Many-Knot Spline Interpolating Curves and Their Applications in Font Design

Many-knot spline interpolating is a class of curves and surfaces fitting method presentedin 1974. Many-knot spline interpolating curves are suitable to computer aided geometric design anddata points interpolation. In this paped, the properties of many-knot spline interpolating curves arediscussed and their applications in font design are considered. The differences between many-knotspline interpolating curves and the curves genoaed by exceeding-lacking adjuStment algorithm aregiven.

Convexity-preserving interpolation of trigonometric polynomial curves with a shape parameter

In computer aided geometric design(CAGD) ,it is often needed to produce a convexity-preserving interpolating curve according to the given planar data points. However,most existing pertinent methods cannot generate convexity-preserving in-terpolating transcendental curves;even constructing convexity-preserving interpolating polynomial curves,it is required to solve a system of equations or recur to a complicated iterative process. The method developed in this paper overcomes the above draw-backs. The basic idea is:first to construct a kind of trigonometric polynomial curves with a shape parameter,and interpolating trigonometric polynomial parametric curves with C2(or G1) continuity can be automatically generated without having to solve any system of equations or do any iterative computation. Then,the convexity of the constructed curves can be guaranteed by the appropriate value of the shape parameter. Performing the method is easy and fast,and the curvature distribution of the resulting interpolating curves is always well-proportioned. Several numerical examples are shown to substantiate that our algorithm is not only correct but also usable.

C^n-CONTINUOUS B-TYPE SPLINE CURVES WITH ITSLOCALIZATION INTERPOLATION AND APPROXIMATION

This paper presents a class of Cn- continuous B- type spline curves with some paramet- ric factors.The length of their local support is equal to4.Taking the different values of the parametric factors,the curves can become free- type curves or interpolate a set of given points even mix the both cases.When the parametric factors satisfy the certain conditions,the degrees of the curves can be decreased as low as possible.Besides,when all the parametric factors tend to zero,the curves globally approximate to the control polygon.

NURBS CURVE INTERPOLATOR WITH ADAPTIVE ACCELERATION-DECELERATION CONTROL

The feedrate profile of non-uniform rational B-spline （NURBS） interpolation due to the contour errors is analyzed. A NURBS curve interpolator with adaptive acceleration-deceleration control is presented. In interpo- lation preprocessing, the sensitive zones of feedrate variations are processed with acceleration-deceleration control. By using the proposed algorithm, the machining accuracy is guaranteed and the feedrate is adaptively adjusted to he smoothed. The mechanical shock imposed in the servo system is avoided by the first and the second time derivatives of feedrates. A simulation of NURBS interpolation is given to demonstrate the validity and the effectiveness of the algorithm. The proposed interpolator can also be applied to the trajectory planning of the other parametric curves.

A comparison of piecewise cubic Hermite interpolating polynomials,cubic splines and piecewise linear functions for the approximation of projectile aerodynamics

Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics.The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions:they are continuous,differentiable at least once,and have a relatively low degree.The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools,and then compares Piecewise Cubic Hermite Interpolating Polynomial(PCHIP),cubic splines,and piecewise linear functions,and their variant,as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile.A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics,and its evaluation against a set of criteria.Finally,the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.

LOOK-AHEAD ALGORITHM FOR VELOCITY CONTROL BASED ON PARAMETERIZED CURVE INTERPOLATOR

To avoid suffering gouge and transient overshooting in high speed cutting machining, a novel parametefized curve interpolator model with velocity look-ahead algorithm is proposed. Based on a prearrangement step interpolation algorithm for parameterized curves and considering high curvature points, parameterized curve tool path is divided into acceleration segments and deceleration segments by look-ahead algorithm. Under condition of characteristics of acceleration and deceleration stored in control system, deceleration before high curvature points and acceleration after high curvature points are realized in real-time in high speed cutting machining. Based on new parameterized curve interpolator model with velocity look-ahead algorithm, a real cubic spline is machined simulativly. The simulation results show that velocity look-ahead algorithm improves velocity changing more smoothly.

MODIFIABLE QUARTIC AND QUINTIC CURVES WITH SHAPE-PARAMETERS

This paper presents a method for creating modificable quartic and quintic curves with shape parameters. The curves can achieve C 2 even C 3 continuity and unify both interpolation and approximation to the control points without solving a system of equations or inserting additional control points. They have the local properties like the cubic B spline. Besides, the quintic curve would be able globally to tend the control polygon.

