUPs), a natural extension of NURBS that maintains affine invariance. PUPs replace the weighted basis functions of NURBS with arbitrary weight-functions (WFs). By choosing appropriate WFs, PUPs yield a comprehensive geometric modeling framework, accounting for a variety of beneficial properties, such as local support, specified smoothness, arbitrary sharp features and approximating or interpolating curves. Additionally, we consider interactive specification of WFs to fine-tune the character of curves and generate non-trivial effects. This serves as a basis for a system where users model the tools used for modeling, here weight-functions, in tandem with the model itself, which we dub a meta-modeling system. PUP curves and surfaces are considered in detail. Curves illustrate basic concepts that apply directly to surfaces. For surfaces, the advantages of PUPs are more pronounced; permitting non-tensor WFs and direct parameter space manipulations. These features allow us to address two difficult geometric modeling problems (sketching features onto surfaces and converting planar meshes into parametric surfaces) in a conceptually and computationally simple way.     

Existence of stiffness matrix integrals for singularly parameterized domains in isogeometric analysis

Isogeometric analysis is a numerical simulation method which uses the NURBS based representation of CAD models. NURBS stands for non-uniform rational B-splines and is a generalization of the concept of B-splines. The isogeometric method uses the tensor product structure of 2-or 3-dimensional NURBS functions to parameterize domains, which are structurally equivalent to a rectangle or a hexahedron. The special case of singularly parameterized NURBS surfaces and NURBS volumes is used to represent non-quadrangular or non-hexahedral domains without splitting, which leads to a very compact and convenient representation.If the parameterization of the physical domain is available, the test functions for the isogeometric analysis are obtained by composing the inverse of the domain parameterization with the NURBS basis functions. In the case of singular parameterizations, however, some of the resulting test functions are not well defined at the singular points and they do not necessarily satisfy the required integrability assumptions. Consequently, the stiffness matrix integrals which occur in the numerical discretizations may not exist.After summarizing the basics of the isogeometric method, we discuss the existence of the stiffness matrix integrals for 1-, 2- and 3-dimensional second order elliptic partial differential equations. We consider several types of singularities of NURBS parameterizations and derive conditions which guarantee the existence of the required integrals. In addition, we present cases with diverging integrals and show how to modify the test functions in these situations.     

空间代数曲线的参数化逼近

提出一种用三次B样条曲线逼近空间代数曲线的方法.对非奇异的情况,先用随机微分方程方法采样,然后对采样点进行聚类排序,最后用三次B样条曲线逼近有序点列;而对包含奇异点的情况,则将空间曲线双有理映射成平面曲线,采用已有的含奇异点的平面代数曲线的采样及排序方法来实现对应空间曲线的采样及排序.两种情况都获得了优于其它方法的逼近效果.     

Direct boolean intersection between acquired and designed geometry

In this paper, a new shape modeling approach that can enable direct Boolean intersection between acquired and designed geometry without model conversion is presented. At its core is a new method that enables direct intersection and Boolean operations between designed geometry (objects bounded by NURBS and polygonal surfaces) and scanned geometry (objects represented by point cloud data).We use the moving least-squares (MLS) surface as the underlying surface representation for acquired point-sampled geometry. Based on the MLS surface definition, we derive closed formula for computing curvature of planar curves on the MLS surface. A set of intersection algorithms including line and MLS surface intersection, curvature-adaptive plane and MLS surface intersection, and polygonal mesh and MLS surface intersection are successively developed. Further, an algorithm for NURBS and MLS surface intersection is then developed. It first adaptively subdivides NURBS surfaces into polygonal mesh, and then intersects the mesh with the MLS surface. The intersection points are mapped to the NURBS surface through the Gauss-Newton method.Based on the above algorithms, a prototype system has been implemented. Through various examples from the system, we demonstrate that direct Boolean intersection between designed geometry and acquired geometry offers a useful and effective means for the shape modeling applications where point-cloud data is involved.     

D-NURBS: a physics-based framework for geometric design

Presents dynamic non-uniform rational B-splines (D-NURBS), a physics-based generalization of NURBS. NURBS have become a de facto standard in commercial modeling systems. Traditionally, however, NURBS have been viewed as purely geometric primitives, which require the designer to interactively adjust many degrees of freedom-control points and associated weights-to achieve the desired shapes. The conventional shape modification process can often be clumsy and laborious. D-NURBS are physics-based models that incorporate physical quantities into the NURBS geometric substrate. Their dynamic behavior, resulting from the numerical integration of a set of nonlinear differential equations, produces physically meaningful, and hence intuitive shape variation. Consequently, a modeler can interactively sculpt complex shapes to required specifications not only in the traditional indirect fashion, by adjusting control points and setting weights, but also through direct physical manipulation, by applying simulated forces and local and global shape constraints. We use Lagrangian mechanics to formulate the equations of motion for D-NURBS curves, tensor-product D-NURBS surfaces, swung D-NURBS surfaces and triangular D-NURBS surfaces. We apply finite element analysis to reduce these equations to efficient numerical algorithms computable at interactive rates on common graphics workstations. We implement a prototype modeling environment based on D-NURBS and demonstrate that D-NURBS can be effective tools in a wide range of computer-aided geometric design (CAGD) applications