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A Family of 4-Point n-Ary Interpolating Scheme Reproducing Conics  [PDF]
Mehwish Bari, Ghulam Mustafa
American Journal of Computational Mathematics (AJCM) , 2013, DOI: 10.4236/ajcm.2013.33031

The n-ary subdivision schemes contrast favorably with their binary analogues because they are capable to produce limit functions with the same (or higher) smoothness but smaller support. We present an algorithm to generate the 4-point n-ary non-stationary scheme for trigonometric, hyperbolic and polynomial case with the parameter for describing curves. The performance, analysis and comparison of the 4-point ternary scheme are also presented.

A New 5-Point Ternary Interpolating Subdivision Scheme and Its Differentiability  [PDF]
Ghulam Mustafa,Jayyada Irum,Mehwish Bari
ISRN Computational Mathematics , 2012, DOI: 10.5402/2012/924839
Abstract: A new 5-point ternary interpolating scheme with a shape parameter is introduced. The resulting curve is for a certain range of parameters. The differentiable properties of the proposed scheme to extend its application in the generation of smooth curves are explored. Application of the proposed scheme is given to show its visual smoothness. The scheme is also extended to a 5-point tensor product ternary interpolating scheme, and its numerical examples are also included. 1. Introduction Geometric modeling plays a significant role to cover up the gap between computer and industry. It has a pivotal importance in the fields of aircraft manufacturing, automobile industry, and general product design. One of the most important tools of computer aided geometric design is “Subdivision.” Subdivision is a well flourished field. It is a process of taking unrefined shape and to polish it up to produce another shape that is more visually tempting. Due to the comprehensibility and simplicity of this method, it is used in the fields of 3D geometrical measurement, computer graphics, computer animation, and computer aided geometric design. In 1986, Dubuc [1] presented a interpolation through an iterative scheme. Dyn et al. [2] introduced a 4-point interpolating subdivision scheme for curve design. Later on, Deslauriers and Dubuc [3] introduced a symmetric iterative interpolation process. Weissman [4] also offered a 6-point interpolating scheme in 1990. In 2002, Hassan et al. [5, 6] gave ternary three-point and 4-point interpolatory schemes. Further analysis of ternary three-point univariate scheme was given in technical report by Hassan and Dodgson [7] in 2004. Dyn [8] has given the analysis of the convergence and smoothness of interpolating and approximating schemes by Laurent’s polynomial method. In 2007, Beccari et al. [9] presented an interpolating 4-point ternary nonstationary scheme with tension control. They also offered a nonstationary uniform tension controlled interpolating 4-point scheme reproducing conics [10] in 2007. Ko [11] in his Ph.D. thesis presented a detailed study on subdivision scheme. Zheng et al. [12] presented the method to find the differentiability of a four-point ternary scheme. Lian [13] extended 3-point and 5-point interpolating schemes into -ary subdivision scheme for curve design. Conti et al. [14] derived symmetric subdivision masks of the Hurwitz type to the interpolating scheme masks. In this paper, we present a new 5-point ternary interpolating subdivision scheme with one parameter. 2. Preliminaries Let , , denote a sequence of points
The Mask of Odd Points -Ary Interpolating Subdivision Scheme
Ghulam Mustafa,Jiansong Deng,Pakeeza Ashraf,Najma Abdul Rehman
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/205863
Abstract: We present an explicit formula for the mask of odd points -ary, for any odd , interpolating subdivision schemes. This formula provides the mask of lower and higher arity schemes. The 3-point and 5-point -ary schemes introduced by Lian, 2008, and ()-point -ary schemes introduced by, Lian, 2009, are special cases of our explicit formula. Moreover, other well-known existing odd point -ary schemes including the schemes introduced by Zheng et al., 2009, can easily be generated by our formula. In addition, error bounds between subdivision curves and control polygons of schemes are computed. It has been noticed that error bounds decrease when the complexity of the scheme decreases and vice versa. Also, as we increase arity of the schemes the error bounds decrease. Furthermore, we present brief comparison of total absolute curvature of subdivision schemes having different arity with different complexity. Convexity preservation property of scheme is also presented.
Four-Point -Ary Interpolating Subdivision Schemes  [PDF]
Ghulam Mustafa,Robina Bashir
International Journal of Mathematics and Mathematical Sciences , 2013, DOI: 10.1155/2013/893414
Abstract: We present an efficient and simple algorithm to generate 4-point n-ary interpolating schemes. Our algorithm is based on three simple steps: second divided differences, determination of position of vertices by using second divided differences, and computation of new vertices. It is observed that 4-point n-ary interpolating schemes generated by completely different frameworks (i.e., Lagrange interpolant and wavelet theory) can also be generated by the proposed algorithm. Furthermore, we have discussed continuity, H?lder regularity, degree of polynomial generation, polynomial reproduction, and approximation order of the schemes. 1. Introduction In general, subdivision schemes can be divided into two categories: approximating and interpolating. For interpolating curve subdivision, new vertices are computed and added to the old polygons for each time of subdivision and the limit curve passes through all the vertices of the original control polygon. Interpolating subdivision schemes are more attractive than approximating schemes in computer aided geometric designs because of their interpolation property. In addition, the interpolation subdivisions are more intuitive to the users. Initial work on interpolating subdivision schemes was started by Dubuc [1]. Later on, Deslauriers and Dubuc [2] have introduced a family of schemes by using Lagrange polynomials indexed by the size of the mask and the arity. In [3], Dyn et al. have studied a family of interpolating schemes with mask size of four. Consequent to this, the research communities are interested in introducing higher arity schemes (i.e., ternary, quaternary -ary) which give better results and less computational cost. Lian [4] has constructed both the -point -ary for any and -point -ary for any odd interpolatory subdivision schemes for curve design by using wavelet theory. Mustafa and Rehman [5] have presented general formulae for the mask of -point -ary interpolating and approximating schemes for any integer and . These formulae provide mask of higher arity schemes and generalize lower arity schemes. Mustafa et al. [6] have presented an explicit formula for the mask of odd points -ary, for any odd , interpolating subdivision schemes. In [7], it has been proved that the large support and higher arity schemes may outperform than small support and lower arity schemes. Even though these schemes are not in practice. It has been suggested that the research on large support and higher arity schemes may continue. The multistage approach is very handy to construct subdivision schemes. This idea is variously used by
Some remarks about interpolating sequences in reproducing kernel Hilbert spaces  [PDF]
Mrinal Raghupathi,Brett D. Wick
Mathematics , 2011,
Abstract: In this paper we study two separate problems on interpolation. We first give some new equivalences of Stout's Theorem on necessary and sufficient conditions for a sequence of points to be an interpolating sequence on a finite open Riemann surface. We next turn our attention to the question of interpolation for reproducing kernel Hilbert spaces on the polydisc and provide a collection of equivalent statements about when it is possible to interpolation in the Schur-Agler class of the associated reproducing kernel Hilbert space.
A Unified Interpolating Subdivision Scheme for Curves/Surfaces by Using Newton Interpolating Polynomial  [PDF]
Faheem Khan, Irem Mukhtar, Noreen Batool
Open Journal of Applied Sciences (OJAppS) , 2013, DOI: 10.4236/ojapps.2013.33033

