基于B样条的平面轮廓重构闭合曲面算法

由一组平行轮廓线重构三维闭合表面是三维可视化研究的主要内容之一。文中通过对B样条插值算法的研究,提出了一种新的公共节点矢量确定方法,利用该方法首先对经过预处理的CT牙齿图片提取轮廓线获得三维数据点,之后对轮廓线数据点进行B样条曲线的拟合,在每条拟合曲线上根据所确定的节点矢量值重新采样,由重新采样的三维数据点利用B样条曲面插值算法构造闭合曲面．所构造的闭合曲面是对原始轮廓数据的拟合。通过实例验证可看出该方法可获得较好的拟合曲面,经过误差分析检测,满足拟合条件,因此该方法可以保证几何重建的准确性。     

基于B样条的平面轮廓重构闭合曲面算法

由一组平行轮廓线重构三维闭合表面是三维可视化研究的主要内容之一.文中通过对B样条插值算法的研究,提出了一种新的公共节点矢量确定方法,利用该方法首先对经过预处理的CT牙齿图片提取轮廓线获得三维数据点,之后对轮廓线数据点进行B样条曲线的拟合,在每条拟合曲线上根据所确定的节点矢量值重新采样,由重新采样的三维数据点利用B样条曲面插值算法构造闭合曲面,所构造的闭合曲面是对原始轮廓数据的拟合.通过实例验证可看出该方法可获得较好的拟合曲面,经过误差分析检测,满足拟合条件,因此该方法可以保证几何重建的准确性.     

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。     

改变B样条曲面与NURBS曲面的节点

通过改变k×h阶B样条曲面和NURBS(Non UniformRationalB spline)曲面的若干节点,分别产生一个B样条曲面族和NURBS曲面族,并指出:曲面族的包络是用相同控制顶点定义的(k -a)×(h -b)阶B样条曲面和NURBS曲面,其中a ,b分别是两个方向上所改变的节点的重数 对于B样条曲面来说,曲面族与其包络的任意阶相同偏微分之间只相差一个因子,文中所得结果可以作为计算机辅助设计系统中曲面造型和形状修改的理论参考     

二次B样条曲面顶点及法向插值

顶点位置插值是自由曲面造型的基本方法,法向插值在一些CAD／CAM系统中也有重要应用,文中利用子分曲面理论研究双二次B样条曲面的性质,在此基础上利用Doo-Sabin分子模式构造插值顶点位置和法向的双二次B样条曲面控制网络,得到插值曲面的参数表示,为了提高效率,对规模较大的网络数据,先把它成若干片子网格,分别求出满足子网格相关的插值条件的控制网格,最后再反它们整合在一起形成完整的控制网格,使得相应的二次B样条曲面插值所有顶点及法向。     

CATIA建模教程(三)——扫掠曲面在曲面造型中的应用

在曲面造型中,通常是首先进行线框造型得到一系列的线条,再进行曲面造型构建基本曲面,最后还需要对这些基本曲面进行裁剪、圆角过渡等处理.扫掠曲面就是一种重要的曲面类型,在曲面造型中具有相当广泛的应用.扫掠曲面是以若干线条为截面线(可以看作是纬线),以另外若干条线条为导引线(可以看作是经线),截面线沿着导引线移动,形成了一张曲面.     

基于曲率指导的B样条飞机进气道几何设计

进气道能够捕获来流,按一定流量、流速和畸变向发动机供气.作为飞机推进系统的起始部件,进气道直接影响飞机发动机的工作效率、性能和安全性.针对进气道自由曲面构型的特点,提出基于曲率指导的B样条进气道几何设计方法.首先给定控制点、节点数据,构造C²光滑约束的B样条关键截线,避免产生台阶效应而影响进气道性能;其次将计算过渡截线的离散点坐标,转化为利用过渡截线导数值反求所对应的B样条曲线,减小在复原曲线过程中计算离散点带来的累积误差;最后将上述B样条曲线拟合为B样条张量积曲面结构,减少格式转换造成的信息损失.该方法已集成在某型号进气道建模系统中,可快速、交互式地完成由任意进气口截线到出气口截线的光滑过渡.实验结果表明,建模得到的进气道曲面结构精确性较高、通用性较好,一定程度上减少了对国外相关CAD航空工业软件的依赖,可以为后续基于数值优化的几何设计与流体仿真一体化提供技术支撑.     

