细分法求解点投影问题时的剪枝算法

点到NURBS曲线/曲面的投影是CAD中的基础问题.为了避免数值迭代法陷入局部最优,通常把曲线/曲面细分为多段.文中提出排除无用曲线段/曲面片的剪枝算法,以提高细分求解投影的运行效率.在投影算法中将曲线/曲面递归细分,分别计算测试点到各个曲线段/曲面片的投影值;在递归过程中,对当前曲线段/曲面片计算测试点到控制点凸包距离的近似距离,如果该距离大于投影距离上界值,根据NURBS曲线/曲面的凸包性,测试点到当前曲线段/曲面片的投影距离也一定大于上界值,无需计算投影结果,直接排除该曲线/曲面.最后通过测试实例验证了该算法的有效性.     

NURBS边界曲面直接生成法

由于非均匀有理B样条(NURBS)曲面的复杂性,传统NURBS边界曲面的生成是先构造孔斯曲面,再由孔斯曲面向NURBS曲面转换得到,其操作过程比较烦琐。针对此问题,提出了NURBS边界曲面直接生成算法,该算法根据给定的四条NURBS边界曲线,结合孔斯曲面生成方法直接插值生成NURBS曲面,从而避免了通过孔斯曲面向NURBS曲面转换所带来的计算代价,因此同传统方法相比,具有较低的计算代价。实验结果表明:该算法简化了曲面生成步骤,减少了曲面转换过程的计算量,生成的曲面边界信息明确,且连续性好。     

基于升阶矩阵的有理曲面之间L2距离计算

计算曲线曲面之间的距离是几何设计与几何逼近的一个重要课题,如估计有理曲线曲面的降阶逼近和多项式逼近的误差时,需要一种简洁有效的方法来计算原曲线曲面和逼近曲线曲面间的距离.首先给出了基于升阶矩阵的两张有理Bézier曲面的L2距离表示,然后利用这个L2距离表示和最小二乘法,对有理Bézier曲面多项式逼近的误差作了明确而统一的度量.最后,基于Bernstein基与B样条基的相互转换,把有理Bézier曲线曲面的L2距离表示简洁地推广到有理B样条曲线曲面.所得到的几个计算曲线曲面之间的L2距离的公式均可通过矩阵运算表示,十分利于程序的实现,有应用价值.最后还给了几个实例.     

Bézier曲线/曲面间最近距离的几何裁剪算法

曲线、曲面间距离的计算问题在CAD/CAM、计算机图形学中有着广泛的应用.为了精确计算Bézier曲线/曲面间的最近距离,结合稳定的曲线、曲面分裂技术提出一种基于offset滚动球裁剪的几何算法.首先给出判定条件来裁剪去落在曲面的滚动球外的曲线段,或者落在曲线的滚动球外的曲面片,以摒弃大部分不包含最近点的曲线段或曲面片,为后续可能的Newton方法提供较好的初始点;然后给出判定最近点是否落在曲线的端点或曲面的边界曲线上的条件,将曲线/曲面间的距离计算问题转化为点/曲面或曲线/曲线间的距离计算问题,简化了问题的复杂度,提高了计算效率.实例结果表明,文中算法具有较好的稳定性和较高的效率.     

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.     

基于法矢控制的B样条曲面逼近的PIA方法

提出了一种基于法矢控制的 B 样条曲面逼近的渐进迭代逼近(PIA)算法。一方面该方法将离散数据点的切失、曲率、法矢等几何特征充分应用到离散数据点的逼近问题上,利用数据点两个方向的切矢构造出数据点的法矢约束来控制逼近曲面形状,相比于无法矢控制的 B 样条曲面逼近的渐进迭代逼近(PIA)方法,逼近曲面更光顺,可获得更好的逼近效果。另一方面由于该算法选取主特征点作为控制顶点,所以允许在曲面拟合中控制顶点的数目小于数据点的数目。而且PIA算法的每次迭代过程中的各个步骤都是独立的,很容易被应用到并行计算上,可提高计算效率。本文还给出了一些实例来验证该算法的有效性。     

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.     

