Efficient Construction of B-Spline Curves with Minimal Internal Energy

In this paper,we propose an efficient method to construct energy-minimizing B-spline curves by using discrete mask method.The linear relations between control points are firstly derived for different energy-minimization problems,then the construction of B-spline curve with minimal internal energy can be addressed by solving a sparse linear system.The existence and uniqueness of the solution for the linear system are also proved.Experimental results show the efficiency of the proposed approach,and its application in 1 G blending curve construction is also presented.

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.

One Fairing Method of Cubic B-spline Curves Based on Weighted Progressive Iterative Approximation

A new method to the problem of fairing planar cubic B-spline curves is introduced in this paper. The method is based on weighted progressive iterative approximation （WPIA for short） and consists of following steps： finding the bad point which needs to fair, deleting the bad point, re-inserting a new data point to keep the structm-e of the curve and applying WPIA method with the new set of the data points to obtain the faired curve. The new set of the data points is formed by the rest of the original data points and the new inserted point. The method can be used for shape design and data processing. Numerical examples are provided to demonstrate the effectiveness of the method.

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.

THE RELATIONSHIP BETWEEN PROJECTIVE GEOMETRIC AND RATIONAL QUADRATIC B-SPLINE CURVES

Abstract For two rational quadratic B spline curves with same control vertexes, the cross ratio of four collinear points are represented: which are any one of the vertexes, and the two points that the ray initialing from the vertex intersects with the corresponding segments of the two curves, and the point the ray intersecting with the connecting line between the two neighboring vertexes. Different from rational quadratic Bézier curves, the value is generally related with the location of the ray, and the necessary and sufficient condition of the ratio being independent of the ray's location is showed. Also another cross ratio of the following four collinear points are suggested, i.e. one vertex, the points that the ray from the initial vertex intersects respectively with the curve segment, the line connecting the segments end points, and the line connecting the two neighboring vertexes. This cross ratio is concerned only with the ray's location, but not with the weights of the curve. Furthermore, the cross ratio is projective invariant under the projective transformation between the two segments.

The quadratic B-spline curve fitting for the shape of log cross sections

This paper describes a new method for simulation of the cross section shape of log. The self-developed MQK3102 log shape recognizing machine was used to acquire the finite discrete sampling points on the cross section of log and those points were fitted with the quadratic B-spline parametric curve. This method can clearly stimulate the real shape of the log cross section and is characterized by limited sampling points and high speed computing. The computed result of the previous curve does not affect the next one, which may avoid the graphic distortion caused by the accumulative error. The method can be used to simulate the whole body shape of log approximately by sampling the cross sections along the length direction of log, thus providing a reference model for optimum saw cutting of log.

ANTIPLANE CIRCULAR INCLUSION WITH A CURVED CRACK CROSSING THE BOUNDARY

The weakly singular integral equation sued to solve the problem ofthe curved crack crossing the boundary of the antiplane circularinclusion is presented. Using the principal part analysis method ofthe Cauchy type integral equation, the singular stress index at theintersection and the singular stress of angular Regions near theintersection are obtained. By using the singular stress obtained, thestress intensity factor at The intersection is defined. After thenumerical solution of the integral equation, the stress intensityfactors at The end points of the crack and intersection areobtainable.

SOLUTION OF MULTIPLE CURVED RIGID LINE ANDANTIPLANE CIRCULAR INCLUSION PROBLEM BYWEAKLY SINGULAR INTEGRAL EQUATION

Interaction between multiple curved rigid line and circular inclusion in antiplane loading condition was considered. Two kinds of elementary solutions corresponding to a concentrated force applying at inclusion and matrix material respectively were presented. Utilizing the elementary solutions and taking density function of traction difference along curved rigid line, a group of weakly singular integral equations with log kernels can be obtained. After the numerical solution of the integral equations, the discrete values of density functions of traction difference are obtainable. So stress singularity coefficients at rigid line tips can be calculated, and several numerical examples are given.

THE INTERACTION BETWEEN MULTIPLE CURVED RIGID LINE AND ANTIPLANE CIRCULAR INCLUSION

The interaction between multiple curved rigid line and circular inclusion in antiplane loading condition is considered in this paper. By utilizing the point force elementary solutions and taking density function of traction difference along curved rigid lines, a group of weakly singular integral equations with logarithmic kernels can be obtained. After the numerical solution of the integral equations, the discrete values of density functions of traction difference are obtainable. So the stress singularity coefficient at rigid line tips can be calculated, and two numerical examples are given.

A Novel Contour Tracing Algorithm for Object Shape Reconstruction Using Parametric Curves

Parametric curves such as Bézier and B-splines, originally developedfor the design of automobile bodies, are now also used in image processing andcomputer vision. For example, reconstructing an object shape in an image,including different translations, scales, and orientations, can be performedusing these parametric curves. For this, Bézier and B-spline curves can be generatedusing a point set that belongs to the outer boundary of the object. Theresulting object shape can be used in computer vision fields, such as searchingand segmentation methods and training machine learning algorithms. Theprerequisite for reconstructing the shape with parametric curves is to obtainsequentially the points in the point set. In this study, a novel algorithm hasbeen developed that sequentially obtains the pixel locations constituting theouter boundary of the object. The proposed algorithm, unlike the methods inthe literature, is implemented using a filter containing weights and an outercircle surrounding the object. In a binary format image, the starting point ofthe tracing is determined using the outer circle, and the next tracing movementand the pixel to be labeled as the boundary point is found by the filter weights.Then, control points that define the curve shape are selected by reducing thenumber of sequential points. Thus, the Bézier and B-spline curve equationsdescribing the shape are obtained using these points. In addition, differenttranslations, scales, and rotations of the object shape are easily provided bychanging the positions of the control points. It has also been shown that themissing part of the object can be completed thanks to the parametric curves.

