Weighted cubic and biharmonic splines

In this paper we discuss the design of algorithms for interpolating discrete data by using weighted cubic and biharmonic splines in such a way that the monotonicity and convexity of the data are preserved. We formulate the problem as a differential multipoint boundary value problem and consider its finite-difference approximation. Two algorithms for automatic selection of shape control parameters (weights) are presented. For weighted biharmonic splines the resulting system of linear equations can be efficiently solved by combining Gaussian elimination with successive over-relaxation method or finite-difference schemes in fractional steps. We consider basic computational aspects and illustrate main features of this original approach.     

Reconstruction of curves with minimal energy using a blending interpolator

A weighted blending interpolator, a kind of smooth rational spline based only on function values, is constructed using a rational cubic spline and a polynomial spline. In order to meet the needs of practical design, a new control method is employed to control the shape of curves. The advantage of the method is that it can be applied to modify the local shape of an interpolating curve by selecting suitable parameters and weight coefficients simply. Also, when the weight coefficient is in [0,1], the error estimation formula of this interpolator is obtained. Copyright © 2012 John Wiley & Sons, Ltd.     

Bézier曲线和B样条曲线光顺拟合法

§1.引言 在计算机辅助几何设计(CAGD)工作中,适用于曲线造型的方法主要有样条函数、Bezier曲线和B样条曲线等。在实际工作中,几何外形设计又大致可以分成两类: (1)从头设计。按照给定的几个原始设计参数,决定曲线的特征多边形顶点,继而决定曲面的特征网格。在[1],[2]中所作的叶片和船体曲面造型,就是一种从头设计方案。 (2)模型设计。例如,传统的汽车车身设计,首先由美工师塑造一只车身的油泥模     

以节点与权因子修改为基础的4阶NURBS受限形状控制

改变k阶NURBS曲线的节点,会产生一个单参数NURBS曲线族,该曲线族的包络是用相同控制顶点定义的k-a阶NURBS曲线,这里a是所改变的节点的重数.论文运用这项理论结果,提出了几种建立在修改一个节点与两个连续权因子基础上的4阶NURBS形状控制方法,该方法要受一定的位置与切线方向的约束.     

A tension approach to controlling the shape of cubic spline surfaces on FVS triangulations

We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to C¹ cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods.     

参数保形插值中决定切矢的方法

由分段三次参数多项式曲线拼合成的Ｃ１插值曲线的形状与数据点处的切矢有很大关系．基于对保形插值曲线特点的分析，本文提出了估计数据点处切矢的一种方法：采用使构造的插值曲线的长度尽可能短的思想估计数据点处的切矢，并且通过四组有代表性的数据对本方法和已有的三种方法进行了比较．     

The cubic trigonometric Bézier curve with two shape parameters

A cubic trigonometric Bézier curve analogous to the cubic Bézier curve, with two shape parameters, is presented in this work. The shape of the curve can be adjusted by altering the values of shape parameters while the control polygon is kept unchanged. With the shape parameters, the cubic trigonometric Bézier curves can be made close to the cubic Bézier curves or closer to the given control polygon than the cubic Bézier curves. The ellipses can be represented exactly using cubic trigonometric Bézier curves.     

A shape-preserving approximation by weighted cubic splines

This paper addresses new algorithms for constructing weighted cubic splines that are very effective in interpolation and approximation of sharply changing data. Such spline interpolations are a useful and efficient tool in computer-aided design when control of tension on intervals connecting interpolation points is needed. The error bounds for interpolating weighted splines are obtained. A method for automatic selection of the weights is presented that permits preservation of the monotonicity and convexity of the data. The weighted B-spline basis is also well suited for generation of freeform curves, in the same way as the usual B-splines. By using recurrence relations we derive weighted B-splines and give a three-point local approximation formula that is exact for first-degree polynomials. The resulting curves satisfy the convex hull property, they are piecewise cubics, and the curves can be locally controlled with interval tension in a computationally efficient manner.     

An energy-minimization framework for monotonic cubic spline interpolation

This paper describes the use of cubic splines for interpolating monotonic data sets. Interpolating cubic splines are popular for fitting data because they use low-order polynomials and have C² continuity, a property that permits them to satisfy a desirable smoothness constraint. Unfortunately, that same constraint often violates another desirable property: monotonicity. It is possible for a set of monotonically increasing (or decreasing) data points to yield a curve that is not monotonic, i.e., the spline may oscillate. In such cases, it is necessary to sacrifice some smoothness in order to preserve monotonicity.The goal of this work is to determine the smoothest possible curve that passes through its control points while simultaneously satisfying the monotonicity constraint. We first describe a set of conditions that form the basis of the monotonic cubic spline interpolation algorithm presented in this paper. The conditions are simplified and consolidated to yield a fast method for determining monotonicity. This result is applied within an energy minimization framework to yield linear and nonlinear optimization-based methods. We consider various energy measures for the optimization objective functions. Comparisons among the different techniques are given, and superior monotonic C² cubic spline interpolation results are presented. Extensions to shape preserving splines and data smoothing are described.     