计算点到平面NURBS曲线的最小距离

提出了一种利用控制顶点投影法计算点到平面NURBS曲线最小距离的算法。将控制顶点投影到平面NURBS曲线上,投影点将平面NURBS曲线分割成若干曲线段;计算点到各曲线段的最小距离,通过比较这些最小距离值,找出点到平面NURBS曲线的最小距离。该算法实现简单,计算精度较高,具有重要的实用价值。     

等距面的构造新方法

本文提出了一种新的等距面构造方法。其核心思想是：根据给定的曲面，在空间点到该曲面的最小距离的数据场内，利用Marching Cube算法，抽取到该曲面的最小距离为等距距离d的等值面（三角面片的集合）。在构成等值面的三角面片的集合中，去掉到给定曲面最短距离为d且是在曲面边界取得最小距离的三角面片，剩下的三角面片的集合即为两个原曲面的用三角面片逼近的等距面。这种方法最大的优点是自动克服了构造等距面过程中出现的自相交现象。     

Convolution surfaces for arcs and quadratic curves with a varying kernel

A convolution surface is an isosurface in a scalar field defined by convolving a skeleton, comprising of points, curves, surfaces, or volumes, with a potential function. While convolution surfaces are attractive for modeling natural phenomena and objects of complex evolving topology, the analytical evaluation of integrals of convolution models still poses some open problems. This paper presents some novel analytical convolution solutions for arcs and quadratic spline curves with a varying kernel. In addition, we approximate planar higher-degree polynomial spline curves by optimal arc splines within a prescribed tolerance and sum the potential functions of all the arc primitives to approximate the field for the entire spline curve. Published online: November 20, 2002     

Curve fitting and fairing using conic splines

We present an efficient geometric algorithm for conic spline curve fitting and fairing through conic arc scaling. Given a set of planar points, we first construct a tangent continuous conic spline by interpolating the points with a quadratic Bézier spline curve or fitting the data with a smooth arc spline. The arc spline can be represented as a piecewise quadratic rational Bézier spline curve. For parts of the G¹ conic spline without an inflection, we can obtain a curvature continuous conic spline by adjusting the tangent direction at the joint point and scaling the weights for every two adjacent rational Bézier curves. The unwanted curvature extrema within conic segments or at some joint points can be removed efficiently by scaling the weights of the conic segments or moving the joint points along the normal direction of the curve at the point. In the end, a fair conic spline curve is obtained that is G² continuous at convex or concave parts and G¹ continuous at inflection points. The main advantages of the method lies in two aspects, one advantage is that we can construct a curvature continuous conic spline by a local algorithm, the other one is that the curvature plot of the conic spline can be controlled efficiently. The method can be used in the field where fair shape is desired by interpolating or approximating a given point set. Numerical examples from simulated and real data are presented to show the efficiency of the new method.     

Efficient circular arc interpolation based on active tolerance control

In this paper, we present an efficient sub-optimal algorithm for fitting smooth planar parametric curves by G¹ arc splines. To fit a parametric curve by an arc spline within a prescribed tolerance, we first sample a set of points and tangents on the curve adaptively as well as with enough density, so that an interpolation biarc spline curve can be with any desired high accuracy. Then, we construct new biarc curves interpolating local triarc spirals explicitly based on the control of permitted tolerances. To reduce the segment number of fitting arc spline as much as possible, we replace the corresponding parts of the spline by the new biarc curves and compute active tolerances for new interpolation steps. By applying the local biarc curve interpolation procedure recursively and sequentially, the result circular arcs with no radius extreme are minimax-like approximation to the original curve while the arcs with radius extreme approximate the curve parts with curvature extreme well too, and we obtain a near optimal fitting arc spline in the end. Even more, the fitting arc spline has the same end points and end tangents with the original curve, and the arcs will be jointed smoothly if the original curve is composed of several smooth connected pieces. The algorithm is easy to be implemented and generally applicable to circular arc interpolation problem of all kinds of smooth parametric curves. The method can be used in wide fields such as geometric modeling, tool path generation for NC machining and robot path planning, etc. Several numerical examples are given to show the effectiveness and efficiency of the method.     

Error-bounded biarc approximation of planar curves

Presented in this paper is an error-bounded method for approximating a planar parametric curve with a G¹ arc spline made of biarcs. The approximated curve is not restricted in specially bounded shapes of confined degrees, and it does not have to be compatible with non-uniform rational B-splines (NURBS). The main idea of the method is to divide the curve of interest into smaller segments so that each segment can be approximated with a biarc within a specified tolerance. The biarc is obtained by polygonal approximation to the curve segment and single biarc fitting to the polygon. In this process, the Hausdorff distance is used as a criterion for approximation quality. An iterative approach is proposed for fitting an optimized biarc to a given polygon and its two end tangents. The approach is robust and acceptable in computation since the Hausdorff distance between a polygon and its fitted biarc can be computed directly and precisely. The method is simple in concept, provides reasonable accuracy control, and produces the smaller number of biarcs in the resulting arc spline. Some experimental results demonstrate its usefulness and quality.     

