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。     

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。     

细分生成B样条的几何作图法

研究均匀B样条曲线细分生成的几何作图问题,给出了采用p-nary细分法细分生成任意次均匀B样条曲线的递归细分算法。在此基础上,研究了任意次均匀B样条曲线p-nary细分生成的几何作图方法。利用这种几何作图法,可以直观地在计算机上通过编程来快速准确地绘制B样曲线,更重要的是,可以使基于几何方法的任意次B样曲线的手工绘制成为可能。     

基于SOM网络的三次B样条曲线重建

使用散乱点集重建曲线曲面,在逆向工程和计算机视觉中有着广泛的应用。提出基于SOM网络的三次B样条曲线重建算法。给定某一曲线散乱点集和一初始神经网络,优化SOM网络中神经元位置,使网络逼近散乱点和映射散乱点空间特征。用特征点反求三次B样条曲线控制点,利用控制点重建三次B样条曲线。试验结果表明,算法取得的曲线重建效果良好。     

基于二次B样条曲线拟合的新算法

针对由四点拟合成一条三次B样条曲线过程中计算量大的缺点,提出了一种简单的二次B样条曲线拟合算法。即用两条二次B样条曲线近似一条三次B样条曲线,以期达到计算量小,光滑度也达到要求,提高B样条曲线的绘制速度。     

带有给定切线多边形的C-Bézier闭曲线和B-型样条闭曲线

§1.引 言 Bézier曲线和B样条曲线已广泛应用到汽车、航空、造船等许多领域中.Hering讨论了与凸多边形每边相切的分段三(四)次 Bézier闭曲线和三(四)次B样条闭曲线.它的所有Bézier点必须通过求解大型方程组得到,计算量大,且曲线易出现拐点,而B样条闭曲线的控制点要通过反算得到[1].方逵改进了Hering的方法,构造了G2连续的分段三次曲线[2],基本上克服了Hering方法的两个缺点,但局部修改仍然是比较复杂的.方逵等再次研究了与任意多边形相切的分段四次和五次Bézier曲线[3],但五次Béier曲线不能作局部修改.本文的第二节研究了与任意多边形相切的分段C-Bézier曲线,该曲线C1连续的,且对切线多边形具有保形性,每段C-Bézier曲线上的控制点由切线多边形的顶点计算     

不规则网格上的曲面设计方法

提出一种在不规则网格上构造曲面的方法.其基本思想是,通过均匀双三次B样条基函数的分解和子基函数的分类,将B样条曲面方法推广到任意四边形网格.给定一个任意四边形控制网格,首先对每个控制点构造一个基函数;所有控制点加权组合形成整体曲面.构造的曲面是分片双三次有理参数多项式曲面.此方法可以看成是均匀B样条曲面构造方法的扩展,如果控制网格是规则四边形网格,那么构造得到的曲面与均匀双三次B样条曲面是一致的.最后,实例证明此方法能够有效地构造曲面.     

基于三次样条曲线绘制与实现

介绍了在VC++环境中实现计算机标图软件中典型区域一集结地域的绘制方法,重点给出了利用初始控制点生成区域图标新的控制点算法以及利用三次样条曲线进行绘制的相应代码,实现了区域的显示与灵活控制.     

一种任意次非均匀B 样条的细分算法

类似于经典的、应用于任意次均匀B 样条的Lane-Riesenfeld 细分算法, 提出了一种任意次非均匀B 样条的细分算法,算法包含加细和光滑两个步骤,可生成任意 次非均匀B 样条曲线。算法是基于于开花方法提出的,不同于以均匀B 样条基函数的卷积 公式为基础的Lane-Riesenfeld 细分算法。通过引入两个开花多项式,给出了算法正确性的 详细证明。算法的时间复杂度优于经典的任意次均匀B 样条细分算法,与已有的任意次非 均匀B 样条细分算法的计算量相当。     

局部调整插值点的三次样条曲线表示

给出了带局部形状参数的三次样条曲线生成方法．所给方法以Hermite型插值曲线和非均匀三次B样条曲线为特殊情形，将插值于控制点的曲线和逼近于控制多边形的非均匀B样条曲线统一起来．一个形状参数只影响两条曲线段，曲线表达式保持了三次Bezier曲线表达式的简单结构．改变形状参数的值或调整Bezier控制点，可以局部调整曲线的形状．基于所给样条曲线，给出了带局部形状参数的双三次样条曲面．     

Information entropy-based viewpoint planning for 3-D object reconstruction

In this paper, we present an information entropy-based viewpoint-planning approach for reconstruction of freeform surfaces of three-dimensional objects. To achieve the reconstruction, the object is first sliced into a series of cross section curves, with each curve to be reconstructed by a closed B-spline curve. In the framework of Bayesian statistics, we propose an improved Bayesian information criterion (BIC) for determining the B-spline model complexity. Then, we analyze the uncertainty of the model using entropy as the measurement. Based on this analysis, we predict the information gain for each cross section curve for the next measurement. After predicting the information gain of each curve, we obtain the information change for all the B-spline models. This information gain is then mapped into the view space. The viewpoint that contains maximal information gain about the object is selected as the next best view. Experimental results show successful implementation of our view planning method for digitization and reconstruction of freeform objects.     

