三次B样条曲线骨架卷积曲面造型

提出一种基于B样条曲线降阶的三次B样条曲线骨架卷积曲面造型方法．首先通过顶点扰动降阶方法把三次B样条曲线骨架（C^1连续）降阶为C^1连续的二次B样条，然后应用二次B样条曲线骨架的卷积曲面势函数计算方法得到三次B样条曲线骨架的势函数．     

一种简单的二次B样条曲线拟合算法

针对双圆弧拟合算法绘制一条B样条曲线,需要反复多次计算各坐标分量的3次多项式,计算量大,绘制拟合速度极慢,难以满足实际需要等情况,该算法提出了一种简单的二次B样条曲线拟合算法,该算法提高了B样条曲线的绘制速度,有效地解决了4个点以上控制点的拟合问题。     

三次B样条曲线的重合判断算法

曲线和曲线求交计算是CAGD领域的一个基本问题,但现有的求交算法都无法处理曲线重合的情况.在2条三次Bézier曲线重合判断条件的基础上,提出一种判断2条三次B样条曲线是否重合的算法.对于每条B样条曲线,首先将其分割成若干Bézier曲线段,然后判断2条Bézier曲线段是否可以合并为一段;通过合并Bézier曲线段,将2条三次B样条曲线的重合判断问题转化为2组三次Bézier曲线段的重合判断问题.文中在理论上证明了该算法的正确性,并通过若干实例验证了其有效性.     

数控加工中的二次曲线拟合与最优插补控制算法

本文提出一种基于二次B样条曲线对G01代码的拟合及插补方法,先通过自适应方法选取G01代码所描述的加工轨迹形状特征的各个特征点,再用通过所有特征点的二次B样条曲线拟合待加工路径.由于Timar等人提出的最优速度规划算法对于三次及三次以上的高阶次样条曲线的计算极为复杂,本文提出了一种改进的基于二次B样条曲线的时间最优插补计算方法.最后,将所提出的算法应用到实际的图案加工仿真与实际数控加工中.     

二次均匀B样条曲线的双圆弧逼近方法*

提出了一种用双圆弧对二次均匀B样条曲线的分段逼近方法。首先,对一条具有n 1个控制顶点的二次均匀B样条曲线按照相邻两节点界定的区间分成n-1段只有三个控制顶点的二次均匀B样条曲线段;然后对每一曲线段构造一条双圆弧进行逼近。所构造的双圆弧满足端点及端点切向量条件,即双圆弧的两个端点分别是所逼近的曲线段的端点,而且双圆弧在两个端点处的切向量是所逼近的曲线段在端点处的单位切向量。同时,双圆弧的连接点是双圆弧连接点轨迹圆与其所逼近的曲线段的交点。这些新构造出来的双圆弧连接在一起构成了一条圆弧样条曲线,即二次均匀B样条曲线的逼近曲线。另外给出了逼近误差分析和实例说明。     

基于双正交非均匀B样条小波的曲线逼近方法

针对B样条曲线逼近有序数据点在应用最小二乘法时出现的计算量较大问题,提出一种基于双正交非均匀B样条小波的曲线逼近方法。其基本思想是：先用最小二乘法生成初始B样条逼近曲线,再用细节曲线逼近误差向量,接着将细节曲线叠加于原逼近曲线得到新的B样条曲线,这个过程是迭代的。细节曲线的基函数是双正交非均匀B样条小波。与传统最小二乘法相比,该方法仅需计算新增线性系统,避免重复计算原系统,降低了计算量,提高了运算效率;此外,给出了B样条逼近曲线的一种多分辨率表示形式。     

曲率连续的三角B样条曲线与曲面

给出了一种二次参数三角样条曲线,基函数由一组特殊的二次三角多项式组成;曲线的每一段由三个控制顶点生成,不仅具有二次均匀B样条曲线的端点性质,而且具有更好的逼近性、整体达到曲率连续。该曲线(面)可用于曲线曲面的造型。     

二次均匀B样条曲线的扩展

为便于对均匀B样条曲线进行形状修改,利用二次均匀B样条基函数所需满足的条件,扩展二次均匀B样条基函数,构造出三次多项式调配函数．基于给出的调配函数,建立1种带形状参数的分段多项式曲线．调整形状参数可使三次多项式曲线在二次均匀B样条曲线两侧摆动．最后给出实例,构造出带局部调节参数G^1的连续曲线．该方法可以通过调整参数扩大二次均匀B样条曲线的调整范围．     