基于拓扑相似性的等距参数曲面求交算法

等距曲面求交算法通常采用曲面求交算法反复迭代计算交线,没有考虑不同Offset距离等距曲面交线的相似性进行求交简化.提出了一种基于拓扑相似性的等距曲面求交优化算法.算法首先求取曲面的拓扑特征点,根据拓扑特征点分布图,确定交线环拓扑结构,在交线拓扑结构信息的指导下,确定初始点的搜索策略.采用提出的方法可以有效解决等距曲面的子环、奇点遗漏、分支跳跃、乱序跟踪和初始点求取问题,精确、鲁棒地计算出交线.     

Compact implicit surface reconstruction via low-rank tensor approximation

Implicit representations have gained an increasing popularity in geometric modeling and computer graphics due to their ability to represent shapes with complicated geometry and topology. However, the storage requirement, e.g. memory or disk usage, for implicit representations of complex models is relatively large. In this paper, we propose a compact representation for multilevel rational algebraic spline (MRAS) surfaces using low-rank tensor approximation technique, and exploit its applications in surface reconstruction. Given a set of 3D points equipped with oriented normals, we first fit them with an algebraic spline surface defined on a box that bounds the point cloud. We split the bounding box into eight sub-cells if the fitting error is greater than a given threshold. Then for each sub-cell over which the fitting error is greater than the threshold, an offset function represented by an algebraic spline function of low rank is computed by locally solving a convex optimization problem. An algorithm is presented to solve the optimization problem based on the alternating direction method of multipliers (ADMM) and the CANDECOMP/PARAFAC (CP) decomposition of tensors. The procedure is recursively performed until a certain accuracy is achieved. To ensure the global continuity of the MRAS surface, quadratic B-spline weight functions are used to blend the offset functions. Numerous experiments show that our approach can greatly reduce the storage of the reconstructed implicit surface while preserve the fitting accuracy compared with the state-of-the-art methods. Furthermore, our method has good adaptability and is able to produce reconstruction results with high quality.     

Fast and versatile algorithm for nearest neighbor search based on a lower bound tree

In this paper, we present a fast and versatile algorithm which can rapidly perform a variety of nearest neighbor searches. Efficiency improvement is achieved by utilizing the distance lower bound to avoid the calculation of the distance itself if the lower bound is already larger than the global minimum distance. At the preprocessing stage, the proposed algorithm constructs a lower bound tree (LB-tree) by agglomeratively clustering all the sample points to be searched. Given a query point, the lower bound of its distance to each sample point can be calculated by using the internal node of the LB-tree. To reduce the amount of lower bounds actually calculated, the winner-update search strategy is used for traversing the tree. For further efficiency improvement, data transformation can be applied to the sample and the query points. In addition to finding the nearest neighbor, the proposed algorithm can also (i) provide the k-nearest neighbors progressively; (ii) find the nearest neighbors within a specified distance threshold; and (iii) identify neighbors whose distances to the query are sufficiently close to the minimum distance of the nearest neighbor. Our experiments have shown that the proposed algorithm can save substantial computation, particularly when the distance of the query point to its nearest neighbor is relatively small compared with its distance to most other samples (which is the case for many object recognition problems).     

具有多项式密度分布的直线骨架卷积曲面

给出了一个具有多项式密度分布的直线骨架卷积曲面的解析表达式，并提出了基于控制曲线的密度控制方法，实验结果表明，该方法的自然景物，海生物等光滑物体造型中具有很大的应用价值。     

Algorithms on cone spline surfaces and spatial osculating arc splines

Developable surfaces are of considerable importance to many industrial applications, e.g., sheet metal forming processes. The objective of this paper is to provide algorithms on the approximation of developable surfaces with pieces of right circular cones. Special emphasis is devoted to practical choices of free parameters and to error estimation. Furthermore, a new algorithm for the approximation of spatial curves with a circular arc spline is presented which stands in close relation to above algorithms on developable surfaces. The proposed arc spline has contact of order 2 to the given curve in a series of curve points. The investigation includes a segmentation algorithm and error estimation.     

Cone spline approximation via fat conic spline fitting

Fat conic section and fat conic spline are defined. With well established properties of fat conic splines, the problem of approximating a ruled surface by a tangent smooth cone spline can then be changed as the problem of fitting a plane fat curve by a fat conic spline. Moreover, the fitting error between the ruled surface and the cone spline can be estimated explicitly via fat conic spline fitting. An efficient fitting algorithm is also proposed for fat conic spline fitting with controllable tolerances. Several examples about approximation of general developable surfaces or other types of ruled surfaces by cone spline surfaces are presented.