A 3-D contour segmentation scheme based on curvature and torsion

The Darboux vector contains both the curvature and torsion of a three-dimensional (3-D) image as its components. Curvature measures how sharply a curve is turning while torsion measures the extent of its twist in 3-D space. Curvature and torsion completely define the shape of a 3-D curve. A scheme is presented that uses the length of this vector, also called the total curvature, for the segmentation of 3-D contours. A quintic B-spline is used in this formulation to obtain the total curvature for noisy data. Examples of nonnoisy and noisy data illustrate the merit of the scheme     

B-spline wavelet-based motion smoothing

A B-spline wavelet-based algorithm is proposed in this study to solve the motion smoothing problem, which has practical applications in the computer animation and virtual reality (VR) areas. The motion of a rigid body consists of translation and rotation. The former is described by a space curve in three-dimensional Euclidean space, while the latter is represented by a curve in the unit quaternion space. Due to the complexity of rendering realistic rigid-body motions in the computer, rigid-body motions are often sampled from the physical world motions. Yet, sampling processes introduce noise; excessive noise results in tremulous motions in a VR environment. The noise-embedded motion data are decomposed using multiresolution analysis and the noise is detected as components of small magnitude. By eliminating the small-magnitude components, a smooth representation of the motion data is achieved.     

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。     

Direct manipulations of B-spline and NURBS curves

This paper presents a method for the direct manipulations of B-spline and non-uniform rational B-splines (NURBS) curves using geometric constraints. A deformable model is developed to define the deformation energy functional of B-spline and NURBS curves. The finite element method is used to minimize the deformation energy functional and solve for the deformed shape of curves subjected to constraints. This approach results in a set of linear equations for a B-spline curve and a set of non-linear equations for a NURBS curve. A perspective mapping is used to linearize the NURBS formulations. NURBS curves are first mapped from the 3D Cartesian coordinate space to the 4D homogeneous coordinate space, and transformed to 4D B-spline curves. After the manipulation in the 4D homogeneous coordinate space, the modified NURBS curves are then mapped back to the 3D Cartesian coordinate space. The approach is implemented by a prototype program, which is written in C, and runs under WINDOWS. Several examples are presented to demonstrate the capabilities of this approach.     

基于B样条曲线的断层离散数据平滑算法

通过二维断层图像进行三维对象重建是现今较为活跃的研究领域,而二维断层数据的质量将直接影响到三维重建的效果。提出了一种对医学断层离散数据在三维重建前进行预处理的方法,通过单层轮廓平面内平滑以及多层轮廓基于层间关系的B样条曲线拟合对原始数据进行了平滑处理,减少了重建后层与层之间的锯齿现象,使得在此基础上的三维重建效果得到了明显的改善。算法具有一定的通用性,同样适用于其它领域的数据处理。     

Gear Geometric Design by B-Spline Curve Fitting and Sweep Surface Modelling

In this paper, B-spline curve fitting and sweep surface generation are used for the geometric design of involute gears. Tooth profiles are described by a B-spline formulation based on interpolating data points with first and second derivative constrains. Tooth surfaces are generated by sweeping the B-spline profiles along specified trajectories. This representation scheme enables tooth shapes to be inter-actively controlled by manipulating control polygons and sweep trajectories. A CAD–CAE integration allows the analysis of contact and structural three-dimensional problems for various geometric configurations. In the paper, the methodology is applied to the geometric design of involute pinions of face gear drives.     

Constrained shape modification of cubic B-spline curves by means of knots

The effect of the modification of knot values on the shape of B-spline curves is examined in this paper. The modification of a knot of a B-spline curve of order k generates a one-parameter family of curves.This family has an envelope which is also a B-spline curve with the same control polygon and of order k−1. Applying this theoretical result, three shape control methods are provided for cubic B-spline curves, that are based on the modification of three consecutive knots. The proposed methods enable local shape modifications subject to position and/or tangent constraints that can be specified within well defined limits.     

基于全正基的三次均匀B 样条曲线的扩展

为了构造具有保形性的三次均匀B 样条扩展曲线,首先运用拟扩展切比雪夫空间的理论框架证明现有文献中的三次Bézier 曲线的扩展基,简称λ-Bézier 基,恰为相应空间的规范B 基。然后用λ -Bézier 基的线性组合来表示三次均匀B 样条曲线的扩展基,根据预设的曲线性质反推扩展基的性质,进而求出线性组合的系数。扩展基可表示成λ-Bézier 基与一个转换矩阵的乘积,证明了转换矩阵的全正性及扩展基的全正性。由扩展基定义了基于3 点分段的曲线,分析了曲线的性质,扩展基的全正性决定了曲线可以较好的模拟控制多边形的形态。简要介绍了由扩展基定义的基于16 点分片的曲面。