A fast direct point-by-point generating algorithm for B Spline curves and surfaces

Traditional generating algorithms for B Spline curves and surfaces require approximation methods where how to increment the parameter to get the best approximation is problematic; or they take the pixel-based method needing matrix trans- formation from B Spline representation to Bézier form. Here, a fast, direct point-by-point generating algorithm for B Spline curves and surfaces is presented. The algorithm does not need matrix transformation, can be used for uniform or nonuniform B Spline curves and surfaces of any degree, and has high generating speed and good rendering accuracy.

Approximate merging of B-spline curves and surfaces

Applying the distance function between two B-spline curves with respect to the L2 norm as the approximate error, we investigate the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. Then this method can be easily extended to the approximate merging problem of multiple B-spline curves and of two adjacent surfaces. After minimizing the approximate error between curves or surfaces, the approximate merging problem can be transformed into equations solving. We express both the new control points and the precise error of approximation explicitly in matrix form. Based on homogeneous coordinates and quadratic programming, we also introduce a new framework for approximate merging of two adjacent NURBS curves. Finally, several numerical examples demonstrate the effectiveness and validity of the algorithm.

RECONSTRUCTION OF SYMMETRIC B-SPLINE CURVES AND SURFACES

A method to reconstruct symmetric B-spline curves and surfaces is presented. The symmetry property is realized by using symmetric knot vector and symmetric control points. Firstly, data points are divided into two parts based on the symmetry axis or symmetry plane extracted from data points. Then the divided data points are parameterized and a symmetric knot vector is selected in order to get symmetric B-spline basis functions. Constraint equations regarding the control points are deduced to keep the control points of the B-spline curve or surface to be symmetric with respect to the extracted symmetry axis or symmetry plane. Lastly, the constrained least squares fitting problem is solved with the Lagrange multiplier method. Two examples from industry are given to show that the proposed method is efficient, robust and able to meet the general engineering requirements.

Multiresolution for Closed Curves and Surfaces Using Wavelets

Multiresolution modeling is becoming a powerful tool for fast display, and geometric data compression and transmission of complex shapes. Most of the existing literatures investigating the multiresolution for B-spline curves and surfaces are concentrated on open ones. In this paper, we focus on the multiresolution representations and editing of closed B-spline curves and surfaces using wavelets. A repetition approach is adopted for the multiresolution analysis of closed B-spline curves and surfaces. Since the closed curve or surface itself is periodic, it can overcome the drawback brought by the repetition method, i.e. introducing the discontinuities at the boundaries. Based on the models at different multiresolution levels, the multiresolution editing methods of closed curves and surfaces are introduced. Users can edit the overall shape of a closed one while preserving its details, or change its details without affecting its overall shape.

Computing B-Spline Surfaces on Transputer Networks

This paper presents a parallel implementation of computing uniform bicubic B spline surfaces on Transputer networks. The work is essential for building Transputer based CAD and graphics systems.

Research into Freedom Surface Modeling System of Double Beta Spline in Space

In this paper, we present the double Beta spline curved surface which is controlled by double parameters including the algorithm principles, the treatment of boundary conditions, the alternation of projection, the algorithms of elimination hiddle line, the process to display and the primiples to produce the shaded curved surface. Based on all the above, a freedom surface modeling system (FSMS) is designed and some examples developed on FSMS are verified and analyzed.

Bounds on partial derivatives of NURBS surfaces

In this paper, we estimate the partial derivative bounds for Non-Uniform Rational B-spline(NURBS) surfaces. Firstly, based on the formula of translating the product into sum of B-spline functions, discrete B-spline theory and Dir function, some derivative bounds on NURBS curves are provided. Then, the derivative bounds on the magnitudes of NURBS surfaces are proposed by regarding a rational surface as the locus of a rational curve. Finally, some numerical examples are provided to elucidate how tight the bounds are.

