参数曲线近似弧长参数化的三角函数方法

本描述了一种局部的近似弧长参数化插值方法，用三角函数对曲线的弧长函数进行分段逼近，段与段之间是相互独立的，且插值曲线在插值点处的弧长与原参数曲线的真实弧长相等。     

用分段仿射变换生成C^1参数样条曲线

利用分段仿射变换，可将任意函数类型的Ｃ＾１曲线段生成一条Ｃ＾１的参数样条曲线，分而段的保持所选曲线的仿射不变的几何性质。     

一类三阶参数曲线

本提出的一类参数曲线，是通常二次Bézier曲线和B一样条曲线等的一种推广。即用l,t,φ（t）三介函数的线性组合构成的参数曲线，讨论了其几何性质。     

拟合隐函数曲线的GNL法

１参数可线性化的曲线拟合问题假设我们已经获得ｎ组独立观测值（ｘｉ,ｙｉ）,其中是精确观测值,ｉ＝１,２,…,ｎ,下同．现欲在最小二乘意义下,拟合非线性模型并假设观测点都分布在模型函数曲线附近．其中为连续可导函数ｇ（ｙ）的反函数,且在我们所研究的ｙ变化范围内恒有ｇ’（ｙ）≠０,（ｊ＝１,２.....下同）为ｐ（＜ｎ）个待定参数．（ｘ）为区间Ｘ内ｐ个线性无关的已知连续函数．随机变量,即为也就是要求求解残差平方和Ｑ（）的无条件极值问题其中参数向量,残差向量,而残差都是微量．对这类常见问题,有以下几种算法．…     

隐函数求导在曲线、曲面设计中的应用

以隐函数求导为工具进行曲线、曲面设计     

函数解析式及函数曲线的应用探讨

函数是数学最基本的概念之一 ,是进一步学习不可或缺的知识 .由美国国家科学基金会资助 ,以哈佛大学为首的合作组编写的教材———“微积分”(以下简称“哈佛教材”) ,函数部分知识的选材很有新意 ,非常值得我们研究和借鉴 .本文从“哈佛教材”中选取求函数解析式、函数曲线应用的例子 ,供读者感悟“哈佛教材”的特色 ,启迪我们改革教材的思路 .1　利用问题的变化规律求函数解析式函数的某些特性可以直接用来模拟实际问题 ,数学教材如能突出这一点 ,将有利于学生掌握知识、提高能力、发展智力 ,同时这样的教学内容也能使课堂气氛生动、形象…     

NURBS曲线的一个插值性质

本介绍了非均匀有理B样条曲线，并给出了非均匀有理B样条曲线的一个插值性质。     

复二次B样条曲线

本文在复平面单位圆弧上引进了复二次Ｂ样条曲线，讨论了它的一些几何性质．实质上它是分段帕斯卡蜗线段的Ｃ１合成曲线．调整控制点可使某段曲线为圆孤．     

样条曲线光顺的数学模型分析

采用函数三次样条光顺曲线,证明在样条曲线局部转角小,总转角不超过120°情况下,曲线的光顺指示函数y″(1+y′2)3/2可以简化为二阶导数曲线y″(x).由于y″(x)对x是分段折线函数,对y是线性泛函,因而定出不光顺之处及用叠加原理计算调整公式均变得很简单.此样条函数曲线光顺能够采用电脑自动化进行.     

二次带形状参数双曲B样条曲线

在空间Ω_5=span{1,sinh t,cosh t,sinh 2t,cosh 2t}上给出了二次带形状参数双曲B样条的基函数.由这组基组成的二次双曲B样条曲线是C~1连续的,同时具有很多与二次B样条曲线类似的性质和几何结构,并且可以精确表示双曲线.在控制多边形固定的情况下,可以通过调节形状参数的大小来进一步调整曲线的形状.     

Generating approximate parametric roots of parametric polynomials

Built upon a ground field is the parametric field, the Puiseux field, of semi-terminating formal fractional power series. A parametric polynomial is a polynomial with coefficients in the parametric field, and roots of parametric polynomials are parametric. For a parametric polynomial with nonterminating parametric coefficients and a target accuracy, using sensitivity of the Newton Polygon process, a complete set of approximate parametric roots, each meeting target accuracy, is generated. All arguments are algebraic, from the inside out, self-contained, penetrating, and uniform in that only the Newton Polygon process is used, for both preprocessing and intraprocessing. A complexity analysis over ground field operations is developed; setting aside root generation for ground field polynomials, but bounding such, polynomial bounds are established in the degree of the parametric polynomial and the target accuracy.     

Piecewise parametric polynomial fuzzy sets

We present a scheme for tractable parametric representation of fuzzy set membership functions based on the use of a recursive monotonic hierarchy that yields different polynomial functions with different orders. Polynomials of the first order were found to be simple bivalent sets, while the second order polynomials represent the typical saw shape triangles. Higher order polynomials present more diverse membership shapes. The approach demonstrates an enhanced method to manage and fit the profile of membership functions through the access to the polynomials order, the number and the multiplicity of anchor points as wells as the uniformity and periodicity features used in the approach. These parameters provide an interesting means to assist in fitting a fuzzy controller according to system requirements. Besides, the polynomial fuzzy sets have tractable characteristics concerning the continuity and differentiability that depend on the order of the polynomials. Higher order polynomials can be differentiated as many times as the order of the polynomial less the multiplicity of the anchor points. An algorithmic optimization approach using the steepest descent method is introduced for fuzzy controller tuning. It was shown that the controller can be optimized to model a certain output within small number of iterations and very small error margins. The mathematics generated by the approach is consistent and can be simply generalized to standard applications. The recursive propagation was noticed for its clarity and ease of calculations. Further, the degree of association between the sets is not limited to the neighbors as in traditional applications; instead, it may extend beyond.Such approach can be useful in dynamic fuzzy sets for adaptive modeling in view of the fact that the shape parameters can be easily altered to get different profiles while keeping the math unchanged. Hypothetically, any shape of membership functions under the partition of unity constraint can be produced. The significance of the mentioned characteristics of such modeling can be observed in the field of combinatorial and continuous parameter optimization, automated tuning, optimal fuzzy control, fuzzy-neural control, membership function fitting, adaptive modeling, and many other fields that require customized as well as standard fuzzy membership functions. Experimental work of different scenarios with diverse fuzzy rules and polynomial sets has been conducted to verify and validate our results.     