高阶连续的形状可调三角多项式曲线曲面

目的目前使用的B样条曲线曲面存在着高连续阶与高局部调整性两者无法兼而有之的不足,且B样条曲线曲面的形状被控制顶点和节点向量唯一确定,这些因素影响着B样条方法的几何设计效果与方便性。本文旨在克服这种局限,以期构造具有高次B样条方法的高连续阶,低次B样条方法的高局部调整性,以及有理B样条方法权因子决定的形状调整性的曲线曲面。方法在三角函数空间上构造了一组含参数的调配函数,进而定义具有与3次B样条曲线曲面相同结构的新曲线与张量积曲面。结果新曲线曲面继承了B样条方法的凸包性、对称性、几何不变性等诸多性质。不同的是,同样是基于4点分段,3次均匀B样条曲线C2连续,而对于等距节点,在一般情况下,新曲线C5连续,当参数取特殊值时可达C7连续。新曲线在C5连续的情况下存在1个形状参数,能较好地调整曲线的形状同时又无须改变控制顶点。另外,将形状参数设为特定值,新曲线可以自动插值给定点列。新曲面具有与新曲线相应的优点。结论在强局部性下实现高阶连续性的形状可调分段组合曲线曲面,为高阶光滑曲线曲面的设计提供了可能,并且新曲线实现了逼近与插值的统一表示,能较好地应用于工程实际。调配函数的构造方法具有一般性,可用相同方式构造其他具有类似性质的调配函数。     

基于曲率优化的机器人砂带磨削轨迹规划

针对复杂曲面零件砂带磨削编程效率低、精度差的问题,基于B样条曲线曲面重构和机器人离线编程技术,提出了一种根据关键接触点曲率值生成工业机器人磨削轨迹的方法.首先,利用零件表面上需要进行砂带磨削的关键接触点和积累弦长参数化法构造节点矢量,从而计算出磨削轨迹的B样条基函数;其次,根据控制顶点反求矩阵得到全部未知控制点和3次B样条加工曲线;然后,分析关键接触点之间的曲率变化率和弧长,对关键点细化生成符合磨削工艺要求的目标点;最后,通过求解双3次B样条插值曲面方程获得目标点的加工姿态.以水龙头磨削为例进行试验,结果表明曲率优化算法磨削的零件表面轮廓形状明显优于截面法,且其粗糙度值能稳定在0.082 μm左右,可以有效提高工件表面加工质量.     

集逼近插值于一体的分段3次多项式曲线曲面

为了用一种模型实现逼近与插值的统一,在多项式函数空间上构造了含两组参数的混合函数,并由之定义了基于四点分段的多项式曲线和相应的张量积曲面。当参数取特殊值时,新曲线曲面成为3次均匀B样条曲线曲面。除了继承B样条方法的局部性,自动光滑性等优点之外,新曲线曲面还具有局部形状可调性。限制混合函数中参数的取值范围,可以使新曲线曲面位于控制顶点的凸包内。让混合函数中的一组参数取特定值,可以使新曲线曲面自动插值除边界点以外的控制顶点,且插值曲线曲面的形状依然局部可调。给出了一些曲线曲面图例。     

Reducing control points in lofted B-spline surface interpolation using common knot vector determination

A new algorithm for reducing control points in lofted surface interpolation to rows of data points is presented in this paper. The key step of surface lofting is to obtain a set of compatible B-spline curves interpolating each row. Given a set of points and their parameterization, a necessary and sufficient condition is proposed to determine the existence of interpolating B-spline curves defined on a given knot vector. Based on this condition, we first properly construct a common knot vector that guarantees the existence of interpolating B-spline curves to each row of points. Then we calculate a set of interpolating B-spline curves defined on the common knot vector by energy minimization. Using this method, fewer control points are employed while maintaining a visually pleasing shape of the lofted surface. Several experimental results demonstrate the usability and quality of the proposed method.     

Lofted B-spline surface interpolation by linearly constrained energy minimization

This paper proposes a new approach for lofted B-spline surface interpolation to serial contours, where the number of points varies from contour to contour. The approach first finds a common knot vector consisting of fewer knots that contain enough degrees of freedom to guarantee the existence of a B-spline curve interpolating each contour. Then, it computes from the contours a set of compatible B-spline curves defined on the knot vector by adopting B-spline curve interpolation based on linearly constrained energy minimization. Finally, it generates a B-spline surface interpolating the curves via B-spline surface lofting. As the energy functional is quadratic, the energy minimization problem leads to that of solving a linear system. The proposed approach is efficient in computation and can realize more efficient data reduction than previous approaches while providing visually pleasing B-spline surfaces. Moreover, the approach works well on measured data with noise. Some experimental results demonstrate its usefulness and quality.     

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.     