COMPLEX RATIONAL TYPE INTERPOLATION AT DISTURBED FEJR POINTS ON A CURVE

The results of accurate order of uniform approximation and simultaneous approximation in the early work ＂Jackson Type Theorems on Complex Curves＂ are improved from Fejer points to disturbed Fejer points in this article. Furthermore, the stability of convergence of Tn,∈（f,z） with disturbed sample values f（z^＊） ＋ Sk are also proved in this article.

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.

Research on the Arc Interpolation for the Nonstandard Section Curve of the Piston

The transverse section of piston skirt is not a standard circle and is with high precision. So the section curve should be interpolated through the high accuracy method of circular arc interpolation before NC machining. In order to smooth the connection of adjacent arcs and shorten the NC machining program, an interpolation method based on Chebyshev theory of function approximation is proposed here. According to the analysis of the interpolation error, the algorithm is simple and with high precision. By this way the fewest interpolating circular arc segments can be got, and the manufacture requirement is satisfied with the circular arc interpolating curves.

Calculation of Space NURBS Curve Interpolation Error

Aiming at the problem of low accuracy of interpolation error calculation of existing NURBS curves, an approximate method for the distance between a point and a NURBS interpolation curve is proposed while satisfying the accuracy of the solution. Firstly, the minimum parameter interval of the node vector corresponding to the data point under test in the original data point sequence is determined, and the parameter interval is subdivided according to the corresponding step size, and the corresponding parameter value is obtained. Secondly, the distance from the measured point to the NURBS curve is calculated, and the nearest distance is found out. The node interval is subdivided again on one side of the nearest distance. Finally, the distance between the data point to be measured and each subdivision point is calculated again, and the minimum distance is taken as the interpolation error between the point and the NURBS curve. The simulation results of actual tool position data show that this method can more accurately obtain the error of spatial NURBS interpolation curve.

Curve interpolation model for visualising disjointed neural elements

Neuron cell are built from a myriad of axon and denddte structures. It transmits electrochemical signals between the brain and the nervous system. Three-dimensional visualization of neuron structure could help to facilitate deeper understanding of neuron and its models. An accurate neuron model could aid understanding of brain＇s functionalities, diagnosis and knowledge of entire nervous system. Existing neuron models have been found to be defective in the aspect of realism. Whereas in the actual biological neuron, there is continuous growth as the soma extending to the axon and the dendrite; but, the current neuron visualization models present it as disjointed segments that has greatly mediated effective realism. In this research, a new reconstruction model comprising of the Bounding Cylinder, Curve Interpolation and Gouraud Shading is proposed to visualize neuron model in order to improve realism. The reconstructed model is used to design algorithms for generating neuron branching from neuron SWC data. The Bounding Cylinder and Curve Interpolation methods are used to improve the connected segments of the neuron model using a series of cascaded cylinders along the neuron＇s connection path. Three control points are proposed between two adjacent neuron segments. Finally, the model is rendered with Gouraud Shading for smoothening of the model surface. This produce a near-perfection model of the natural neurons with attended realism. The model is validated by a group of bioinformatics analysts＇ responses to a predefined survey. The result shows about 82% acceptance and satisfaction rate.

CONSTRUCTION OF G^2 CONTINUOUS CURVES ON SURFACE WITH PLANAR CUBIC BZIER CURVES

The problem of constructing curve on parametric surface (or surface that canbe parameterized) such that it interpolates a sequence of points with prescribed tangent directionand curvature vector (or geodesic curvature) at every point and the issue of curve blending on thiskind of surface are researched. The mapping and tangent mapping from the surface to its parametricplane are introduced and thus several conclusions with differential geometry are deduced. Based onthose conclusions, the problem of interpolating (or blending) curve on a parametric surface isconverted to a similar one on its parametric plane. The final solution curve of either interpolationor blending issue is explicit and can still be expressed by parametric form. And so, unlikeexisting methods, the presented method needs not to use any surface/ surface intersectionalgorithms, usually a troublesome process, for displaying such interpolation curve. Experimentresults show the presented methods are feasible and applicable to CAD/CAM and computer graphics

A SMOOTH C^1 INTERPOLATION FOR TWO-DIMENSIONAL CONTACT PROBLEMS USING PARAMETRIC CURVE TECHNIQUE

A smooth C^1 interpolation for two-dimensional contact problems using parametric curve technique was developed and implemented.The parametric curve can ensure C^1 continuity of the contact surfaces and provide a unique surface normal vector.Some numerical examples were used to illustrate the advantages of the newly developed representation of contact surface. The results reveal a significant improvement in the prediction of contact stresses and contact area.The predicted contact stresses are less sensitive to the mismatch in meshes of the different contacting bodies.