两种带形状参数的三角曲线

定义了带形状参数的三次三角多项式曲线和三次三角样条曲线。前者具有 与二次Bézier 曲线类似的端点性质,但逼近性比二次Bézier 曲线更好,且在拼接时能达到 更高阶的连续性。而后者与二次B 样条曲线类似,其每一段由相继的三个控制顶点生成。 对于等距节点,在一般情况下曲线C2 连续,在特殊条件下可达C3 连续。     

任意个控制点的三次B样条曲线绘制系统设计

三次B样条曲线在实际工程中得到了广泛应用,在目前的文献中还未查到任意个控制点的三次B样条曲线绘制系统。为了使B样条曲线得到更广泛应用和方便《计算机辅助几何设计》课程关于B样条曲线的可视化教学,本文用MatlabGUI编程设计三次B样条曲线绘制系统。此系统界面简洁,使用方便。可用鼠标在屏幕坐标系上点击输入任意个控制点,也可用文件方式输入任意个控制点。用矩阵描述B样条曲线方程,实现快速计算并绘制出光滑的三次B样条曲线。     

隐式B-样条曲线重建的直接Greville纵标法

提出了一种以隐式B-样条曲线为表达形式,基于直接Greville纵标的曲线重建方法。根据点云建立有向距离场,并作为B-样条函数的Greville纵标,然后根据高影响区内的平均代数误差优化Greville纵标;得到一个隐式B-样条函数,该函数的零点集即为重建曲线。该方法具有模型简单,重建速度快,无多余分支,无需手工调节任何参数的优点。实验结果证实了该直接法的效率明显高于点拟合法和普通场拟合法,以几何误差为准则的精度亦优于普通场拟合方法。     

基于安德森加速的快速 B 样条拟合算法

曲线拟合技术已被广泛地应用于图像处理、工程实验等领域。其中,B 样条曲线拟 合是曲线拟合中最常见的方法,它具有局部性好、连续性好等优点,但拟合精度一般较低。在实 际应用中,B 样条曲线拟合对于精度和速度的要求都较高。为了提升平面 B 样条曲线拟合速度, 将安德森加速的想法应用到曲线拟合的方法之中,提出一种基于安德森加速的拟牛顿方法。首先 设定一个初始形状,然后根据初始形状找到其每个数据点的投影点的位置参数,然后利用安德森 加速计算出控制点的相应位置,迭代进行以上 2 步,直到结果收敛。实验结果表明,该方法在收 敛速度和迭代时间上均优于其他方法。     

B-spline surface fitting based on adaptive knot placement using dominant columns

By expanding the idea of B-spline curve fitting using dominant points (Park and Lee 2007 [13]), we propose a new approach to B-spline surface fitting to rectangular grid points, which is based on adaptive knot placement using dominant columns along u- and v-directions. The approach basically takes approximate B-spline surface lofting which performs adaptive multiple B-spline curve fitting along and across rows of the grid points to construct a resulting B-spline surface. In multiple B-spline curve fitting, rows of points are fitted by compatible B-spline curves with a common knot vector whose knots are computed by averaging the parameter values of dominant columns selected from the points. We address how to select dominant columns which play a key role in realizing adaptive knot placement and thereby yielding better surface fitting. Some examples demonstrate the usefulness and quality of the proposed approach.     

Shape modeling based on specifying the initial B-spline curve and scaled BFGS optimization method

In this paper, we consider the problem of fitting the B-spline curves to a set of ordered points, by finding the control points and the location parameters. The presented method takes two main steps: specifying initial B-spline curve and optimization. The method determines the number and the position of control points such that the initial B-spline curve is very close to the target curve. The proposed method introduces a length parameter in which this allows us to adjust the number of the control points and increases the precision of the initial B-spline curve. Afterwards, the scaled BFGS algorithm is used to optimize the control points and the foot points simultaneously and generates the final curve. Furthermore, we present a new procedure to insert a new control point and repeat the optimization method, if it is necessary to modify the fitting accuracy of the generated B-spline fitting curve. Associated examples are also offered to show that the proposed approach performs accurately for complex shapes with a large number of data points and is able to generate a precise fitting curve with a high degree of approximation.     