Neural Network Methods for NURBS Curve and Surface Interpolation

New algorithms based on artificial neural network models are presented for cubic NURBS curve and surface interpolation.When all th knot spans are identical,the NURBS curve interpolation procedure degenerates into that of uniform rational B-spline curves.If all the weights of data points are identical,then the NURBS curve interpolation procedure degenerates into the integral B-spline curve interpolation.     

Smooth surface approximation to serial cross-sections

The reconstruction of the surface model of an object from 2D cross-sections plays an important role in many applications. In this paper, we present a method for surface approximation to a given set of 2D contours. The resulting surface is represented by a bicubic closed B-spline surface with C² continuity. The method performs the skinning of intermediate contour curves represented by cubic B-spline curves on a common knot vector, each of which is fitted to its contour points within a given accuracy. In order to acquire more compact representation for the surface, the method includes an algorithm for reducing the number of knots in the common knot vector. The proposed method provides a smooth and accurate surface model, yet realizes efficient data reduction. Some experimental results are given using synthetic and MRI data.     

Terrain modelling with B-spline type surfaces defined on curved knot lines

B-spline type surfaces on curved knot lines are useful for interpolation of scattered data while preserving desirable terrain breaks or discontinuities which are not necessarily parallel to a given straight line. This paper considers data scattered along parallel scan profiles. It is shown how the large composite linear system which arises for curved knot lines in two directions may be decomposed into smaller linear systems with subsequent savings in storage and processing time.     

满足数据点切向约束的二次B样条插值曲线

给出一种二次B样条曲线插值方法.利用数据点的参数化和节点向量的自由度,构造在各数据点满足切向约束的二次B样条插值曲线,直观地控制插值曲线达到预期形状.用文中方法构造插值曲线是一个递推过程,不必预先确定数据点参数值和节点向量、不必解线性方程组,而是在插值过程中根据数据点及其切向的约束条件递推地确定数据点的参数值、节点和控制顶点.该文方法允许插值曲线各段的连接点与数据点不一致,以使得二次B样条插值曲线的形状更自然.而且在满足数据点切向约束的条件下,还可利用节点进一步调控插值曲线的形状.另外,用文中方法构造的二次B样条插值曲线对于数据点的改变具有较好的局部性质.文中最后给出一些例子将该文方法与其它一些插值方法进行比较,实验结果表明,该文方法是有效的.     

Target shape design optimization by evolving B-splines with cooperative coevolution

With high reputation in handling non-linear and multi-model problems with little prior knowledge, evolutionary algorithms (EAs) have successfully been applied to design optimization problems as robust optimizers. Since real-world design optimization is often computationally expensive, target shape design optimization problems (TSDOPs) have been frequently used as efficient miniature model to check algorithmic performance for general shape design. There are at least three important issues in developing EAs for TSDOPs, i.e., design representation, fitness evaluation and evolution paradigm. Existing work has mainly focused on the first two issues, in which (1) an adaptive encoding scheme with B-spline has been proposed as a representation, and (2) a symmetric Hausdorff distance based metric has been used as a fitness function. But for the third issue, off-the-shelf EAs were used directly to evolve B-spline control points and/or knot vector. In this paper, we first demonstrate why it is unreasonable to evolve the control points and knot vector simultaneously. And then a new coevolutionary paradigm is proposed to evolve the control points and knot vector of B-spline separately in a cooperative manner. In the new paradigm, an initial population is generated for both the control points, and the knot vector. The two populations are evolved mostly separately in a round-robin fashion, with only cooperation at the fitness evaluation phase. The new paradigm has at least two significant advantages over conventional EAs. Firstly, it provides a platform to evolve both the control points and knot vector reasonably. Secondly, it reduces the difficulty of TSDOPs by decomposing the objective vector into two smaller subcomponents (i.e., control points and knot vector). To evaluate the efficacy of the proposed coevolutionary paradigm, an algorithm named CMA-ES-CC was formulated. Experimental studies were conducted based on two target shapes. The comparison with six other EAs suggests that the proposed cooperative coevolution paradigm is very effective for TSDOPs.     

基于最佳平方逼近的B样条曲线降阶

提出了一种基于带约束的最佳平方逼近的B样条曲线降阶的方法.首先讨论了降阶后曲线控制顶点个数以及节点向量的取法、保端点的B样条曲线降阶方法,并把带约束的最佳平方逼近技术引入到B样条曲线的降阶,即误差大的区域施加较大的权函数以降低最大误差.为满足给定误差限制下的降阶,提出了对原曲线插入节点的准则,即对不满足误差限制的区域插入节点.并用实例对新方法和基于扰动约束技术的降阶方法进行了比较.