Least eccentric ellipses for geometric Hermite interpolation

We present a rational Bézier solution to the geometric Hermite interpolation problem. Given two points and respective unit tangent vectors, we provide an interpolant that can reproduce a circle if possible. When the tangents permit an ellipse, we produce one that deviates least from a circle. We cast the problem as a theorem and provide its proof, and a method for determining the weights of the control points of a rational curve. Our approach targets ellipses, but we also present a cubic interpolant that can find curves with inflection points and space curves when an ellipse cannot satisfy the tangent constraints.     

A meshfree Hermite‐type radial point interpolation method for Kirchhoff plate problems

A meshfree computational method is proposed in this paper to solve Kirchhoff plate problems of various geometries. The deflection of the thin plate is approximated by using a Hermite‐type radial basis function approximation technique. The standard Galerkin method is adopted to discretize the governing partial differential equations which were derived from using the Kirchhoff's plate theory. The degrees of freedom for the slopes are included in the approximation to make the proposed method effective in enforcing essential boundary conditions. Numerical examples with different geometric shapes and various boundary conditions are given to verify the efficiency, accuracy, and robustness of the method. Copyright © 2005 John Wiley & Sons, Ltd.     

Object representation and recognition in shape spaces

In this paper, we describe a shape space based approach for invariant object representation and recognition. In this approach, an object and all its similarity transformed versions are identified with a single point in a high-dimensional manifold called the shape space. Object recognition is achieved by measuring the geodesic distance between an observed object and a model in the shape space. This approach produced promising results in 2D object recognition experiments: it is invariant to similarity transformations and is relatively insensitive to noise and occlusion. Potentially, it can also be used for 3D object recognition.     

Polynomial approximation to clothoids via s-power series

Given an arbitrary segment of a clothoid over a finite interval, we propose a novel method for generating a polynomial approximation, based on employing s-power series, the two-point analogue of Taylor expansions. Truncating at the kth term the s-power series furnishes the order-k Hermite interpolant, i.e. the degree-(2k+1) polynomial curve that reproduces up to the kth derivative of the original curve at the endpoints of a given interval. By piecing these approximations we obtain a Hermitian spline that exhibits C^k continuity at the joints and enjoys almost arc-length parameterization. This is a more suitable alternative than the truncated Taylor series or the blending of Taylor expansions advocated by Wang et al. [Computer-Aided Design 33 (2001) 1049] in a recent article.     

C-Coons曲面片及其性质

讨论由C-Herrnite函数构造的C-Coons曲面片．C-Coons曲面是双三次Coons曲面的推广，它的形状调整还依赖于参数α，β，当α，β→0时，Coons曲面的极限是双三次Coons曲面．     

Numerical inversion of multidimensional Laplace transforms by the Laguerre method

Numerical transform inversion can be useful to solve stochastic models arising in the performance evaluation of telecommunications and computer systems. We contribute to this technique in this paper by extending our recently developed variant of the Laguerre method for numerically inverting Laplace transforms to multidimensional Laplace transforms. An important application of multidimensional inversion is to calculate time-dependent performance measures of stochastic systems. Key features of our new algorithm are: (1) an efficient FFT-based extension of our previously developed variant of the Fourierseries method to calculate the coefficients of the multidimensional Laguerre generating function, and (2) systematic methods for scaling to accelerate convergence of infinite series, using Wynn's ε-algorithm and exploiting geometric decay rates of Laguerre coefficients. These features greatly speed up the algorithm while controlling errors. We illustrate the effectiveness of our algorithm through numerical examples. For many problems, hundreds of function evaluations can be computed in just a few seconds.     

On Pattern Frequency Occurrences in a Markovian Sequence

Consider a given pattern H and a random text T generated by a Markovian source. We study the frequency of pattern occurrences in a random text when overlapping copies of the pattern are counted separately. We present exact and asymptotic formulae for moments (including the variance), and probability of r pattern occurrences for three different regions of r , namely: (i) r=O(1) , (ii) central limit regime, and (iii) large deviations regime. In order to derive these results, we first construct certain language expressions that characterize pattern occurrences which are later translated into generating functions. We then use analytical methods to extract asymptotic behaviors of the pattern frequency from the generating functions. These findings are of particular interest to molecular biology problems (e.g., finding patterns with unexpectedly high or low frequencies, and gene recognition), information theory (e.g., second-order properties of the relative frequency), and pattern matching algorithms (e.g., q -gram algorithms).     

Multi-degree reduction of NURBS curves based on their explicit matrix representation and polynomial approximation theory

NURBS curve is one of the most commonly used tools in CAD systems and geometric modeling for its various specialties, which means that its shape is locally adjustable as well as its continuity order, and it can represent a conic curve precisely. But how to do degree reduction of NURBS curves in a fast and efficient way still remains a puzzling problem. By applying the theory of the best uniform approximation of Chebyshev polynomials and the explicit matrix representation of NURBS curves, this paper gives the necessary and sufficient condition for degree reducible NURBS curves in an explicit form. And a new way of doing degree reduction of NURBS curves is also presented, including the multi-degree reduction of a NURBS curve on each knot span and the multi-degree reduction of a whole NURBS curve. This method is easy to carry out, and only involves simple calculations. It provides a new way of doing degree reduction of NURBS curves, which can be widely used in computer graphics and industrial design.     

求解线性时变系统状态方程的一种新方法

本文给出了移位雅可比多项式的乘积运算矩阵,利用该矩阵,将时变系统的状态方程等效变换为一个简单的矩阵代数方程,使其求解大为简化。通过对两个实例的计算,获得了相当令人满意的结果。     

龙格现象难题破解之系数与阶次双确定方法

龙格现象指出,使用基于等距节点的高阶插值多项式逼近龙格函数时,插值多项式在逼近区间两端会产生明显的振荡现象。因此,传统认为,不适宜用基于等距节点的高阶多项式逼近龙格函数。针对龙格现象,展示一种新型的多项式系数与阶次双确定方法。该方法可快速构造出基于等距节点的不会振荡且有较高逼近精度的高阶多项式,良好地逼近龙格函数。计算机数值实验表明该方法是有效的,即运用基于等距节点的高阶多项式可以很好地消解龙格现象。