Piecewise quadratic trigonometric polynomial curves

Analogous to the quadratic B-spline curve, a piecewise quadratic trigonometric polynomial curve is presented in this paper. The quadratic trigonometric polynomial curve has continuity, while the quadratic B-spline curve has continuity. The quadratic trigonometric polynomial curve is closer to the given control polygon than the quadratic B-spline curve.

     

Dynamic PDE parametric curves

Dynamic partial differential equation (PDE) parametric curves which can be expressed as a coupled system of two hyperbolic equations are developed. In curve design, dynamic PDE parametric curves can be modified intuitively and are more flexible than ordinary differential equation (ODE) curves. The calculation of dynamic PDE parametric curves must recur to numerical methods and a three-level finite difference scheme is proposed. Approximation and stability properties for the scheme are proved and convergence property is derived. An example of interpolating PDE curves is presented as an application of dynamic PDE parametric curves.     

Robust fitting of parametric curves

We consider the problem of fitting a parametric curve to a given point cloud (e.g., measurement data). Least-squares approximation, i.e., minimization of the ℓ₂ norm of residuals (the Euclidean distances to the data points), is the most common approach. This is due to its computational simplicity [1]. However, in the case of data that is affected by noise or contains outliers, this is not always the best choice, and other error functions, such as general ℓ_p norms have been considered [2]. We describe an extension of the least-squares approach which leads to Gauss-Newton-type methods for minimizing other, more general functions of the residuals, without increasing the computational costs significantly. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Continuous approximate solutions in parametric optimization

The question of the existence of approximate solutions in parametric optimization is considered. Most results show that (under hypotheses) if a certain optimization problem has an approximate solution x ₀ for a value p ₀ of a parameter, then an approximate solution x=b(p) can be found for p in P, with b continuous, b(p ₀)=x₀, and any two such bs are homotopic. Some topological methods (use of fibrations) are used to weaken the usual convex hypotheses of such results. An equisemicontinuity condition (relative to a constraint) is introduced to allow some noncompactness. The results are applied to get approximate Nash equilibrium results for games with some nonconvexity in the strategy sets.     

Piecewise polynomial spaces and geometric continuity of curves

Summary We say that a curve has geometric continuity if its curvatures and Frenet frame are continuous. In this paper we introduce spaces of piecewise polynomials which can be used to model space curves which have geometric continuity. We show that the basic theoretical properties of ordinary spline functions also hold for these spaces. These results extend and unify recent work on Beta-splines and Nu-splines which are used as a design tool in computer-aided geometric design of free form curves and surfaces.This work was initiated when the first author was on Sabbatical at Thomas J. Watson IBM Research Center, and was partially supported by the U.S.-Israel Binational Foundation, grant no. 86-00243/1.     

On fair parametric rational cubic curves

First we derive conditions that a parametric rational cubic curve segment, with a parameter, interpolating to plane Hermite data {(x _i ^(k) ,y _i ^(k) ),i = 0, 1;k = 0, 1} contains neither inflection points nor singularities on its segment. Next we numerically determine the distribution of inflection points and singularities on a segment which gives conditions that aC ² parametric rational cubic curve interpolating to dataS = {(x _i ^(k) ,y _i ^(k) ), 0 i n} is free of inflection points and singularities. When the parametric rational cubic curve reduces to the well-known parametric cubic one, we obtain a theorem on the distribution of the inflection points and singularities on the cubic curve segment which has been widely used for finding aC ¹ fair parametric cubic curve interpolating toS.     

Computing parallel curves on parametric surfaces

Generating parallel curves on parametric surfaces is an important issue in many industrial settings. Given an initial curve (called the base curve or generator) on a parametric surface, the goal is to obtain curves on the surface that are parallel to the generator. By parallel curves we mean curves that are at a given distance from the generator, where distance is measured point-wise along certain characteristic curves (on the surface) orthogonal to the generator. Except for a few particular cases, computing these parallel curves is a very difficult and challenging problem. In fact, only partial, incomplete solutions have been reported so far in the literature. In this paper we introduce a simple yet efficient method to fill this gap. In clear contrast with other existing techniques, the most important feature of our method is its generality: it can be successfully applied to any differentiable parametric surface and to any kind of characteristic curves on surfaces. To evaluate our proposal, some illustrative examples (not addressed with previous methods) for the cases of section, vector-field, and geodesic parallels are discussed. Our experimental results show the excellent performance of the method even for the complex case of NURBS surfaces.     

Continuity of approximate solution mappings for parametric equilibrium problems

In this paper, we obtain sufficient conditions for Hausdorff continuity and Berge continuity of an approximate solution mapping for a parametric scalar equilibrium problem. By using a scalarization method, we also discuss the Berge lower semicontinuity and Berge continuity of a approximate solution mapping for a parametric vector equilibrium problem.