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.     

Total curvature variation fairing for medial axis regularization

We present a new fairing method for planar curves, which is particularly well suited for the regularization of the medial axis of a planar domain. It is based on the concept of total variation regularization. The original boundary (given as a closed B-spline curve or several such curves for multiply connected domains) is approximated by another curve that possesses a smaller number of curvature extrema. Consequently, the modified curve leads to a smaller number of branches of the medial axis. In order to compute the medial axis, we use the state-of-the-art algorithm from [1] which is based on arc spline approximation and a domain decomposition approach. We improve this algorithm by using a different decomposition strategy that allows to reduce the number of base cases from 13 to only 5. Moreover, the algorithm reduces the number of conic arcs in the output by approx. 50%.     

3维碎片拼合的算法研究

为实现艺术珍品保护或古器物修复中3维碎片的比对和拼接,提出了一种空间任意形状的3维碎片拼合算法。该算法首先对由实物测量数据建立的三角网格模型进行预处理,即去除模型中所存在的一些缺陷,以提取模型边界轮廓曲线;然后对碎片的边界轮廓离散点进行五次B样条插值,同时计算轮廓曲线上各个点的曲率、挠率和法矢,并据此分析3维碎片轮廓曲线的几何特性;接着根据轮廓曲线上各个点的总曲率来检测轮廓的特征点,并对轮廓进行分段;最后根据曲率、挠率的变化对不同轮廓上的特征段进行相似性度量,并运用法矢对相似性程度较高的轮廓段进行可匹配性验证,同时计算出可匹配轮廓段的坐标转换关系,以实现碎片的拼合。实例表明,该算法是高效、稳定的。     

基于Freeman链码的B样条曲线轮廓拟合

提出了一种Freeman链码与B样条曲线误差控制相结合实现轮廓拟合的算法,首先利用Freeman链码法进行边界跟踪,根据相邻像素点间的不同的链码变化关系,排除伪特征点,提取出轮廓中绝大多数特征点,然后结合基于误差控制的B样条曲线法,取得能够精确表示轮廓信息的特征点。本文算法即避免了使用曲率来进行求取特征点的复杂计算,提高了特征点检测速度,又提取出能够精确拟合轮廓的局部支撑点,实现了基于误差控制的轮廓曲线拟合。实验结果证明了本文算法的正确性。     

Data reduction using cubic rational B-splines

A geometric method for fitting rational cubic B-spline curves to data representing smooth curves, such as intersection curves or silhouette lines, is presented. The algorithm relies on the convex hull and on the variation diminishing properties of Bezier/B-spline curves. It is shown that the algorithm delivers fitting curves that approximate the data with high accuracy even in cases with large tolerances. The ways in which the algorithm computes the end tangent magnitudes and inner control points, fits cubic curves through intermediate points, checks the approximate error, obtains optimal segmentation using binary search, and obtains appropriate final curve form are discussed     

Reparameterization of piecewise rational Bezier curves and its applications

degree . Although the curve segments are C ¹ continuous in three dimensions, they may be C ⁰ continuous in four dimensions. In this case, the multiplicity of each interior knot cannot be reduced and the B-spline basis function becomes C ⁰ continuous. Using a surface generation method, such as skinning these kinds of rational B-spline curves to construct an interpolatory surface, may generate surfaces with C ⁰ continuity. This paper presents a reparameterization method for reducing the multiplicity of each interior knot to make the curve segments C ¹ continuous in four dimensions. The reparameterized rational B-spline curve has the same shape and degree as before and also has a standard form. Some applications in skinned surface and ruled surface generation based on the reparameterized curves are shown. Published online: 19 July 2001     

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.     

带形状参数的C~2连续类三次三角样条曲线

传统的三次均匀B样条曲线在给定控制顶点时其形状不能调整,以及不能精确表示圆锥曲线。针对三次均匀B样条曲线的不足,提出了一种带形状参数的C2连续的类三次三角样条曲线。该曲线不仅与三次均匀B样条曲线具有相似的性质,而且在控制顶点保持不变时其形状可通过形状参数的取值进行调整。在适当条件下,类三次三角样条曲线比三次均匀B样条曲线更能逼近于控制多边形,且能精确表示圆、椭圆、抛物线等圆锥曲线。     

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.     

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。     

基于曲率单调变化的空间非均匀三次B样条曲线的构造方法

B样条曲线在目前CAD系统中得到广泛应用,针对B样条曲线的光顺问题,给出并证明了具有曲率单调变化的非均匀三次B样条曲线的构造方法.首先通过给定非均匀三次B样条曲线的中间控制边矢量及相关初始条件,然后计算初始和结尾控制边矢量,由此得到的非均匀三次B样条曲线具有单调变化的曲率.实验在Windows系统下基于VC++语言实现,相关实例验证了该构造方法的有效性及实用性.     