B-spline curve-based routing in ad hoc networks

Routing algorithms capable of providing quality of service （QoS） will play an important role in future communication networks. For the trajectory-based routing （ TBR）, An effective method of en- coding trajectories into packets is proposed. The method uses a B-spline curve, which provides a lot of flexibility. The simulation results show that the performance of the proposed algorithms is im- proved significantly compared with the existing algorithm.

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.

An adaptive generation method for free curve trajectory based on NURBS

To realize the high precision and real-time interpolation of the NURBS （non-uniform rational B-spline） curve, a kinetic model based on the modified sigmoid function is proposed. The constraints of maximum feed rate, chord error, curvature radius and interpolator cycle are discussed. This kinetic model reduces the cubic polynomial S-shape model and the trigonometry function S-shape model from 15 sections into 3 sections under the precondition of jerk, acceleration and feedrate continuity. Then an optimized Adams algorithm using the difference quotient to replace the derivative is presented to calculate the interpolator cycle parameters. The higher-order derivation in the Taylor expansion algorithm can be avoided by this algorithm. Finally, the simplified design is analyzed by reducing the times of computing the low-degree zero-value B-spline basis function and the simplified De Boor-Cox recursive algorithm is proposed. The simulation analysis indicates that by these algorithms, the feed rate is effectively controlled according to tool path. The calculated amount is decreased and the calculated speed is increased while the machining precision is ensured. The experimental results show that the target parameter can be correctly calculated and these algorithms can be applied to actual systems.

基于B-spline曲线的非圆齿轮节曲线设计

重点探讨一种利用B-spline曲线合成非圆齿轮节曲线的设计方法。通过实例说明利用开放均匀分布B-spline合成非圆齿轮节曲线的方法,并通过对控制点位置向量值的设定从而改变节曲线形状。

基于Circular模型的大剪应变率点接触弹流界面滑移数值分析

采用Circular流变模型,将假定的流体的极限剪应力特性模型拟合入Reynolds方程,编写Fortran语言程序,数值模拟界面滑移效应.计算从弹流润滑(EHL)延展到动压润滑(HL)区域.随着速度增加,摩擦系数曲线出现反常波动,表现为两个异常拐点.在中、高速度下,模拟获得的接触轮廓等值线图中观测到入口凹陷及中心区下凸.进一步讨论了载荷、速度、综合弹性模量、滑滚比等因素对滑移的影响.界面滑移效应被认为是产生反常接触轮廓和摩擦力波动的主因,与试验结果互为验证.

Ship hull plate processing surface fairing with constraints based on B-spline

The problem of ship hull plate processing surface fairing with constraints based on B-spline is solved in this paper. The algorithm for B-spline curve fairing with constraints is one of the most common methods in plane curve fairing. The algorithm can be applied to global and local curve fairing. It can constrain the perturbation range of the control points and the shape variation of the curve, and get a better fairing result in plane curves. In this paper, a new fairing algorithm with constraints for curves and surfaces in space is presented. Then this method is applied to the experiments of ship hull plate processing surface. Finally numerical results are obtained to show the efficiency of this method.

Study on Real-time Look-ahead and Adaptive Parametric Curve Interpolator

The principle of real-time look-ahead was introduced and analysed. An adaptive parametric curve interpolator with a real-time look-ahead function was developed for non-uniform rational B-spline (NURBS) curves interpolation, which considering the maximum acceleration/deceleration of the machine tool. In order to deal with the acceleration/deceleration around the feedrate sensitive corners, the look-ahead function was designed and illustrated. It can detect and adjust the feedrate adaptively. With the help of real-time look-ahead, the acceleration/deceleration can be limited to the range of the machine tool capacity. Thus, feedrate fluctuation is reduced. A NURBS curve interpolation experiment was provided to verify the feasibility and advantages of the proposed interpolator with a real-time look-ahead function.

Performance Investigation of 2D Lattice Boltzmann Simulation of Forces on a Circular Cylinder

The lattice Boltzmann method (LBM) is employed to simulate the uniform flow past a circular cylinder. The performance of the two-dimensional LBM model on the prediction of force coefficients and vortex shedding frequency is investigated. The local grid refinement technique and second-order boundary condition for curved walls are applied in the calculations. It is found that the calculated vortex shedding frequency, drag coefficient and lift coefficient are consistent with experimental results at Reynolds nu...

Adaptive B-snake for planar curve approximation

An adaptive B-spline active contour model for planar curve approximation is proposed. Starting with an initial B-spline curve, the finite element method is adopted to make the active B-spline curve converge towards the target curve without the need of data points parameterization. A strategy of automatic control point insertion during the B-spline active contour deformation, adaptive to the shape of the planar curve, is also given. Experimental results show that this method is efficient and accurate in planar curve approximation.

TRIPLET CIRCULAR HOUGH TRANSFORM FOR CIRCLE DETECTION

A new method, triplet circular Hough transform, is proposed for circle detection in image processing and pattern recognition. In the method, a curve in an image is first detected. Next, a sequence of three points on the curve are selected, a sequence of parameters (a,b,r) corresponding to the three points are calculated by solving the circle equation of the curve, and two 2-D accumulators A(a,b) and R(a,b) are accumulated with 1 and r, respectively. Then the parameters {(a, b, r)} of the circles fitting the curve are determined from A(a,b) and R(a,b) by searching for the local maximum over A(a,b). Because no computation loops over center (a, 6) and/or radius r are needed, the method is faster than the basic and directional gradient methods. It needs also much smaller memory for accumulation.