Research on a toolpath generation method of NC milling based on space-filling curve

Tool path generated by space-filling curve always turns frequently causing trembling to machine,reducing toollife and affecting workpiece quality. Length and generation time of tool paths are both relatively long. In order to solve these problems,a toolpath generation method of NC milling based on space-filling curve is proposed. First,T-spline surface is regarded as the modeling surface,the grid,which is based on the limited scallop-height,can be got in the parameter space,and the influence value of grid node is determined. Second,a box is defined and planned,and the tool paths are got preliminarily,which is based on minimal spanning tree; Finally,based on an improved chamfering algorithm,the whole tool paths are got. A simulation system is developed for computer simulation,and an experiment is carried out to verify the method. The results of simulation and experiment show that the method is effective and feasible,and length and time of the tool paths are reduced.

Constructing iterative non-uniform B-spline curve and surface to fit data points

In this paper, based on the idea of profit and loss modification, we presentthe iterative non-uniform B-spline curve and surface to settle a key problem in computeraided geometric design and reverse engineering, that is, constructing the curve (surface)fitting (interpolating) a given ordered point set without solving a linear system. We startwith a piece of initial non-uniform B-spline curve (surface) which takes the given point setas its control point set. Then by adjusting its control points gradually with iterative formula,we can get a group of non-uniform B-spline curves (surfaces) with gradually higherprecision. In this paper, using modern matrix theory, we strictly prove that the limit curve(surface) of the iteration interpolates the given point set. The non-uniform B-spline curves(surfaces) generated with the iteration have many advantages, such as satisfying theNURBS standard, having explicit expression, gaining locality, and convexity preserving,etc

Generation of Discrete Bicubic G^1 B-Spline Ship Hullform Surfaces from a Given Curve Notwork Using Virtual Iso-Parametric Curves

We propose a method that automatically generates discrete bicubic G^1 continuous B-spline surfaces that interpolate the curve network of a ship huliform.First,the curves in the network are classified into two types;boundary curves and ＂reference curves＂,The boundary curves correspond to a set of rectangular（or triangular）topological type that can be representes with tensot-product （or degenerate）B-spline surface patches.Next,in the interior of the patches,surface fitting points and cross boundary derivatives are estimated from the reference curves by constructing ＂virtual＂isoparametric curves.Finally,a discrete G^1 continuous B-spline surface is gencrated by a surface fitting algorithm.Several smooth ship hullform surfaces generated from curve networks corresponding to actual ship hullforms demonstrate the quality of the method.

C^2-(C^3-) Continuous Interpolation Spline Curve and Surface

Free-formed or sculptured surfaces in engineering products are frequently constructed from a set of measured 3D data points. C2- (C3-) continuity approach is important in this field. This paper presents a method of rectangular interpolation of given 3D data array which is regularly arranged. The interpolation surface which is constructed by tensor product has desirable properties (second-order or third-order continuity locality) and is implemented and adjusted easily. Higher order continuity methods are also briefly discussed.

基于空间点阵的齿轮精确修形方法

齿轮修形是改善啮合状态、提高传动精度和效率的一种重要工程技术,但现有的齿轮修形方式和工具性软件难以确保修形的齿面几何质量和效率。因此,提出了一种空间点阵的齿轮修形方法。首先,根据修形精度要求,将齿轮的齿面离散为规则空间点阵,并将修形量映射到对应的空间点上,再将调整后的空间点阵构建成规则的二次B样条曲线;然后,通过优化网格曲线,得到修形后的齿面,并自适应更新整个齿轮三维模型的齿面;最后,将该方法无缝集成到三维设计软件NX平台上,开发了齿轮精确修形设计系统。实例证明,该方法通过修形位置和修形量的双重控制,实现了齿面的精确修形,获得了高质量齿廓曲面,在保证精度的同时提高了齿轮修形的高效性和实用性。