集多种特性的三次三角伪B样条

目的 为了同时解决传统多项式B样条曲线在形状调控、精确表示常见工程曲线以及构造插值曲线时的不足,提出了一类集多种特性的三次三角伪B样条。方法 首先构造了一组带两个参数的三次三角伪B样条基函数,然后在此基础上定义了相应的参数伪B样条曲线,并讨论了该曲线的特性及光顺性问题,最后研究了相应的代数伪B样条,并给出了最优代数伪B样条的确定方法。结果 参数伪B样条曲线不仅满足C²连续,而且无需求解方程系统即可自动插值于给定的型值点。当型值点保持不变时,插值曲线的形状还可通过自带的两个参数进行调控。在适当条件下,该参数伪B样条曲线可精确表示圆弧、椭圆弧、星形线等常见的工程曲线。相应的代数伪B样条具有参数伪B样条曲线类似的性质,利用最优代数伪B样条可获得满意的插值效果。结论 所提出的伪B样条同时解决了传统多项式B样条曲线在形状调控、精确表示常见工程曲线以及构造插值曲线时的不足,是一种实用的曲线造型方法。     

基于曲率分析的三次Bezier曲线采样方法的研究

针对三维建模过程中数据量大的缺点,提出一种简单的基于曲率分析的三次Bezier曲线采样方法。该方法采用每个分段的三次Bezier曲线的特征点和该段曲率半径的极小值作为采样密度的判断标准,曲线采样主要分为多层轮廓和单一轮廓两种情况,对于多层轮廓,采样密度涉及到的因素有曲线特征点,曲率半径极小值,轮廓之间的间距,曲线的长度。而对于单一轮廓,采样密度涉及到的因素有曲线特征点,曲率半径极小值,曲线的长度。通过以上因素,计算出采样点的数目。实验结果证明,提出的方法可行有效,可用于三维建模的数据点采样。     

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．     

一种实用的汉字字形合成方法

提出一种实用的汉字字形合成方法。本方法用三次Ｂ－样条插值曲线表示字形笔划的轮廓线。根据Ｂ－样条插值曲线的连续性与唯一性，在待合成字形笔划曲线之间建立对应关系。给定合成比例系数，统一曲线控制点长度与参数取值范围，得到新的字形笔划曲线表达式，产生新的汉字字形。从实验结果看，该方法简洁，容易实现，不失为一种有效的汉字字形合成方法。     

G 2 blends of linear segments with cubics and Pythagorean-hodograph quintics

Fillets, also known as blend arcs, are used in CNC machining to round corners. Fillets are normally circular arcs, which have G ¹ contact with the straight line segments to which they are joined. Recent advances in machining technology allow NURBS, including Pythagorean-hodograph (PH) curve segments, to be incorporated in CNC tool paths. This article examines the use of cubic and PH quintic Bézier curve segments that have a single curvature extremum, and which have G ² contact with the straight line segments to which they are joined, as fillets. It is shown how the extreme circle of curvature can be determined. The point of curvature extremum and the corresponding value of the curvature can be changed by adjusting the joining points of the blending curve with the neighbouring straight lines. These blending curves can also be incorporated in computer-aided design packages for curve or surface design.     

基于四点分段的一类三角多项式曲线

提出了一类m(m=1,2,3)次分段三角多项式曲线，通过引入形状参数，给出了加权三角多项式曲线，与三次B样条曲线类似。每段三角多项式曲线由4个相继的控制点生成，对于等距节点的情形，所提出的三角多项式曲线是C^2m-1连续；给出了三角开曲线和闭曲线的构造方法。论述了椭圆的表示方法，给出了三角多项式曲线与三次B样条曲线的对比，通过改变次数m或调整形状参数，可以得到不同程度地接近于控制多边形的曲线，因此，所给曲线的生成方法是一种结构简单和使用方便的曲线生成方法。     

基于渐进迭代逼近的矢量地图曲线化简方法

矢量地图化简在地形仿真、制图综合等研究中具有重要应用。针对已有算法难以兼顾化简曲线 的整体形态和局部特征点精度的问题,提出一种基于 B 样条曲线渐进迭代逼近(PIA)的矢量地图曲线化简方法。 首先筛选出能保持曲线轮廓、具有最大信息量的特征点列,将其作为初始控制点列,得到相应的非均匀 3 次 B 样条拟合曲线;然后根据拟合曲线与特征点的误差进行迭代调整控制点,逐步得到一系列逼近曲线,直至最终 满足精度要求。实验表明,PIA 方法不仅保持了化简曲线的整体几何形态,而且能在满足全局误差要求的情况 下,实现特征点处的高精度逼近。     

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.