On the quartic curve of Han

The quartic curve of Han [X. Han, Piecewise quartic polynomial curves with shape parameter, Journal of Computational and Applied Mathematics 195 (2006) 34–45] can be considered as the generalization of the cubic B-spline curve incorporating shape parameters into the polynomial basis functions. We show that this curve can be considered as the linear blending of the original cubic B-spline curve and a fixed quartic curve. Moreover, we present the Bézier form of the curve, which is useful in terms of incorporating the curve into existing CAD systems. Geometric effects of the alteration of shape parameters is also discussed, including design oriented computational methods for constrained shape control of the curve.     

Practical selection of neighbourhoods for local regression in the bivariate case

The use of supports that are variable in shape and size are considered here, in dimension two, for a weighted local average of the data measured on a grid. The possibilities of capturing local features of the function with local methods are explored. Such a scheme is in line with the approximation schemes with parameters regulated adaptively to the data.     

Galerkin's finite element formulation of the system of fourth‐order boundary‐value problems

In this article, a Galerkin's finite element approach based on weighted‐residual is presented to find approximate solutions of a system of fourth‐order boundary‐value problems associated with obstacle, unilateral and contact problems. The approach utilizes a piece‐wise cubic approximations utilizing cubic Hermite interpolation polynomials. Numerical studies have shown the superior accuracy and lesser computational cost of the scheme in comparison to cubic spline, non‐polynomial spline and cubic non‐polynomial spline methods. Numerical examples are presented to illustrate the applicability of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1551–1560, 2011     

Dog and tricks: some mathematics for students of graphics

This paper describes how parametric cubic splines and cubicBezier curves may be used in designing a two dimensional shape.A simple aerofoil shape is designed using both methods. Themathematics is described and the shape drawn using Excel. Theeffect of varying parameters is shown in both methods.     

Monotone and convex interpolation by weighted quadratic splines

In this paper we discuss the design of algorithms for interpolating discrete data by using weighted C ¹ quadratic splines in such a way that the monotonicity and convexity of the data are preserved. The analysis culminates in two algorithms with automatic selection of the shape control parameters: one to preserve the data monotonicity and other to retain the data convexity. Weighted C ¹ quadratic B-splines and control point approximation are also considered.     

Monotone and convex interpolation by weighted cubic splines

Algorithms for interpolating by weighted cubic splines are constructed with the aim of preserving the monotonicity and convexity of the original discrete data. The analysis performed in this paper makes it possible to develop two algorithms with the automatic choice of the shape-controlling parameters (weights). One of them preserves the monotonicity of the data, while the other preserves the convexity. Certain numerical results are presented.     

广义Pareto分布的广义有偏概率加权矩估计方法

广义Pareto分布(GPD)是统计分析中一个极为重要的分布,被广泛应用于金融、保险、水文及气象等领域.传统的参数估计方法如极大似然估计、矩估计及概率加权矩估计方法等已被广泛应用,但使用中存在一定的局限性.虽然提出很多改进方法如广义概率加权矩估计、L矩和LH矩法等,但都是研究完全样本的估计问题,而在水文及气象等应用领域常出现截尾样本.本文基于概率加权矩理论,利用截尾样本对三参数GPD提出一种应用范围广且简单易行的参数估计方法,可有效减弱异常值的影响.首先求解出具有较高精度的形状参数的参数估计,其次得出位置参数及尺度参数的参数估计.通过Monte Carlo模拟说明该方法估计精度较高.     

Inflections and singularity on parametric rational cubic curves

Summary. This paper considers the distribution of inflection points and singularity on the parametric rational cubic curve, using much algebraic manipulation. Its use allows one to find a shape preserving interpolatory rational cubic curve of a planar data. Some numerical examples are given to illustrate usefulness of the method. Received April 30, 1995 / Revised version received January 15, 1996     

C^3连续的保形插值三角样本曲线

本给出了构造保形插值曲线的三角样条方法，即在每两个型值点之间构造两段三次参数三角样条曲线。所构造的插值曲线是局部的，保形的和C^3连续的而且曲线的形状可由参数调节。     

<Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> positivity preserving scattered data interpolation using rational Bernstein-Bézier triangular patch

A local C ¹ positivity preserving scheme is developed using Bernstein-Bézier rational cubic function. The domain is triangulated by Delaunay triangulation method. Simple sufficient conditions are derived on the inner and boundary Bézier ordinates to preserve the shape of positive data. These inner and boundary Bézier ordinates involve weights in their definition. In any triangular patch if the Bézier ordinates do not satisfy the derived conditions of positivity, then these are modified by the weights (free parameters) involved in the construction of Bernstein-Bézier rational cubic function to preserve the shape of positive scattered data.     

有理B样条曲线的光顺拟合法

研究了用三次均匀有理Ｂ样条样曲线光顺拟合一组平面点列的问题,其中光顺性由曲线的能量积分与扰动的权平均来确定。