G^(1)连续组合曲线曲面的构造

为了在保留B样条方法自动光滑性的基础上突破其要求各部分曲线曲面的次数必须相等的限制,同时赋予曲线曲面独立于控制顶点的形状可调性,提出一种G^(1)连续组合曲线曲面的构造方法.首先构造一组含2个自由参数的n(n≥2)次基函数并分析其性质,基于该基函数,定义了结构与n次Bézier曲线曲面相同的新曲线曲面,其包含n次Bézier曲线曲面为特例;然后分析新曲线曲面的G^(1)光滑拼接条件,根据拼接条件,采用与B样条方法相同的组合思想但是不同的组合方式,定义基于新曲线曲面的分段组合曲线与分片组合曲面,其包含2次均匀和2次准均匀B样条为特例.实例结果表明,定义方式自动保证了组合曲线曲面在连接处的G^(1)连续性,组合曲线曲面的端点与角点位置以及内部形状都可以通过改变自由参数的取值来进行调整,且调整方式既可以是全局的又可以是局部的;所提方法为复杂曲线曲面的造型设计提供了便利.

基于“D”型深孔测斜曲线的滑坡滑动面位置确定方法研究

在实际的深孔位移监测中,测斜曲线的突变特征是滑动面辨识的关键依据,前人通过大量的研究总结将滑动面迹象显著的测斜曲线类型分为了“B”型、“D”型、“r”型等几种。其中,对于“D”型测斜曲线通常是将曲线的鼓包凸起点作为滑动面的位置,但这种方法容易受到测点布置间隔和横纵坐标观测尺度的影响,存在滑面位置定义不清晰、数值不确定的问题。为了能够有效地克服这些缺点,提升测斜曲线滑动面辨识的准确度,基于“D”型测斜曲线变化特征,将滑坡抽象为由“滑动体”“滑动区间”以及“不动体”三者组成的概化模型,根据三者抗弯刚度的差异建立外界荷载作用下的杆件力学模型,深入分析滑坡运动过程中不同深度处土体的变形特点。研究表明,由于“D”型曲线滑动面并未完全贯通,使得土体沿深度方向变形连续无突变,力学模型中杆件正负弯矩的分界点是变形曲线水平位移最大处,能够真实地反映滑坡变形特点以及滑动面的位置。将土体累计位移转化为相对位移,则“D”型深孔测斜曲线变为了“S”型相对位移-深度曲线,且“S”型曲线的拐点与滑动面的位置相近;通过提取监测期内不同深度处土体的平均相对位移,运用三次样条插值法计算“S”型区段内拐点的深度值,能够更加精准地确定滑动面位置,更好地提升深孔位移监测的可靠度和准确度,具有较大的实用价值。

基于B型深孔测斜曲线的滑坡滑动面位置确定方法

在实际的深孔位移监测中,测斜曲线的突变特征是滑动面辨识的关键依据,对于“B”型测斜曲线通常是将曲线的鼓包凸起点作为滑动面的位置,但这种方法容易受到测点布置间隔和横纵坐标观测尺度的影响,存在滑面位置定义不清晰、数值不确定的问题。为了能够有效地克服这些缺点,提升测斜曲线滑动面辨识的准确度,基于“B”型测斜曲线变化特征,研究将滑坡抽象为由“滑动体”、“滑动区间”以及“不动体”三者组成的概化模型,并对滑坡运动过程中不同深度处土体的位移速率和加速度变化特点进行了深入分析。结果表明:土体位移速率沿深度方向会形成多道曲线簇,且不同曲线簇分别对应各滑面所在的滑动区间及非滑动区间。此外,将位移速率转化为加速度,则土体加速度曲线具有更加显著的分簇特征,能够为准确区分滑动体和不动体的区间范围提供重要依据。进一步地,提取监测期内加速度曲线分簇特征显著的当日土体加速度并绘制散点图,能够快速准确地确定坡体不同区间的深度范围。基于当日各滑面所在滑动区间内的加速度数据,运用三次样条插值法计算该区间内的位移加速度最大值,由于滑面处的土体加速度最大,则该值对应的土体深度可作为滑面位置的计算深度。研究方法能够更加精准地确定滑动面位置,并使滑面位置的确定可计算化,有效提升了深孔位移监测数据的利用程度和可靠度,具有较大的实用价值。