C2 Continuous Quartic Hermite Spline Curves with Shape Parameters

In order to relieve the deficiency of the usual cubic Hermite spline curves,the quartic Hermite spline curves with shape parameters is further studied in this work. The interpolation error and estimator of the quartic Hermite spline curves are given. And the characteristics of the quartic Hermite spline curves are discussed.The quartic Hermite spline curves not only have the same interpolation and continuity properties of the usual cubic Hermite spline curves, but also can achieve local or global shape adjustment and C;continuity by the shape parameters when the interpolation conditions are fixed.

(G^2-continuous) SHAPE PRESERVING CUBIC INTERPOLATION SPLINE CURVE

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CURVE AND SURFACE INTERPOLATIONBY SUBDIVISION ALGORITHMS

Interpolatory subdivision algorithms for the generation of curves and surfaces play a veryimportant rule in shape design and modelling in CAD/CAM systems. In this paper, by using the dif-ference and divided difference analysis, a systematic method to construct Cn (n≥ 0) interpolatorycurves by subdivision from given data is described and the mask (filter) of the algorithm is presentedexplicitly. This algorithm generates a Cn smooth curve which interpolates the initial control points.Control parameters are also provided so that the shape of the final curve can be adjusted according torequirements. An immediate generalisation of the method is the construction of smooth interpolatorysubdivision algorithms over uniform triangular networks (tensor product type data) in Rm. The mainresults of this algorithm for smooth interpolatory surface subdivision algorrthm are also included.AMS(MOS) : 65D05 , 65D15 , 65D17.

面向曲率及边界距离约束的弯曲管道内路径规划

弯曲管道结构是工业探测或医用诊察中常见的一种场景,传统机器人的路径规划大多基于管道中心线的提取,而忽略了在狭窄空间中作业的真实参数约束,易使得作业工具末端受到弯折或挤压,影响作业效率,增加作业风险.针对以上问题,提出了一种面向真实曲率和边界距离参数约束的路径规划方法,在可视化的三维模型中生成管道的中心稠密点集,针对该点集提出两步走的空间特征点识别策略.在选定作业路径的初始点、终止点后,依据邻近点间的曲率均值、极值及差分关系识别曲率极值点、分段点,通过对粗扫描特征点的遍历进一步识别出拐点和宽域节点以及过密点.将识别出的特征点作为型值点,反算得到NURBS曲线及对应插值点的曲率值,通过模型切片搜索得到曲线插值点的边界距离不合格点集.将极值点添加至特征点集中,再通过搜索得到的曲率不合格点集,将极值点进行拟合圆调整至后更新特征点集、在完成自适应轮次的特征点迭代后,检测并标记为空或连续不变的全区间参数不合格点区间集.实验结果证明:弯曲圆柱仿真模型和CT重建肾腔模型的规划路径经由迭代分别满足曲率10 mm^(-1)、8 mm^(-1)的约束.同时最短距离的全区间检测及标记结果也表明本文方法可以在满足曲率要求的前提下尽可能不碰撞管道内壁,验证了本文方法的有效性和可行性.

数控系统样条曲线插补技术研究综述

传统数控系统加工复杂曲线时,一般采用微小线段或圆弧段逼近的方法,加工精度低,易引发刀具抖动、机床振动等问题。随着CAD/CAM技术的发展,现代数控机床向高速度、高柔性方向发展,基于高质量加工的要求,提出样条曲线插补方法。对样条曲线插补方法进行综述,首先阐述了插补技术的重要性;其次从进给速度、加工精度等角度系统分析了样条曲线插补方法,分别介绍了A样条插补、B样条插补和C样条插补;最后描述了三种插补方法的发展过程和最新研究动态,对其未来趋势做了总结和展望。