基于遗传算法的B样条曲线和Bézier曲线的最小二乘拟合

考虑用B样条曲线拟合平面有序数据使得最小二乘拟合误差最小.一般有两种考虑,一种是保持B样条基函数的节点不变,选择参数使得拟合较优.参数的选择方法包括均匀取值、累加弦长法、centripetal model、Gauss-Newton迭代法等.另一种则是先确定好参数值(一般用累加弦长法),然后再用.某一算法计算出节点,使得拟合较优.同时把两者统一考虑,用遗传算法同时求出参数、节点使得拟合在最小二乘误差意义下最优.与Gauss-Newton迭代法、Piegl算法相比,本方法具有较好的鲁棒性(拟合曲线与初始值无关)、较高的精度及控制顶点少等优点.实验结果说明采用遗传算法得到的曲线逼近效果更好.用遗传算法对Bezier曲线拟合平面有序数据也进行了研究.     

基于B样条隶属函数的模糊推理系统

隶属函数和推理规则的确定是模糊推理的难点。通过研究模糊推理过程和B样条函数的特性,对应用B样条函数拟合模糊隶属函数进行推理的方法进行改进。通过对误差极值点、曲率极值点的计算和筛选,得到B样条函数的型值点。反算求得控制点之后,通过自适应增加控制点对曲线进行调整,增加曲线对隶属函数的拟合度,解决了B样条函数对隶属函数的拟合问题。建立B样条推理规则,构造实现了B样条推理系统,并求出该系统的最终结果为B样条超曲面。最后,通过实验验证了该方法的有效性和可行性。     

Contour curve reconstruction from cloud data for rapid prototyping

In this study, a method for generation of sectional contour curves directly from cloud point data is given. This method computes contour curves for rapid prototyping model generation via adaptive slicing, data points reducing and B-spline curve fitting. In this approach, first a cloud point data set is segmented along the component building direction to a number of layers. The points are projected to the mid-plane of the layer to form a 2-dimensional (2D) band of scattered points. These points are then utilized to construct a boundary curve. A number of points are picked up along the band and a B-spline curve is fitted. Then points are selected on the B-spline curve based on its discrete curvature. These are the points used as centers for generation of circles with a user-define radius to capture a piece of the scattered band. The geometric center of the points lying within these circles is treated as a control point for a B-spline curve fitting that represents a boundary contour curve. The advantage of this method is simplicity and insensitivity to common small inaccuracies. Two experimental results are included to demonstrate the effectiveness and applicability of the proposed method.     

An error-bounded B-spline curve approximation scheme using dominant points for CNC interpolation of micro-line toolpath

This paper presents a unified framework for computing a B-spline curve to approximate the micro-line toolpath within the desired fitting accuracy. First, a bi-chord error test extended from our previous work is proposed to select the dominant points that govern the overall shape of the micro-line toolpath. It fully considers the geometric characteristics of the micro-line toolpath, i.e., the curvature, the curvature variation and the torsion, appropriately determining the distribution of the dominant points. Second, an initial B-spline curve is constructed by the dominant points in the least square sense. The fitting error is unpredictable and uncontrollable. It is classified into two types: (a) the geometric deviations between the vertices of the polygon formed by the data points and the constructed B-spline curve; (b) those between the edges of the polygon and the constructed B-spline curve. Herein, an applicable dominant point insertion is employed to keep the first geometric deviation within the specified tolerance of fitting error. A geometric deviation model extended from our previous work is developed to estimate the second geometric deviation. It can be effectively integrated into global toolpath optimization. Computational results demonstrate that the bi-chord error test applies to both the planar micro-line toolpath and the spatial micro-line toolpath, and it can greatly reduce the number of the control points. Simulation and experimental results demonstrate that the proposed B-spline approximation approach can significantly improve machining efficiency while ensuring the surface quality.     

基于遗传算法的B样条曲线在气象探测中的应用

针对高空气象探测数据变化规律复杂、突变情况不可预测、数据量大等特点,采用基于遗传算法确定节点矢量的B样条曲线拟合方法,并提出优化染色体的产生方式,加速算法的收敛效率,实现了在给定误差要求下,用较少控制点的B样条曲线拟合高空气象探测数据曲线,并通过对气温－高度曲线特征点的拟合效果证明了算法的可行性。     

基于差分进化算法的B 样条曲线曲面拟合

应用B 样条曲线曲面拟合内在形状带有间断或者尖点的数据时,最小二乘法得到的 拟合结果往往在间断和尖点处误差较大,原因在于最小二乘法将拟合函数B 样条的节点固定。本 文在利用3 次B 样条曲线和曲面拟合数据时,应用差分进化算法设计出一种能够自适应地设置B 样条节点的方法,同时对节点的数量和位置进行优化,使得B 样条拟合曲线曲面在间断和尖点处 产生拟多重节点,实现高精度地拟合采样于带有间断或尖点的曲线和曲面数据。