基于T-Spline的全自动几何拓扑修复方法

从高质量曲面网格生成的需求出发,提出了一种基于 T-Spline 的全自动几何拓扑修复方法.本文方法创新性主要可归纳为: 1)对原有计算机辅助设计(Computer aided design, CAD)几何模型不进行任何修改保留其本真,自动识别 CAD 几何模型中常见不必要的几何特征,成功解决了 CAD 几何模型中存在的几何瑕疵,如短边、窄面、退化边、退化面、非连续光滑边界及尖锐特征等,利用新生成的\虚边"、\虚面"处理几何瑕疵,同时通过虚拓扑重构 CAD 几何模型的 B-Rep;2)开发了一套 CAD/CAE 集成系统,统一了几何模型与计算分析模型,实现计算机辅助工程(Computer aided engineering, CAE)与CAD 两者的无缝集成,所有拓扑修复操作及后续 CAE 分析计算均在同一环境下进行,避免了几何模型在 CAE 与 CAD 系统间进行转换时造成的数据丢失.该方法能够对复杂实体实现全自动几何拓扑修复及网格生成,实验表明,在保证不失真的前提下,修复后的几何模型能够生成质量良好的网格且能降低网格的生成规模,验证了本文方法的实用性和有效性,以满足工程实际分析的需要.

Extended Cubic Uniform B-spline and α-B-spline

花键曲线和表面在 CAD 和计算机图形起一个重要作用。在这份报纸，我们建议立方的一致 B 花键的几延期。然后，我们在场插入内推的延期 -B-spline 基于新 B 花键和单个相配技术。延期的优点是他们有全球、本地的形状参数。而且，我们也在数据插值和多角形的形状变丑调查他们的应用。

基于Coons曲面与B-spline曲线的有限元网格自动生成

在概述了有限元网格自动生成的基础上,介绍了一种能对复杂曲边区域作有限元网格生成的技术。这种技术以Coons曲面作映射函数,以B-spline曲线作拟合区域边界,充分发挥ICG的交互能力,利用B-spline曲线的形状控制,对多种复杂曲边区域作网格自动生成。用该技术研制的网格生成系统在交互式人机界面支持下,能快速输入几何信息,实现图形交互编辑,边界贴合良好,内部网格点理想。

隐式B样条曲线拟合的加权PIA算法

为了使拟合数据点的曲线生成速度更快、误差更小,提出一种隐式B样条曲线拟合数据点的加权PIA算法.首先,用待拟合数据点以及给定法向量生成偏移点集.然后,通过偏移点集构造差分向量,从而得到需要调整的误差控制系数,为了使迭代效率更高,在迭代过程中对误差控制系数做加权处理.最后,用最新的控制系数矩阵得到拟合数据点的曲线.文中5个数值算例采用均匀节点序列,实验结果表明,在相同迭代次数下,相对于I-PIA算法,该算法得到的拟合曲线误差值更小,曲线能更好保特征.

基于蚁群算法的无人船平滑路径规划

为解决无人驾驶船舶在复杂环境中规划路径时存在的转向角度大、路径拐点多、航行能耗高等问题,文中提出一种基于改进蚁群算法的平滑路径规划方法。该方法采用栅格法进行环境建模,通过在启发函数中引入路径平滑度、距离启发因子以及在路径转移概率中引入障碍物启发因素,提高路径寻优和静态避障能力。结合启发因素改进信息素更新标准,设置可调节信息素挥发因子增加算法的自适应性。提取输出的最优路径关键节点并对其进行平滑处理,进一步保证路径平滑度和安全性。根据不同栅格环境下的避障仿真结果可知,与传统算法相比,文中改进蚁群算法的路径寻优速度提高了45%~62%,转向次数减少了25%~44%,平滑处理后的路径安全性和可行性得到了提升,较好地实现了不同环境下无人船自主路径规划。