This paper presents a general formula for (2m + 2)-point n-ary interpolating subdivision scheme for curves for any integer m 0 and n 2 by using Newton interpolating polynomial. As a consequence, the proposed work is extended for surface case, which is equivalent to the tensor product of above proposed curve case. These formulas merge several notorious curve/surface schemes. Furthermore, visual performance of the subdivision schemes is also presented.  

Interpolating reproducing kernel particle method for elastic mechanics

Li Zhong-Hua,Qin Yi-Xiao,Cui Xiao-Chao,

物理学报 , 2012,
Abstract: The displacement trial function is reconstructed by reproducing kernel particle shape function method with interpolation property on discrete points, then combining the principle of minimum potential energy of elasticity, the new interpolating reproducing kernel particle method to analyze the plane problem of elasticity is obtained. Because interpolation reproducing kernel particle shape function has a point interpolation property and no less than the high-order smoothness of kernel function, the difficulty for most of meshless methods to be used to deal with the essential boundary conditions is already overcome, and the high numerical accuracy is assured as well. Compared with the early meshless methods, this method has a high accuracy and a small scale of solving problem and it can be directly applied to boundary conditions. Numerical results for some typical examples of elasticity prove the proposed method to be valid.
The 4-Point α-Ary Approximating Subdivision Scheme  [PDF]
Abdul Ghaffar, Ghulam Mustafa, Kaihuai Qin
Open Journal of Applied Sciences (OJAppS) , 2013, DOI: 10.4236/ojapps.2013.31B1022

A general formula for 4-point α-Ary approximating subdivision scheme for curve designing is introduced for any arity α≥2. The new scheme is extension of B-spline of degree 6. Laurent polynomial method is used to investigate the continuity of the scheme. The variety of effects can be achieved in correspondence for different values of parameter. The applications of the proposed scheme are illustrated in comparison with the established subdivision schemes.

Discrete Conics  [PDF]
Emmanuel Tsukerman
Mathematics , 2012,
Abstract: In this paper, we introduce discrete conics, polygonal analogues of conics. We show that discrete conics satisfy a number of nice properties analogous to those of conics, and arise naturally from several constructions, including the discrete negative pedal construction and an action of a group acting on a focus-sharing pencil of conics.
Studies on Non-interpolating Semi-Lagrangian Scheme and Numerical Solution to KdV Equation
Studies on Non-interpolating Semi-Lagrangian Scheme and Numerical Solution to KdV Equation

Chen Jiabin,Wang Jun,
Chen Jiabin
,Wang Jun

大气科学进展 , 1996,
Abstract: A new non-interpolating semi-Lagrangian scheme has been proposed, which can eliminate any interpolation, and consequently numerical smoothing of forecast fields. Here the new scheme is applied to KdV equation and its performance is assessed by comparing the numerical results with those produced by Ritchie’s scheme (1986).The comparison shows that the non-interpolating semi-Lagrangian scheme appears to have efficiency advantages. Supported by the National Natural Science Foundation of China (4947266) and LASG at Institute of Atmospheric Physics, Chinese Academy of Sciences. Supported by the National Natural Science Foundation of China (4947266) and LASG at Institute of Atmospheric Physics, Chinese Academy of Sciences.
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