Local shape control of the rational interpolation curves with quadratic denominator

Duan, Qi; Liu, Xipu; Bao, Fangxun

doi:10.1080/00207160802132897

Cited by 6 publications

(27 citation statements)

References 20 publications

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“…Gregory discussed applications in shape preserving interpolation. Duan, et al in [10] extended the work done in [19] with the several properties of interpolation. They developed the methods of value control, convex control and inflection point control of the interpolation along with some numerical examples.…”

Section: Introductionmentioning

confidence: 88%

Two Methods of Object Designing by Rational Splines

Hussain

Sarfraz

Ishaq

et al. 2014

2014 18th International Conference on Information Visualisation

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Object designing has a significance importance in the field of Computer-Aided Design (CAD), Computer-Aided Geometric Design (CAGD) and Computer-Aided Manufacturing (CAM). In this paper, two techniques for the designing of 2D objects are developer. These are based on rational cubic spline interpolant and rational quadratic spline interpolant with shape parameters controlling the shape of the curve in the description of rational functions. It is shown that a piece of curve designed using rational cubic spline interpolant can be presented by two segments of rational quadratic spline interpolant. The developed schemes are simple, local and computationally inexpensive.

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Section: Introductionmentioning

confidence: 88%

Two Methods of Object Designing by Rational Splines

Hussain

Sarfraz

Ishaq

et al. 2014

2014 18th International Conference on Information Visualisation

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“…(i) In this paper a new rational cubic spline (cubic/ quadratic) with the three parameters initiated by Karim and Kong [21][22][23] has been used for local control of the interpolation data. (ii) Our methods have three parameters while in the work of Duan et al [18] they have only two parameters. By having three parameters, the local control of the interpolating function can be achieved with a greater flexibility.…”

Section: Introductionmentioning

confidence: 93%

“…These ideas have been extended by Sun et al [17]. They apply their blending interpolator for value control with minimal strain energy meanwhile Duan et al [18] have used rational cubic spline (cubic/quadratic) of Gregory et al [19] to control the value, convexity, and inflection-point control of the interpolation. Due to the fact that there are some cases where there are no parameters values that can be used in order to control the final shape of the curves, Bao et al [16] and Sun et al [17] have proposed blending interpolator that will enable the local control of the interpolating curves.…”

Section: Introductionmentioning

confidence: 99%

“…If the design requires that the value of the interpolating function at the point * , ( * ) is equal to real number , that is, ( * ) = , the main question is, how to select or choose the value of ? In Duan et al [18] the authors did not mention how to ensure that the user will choose the right . Otherwise the user may need to keep changing the values of each shape parameter that satisfies the sufficient condition for the value-point control of the interpolating function.…”

Section: Introductionmentioning

confidence: 99%

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Local Control of the Curves Using Rational Cubic Spline

Karim

Pang

2014

Journal of Applied Mathematics

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This paper discussed the local control of interpolating function by using rational cubic spline (cubic/quadratic) with three parameters originally proposed by the authors. The rational spline hasC1continuity. The bounded properties of the rational cubic interpolants and shape controls of the rational interpolants are discussed in detail. The value control, inflection point control, and convexity control at a point by using the proposed rational cubic spline are constructed. Several numerical results are presented to show the capability of the method. Numerical comparisons with the existing scheme are also further elaborated. From the results, it was indicated that the scheme works well and it is comparable with the established existing scheme.

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“…The literature on the classical (non-fractal) methods of shape preserving interpolation and their applications in various fields is abundant. For brevity reader is referred to [10][11][12][13][14][15][16]27,28,33].…”

Section: Introductionmentioning

confidence: 99%

A rational iterated function system for resolution of univariate constrained interpolation

Viswanathan

Chand

Navascués

2014

RACSAM

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Iterated Function Systems (IFSs) provide a standard framework for generating Fractal Interpolation Functions (FIFs) that yield smooth or non-smooth approximants. Nevertheless, the most widely studied FIFs so far in the literature that are obtained through polynomial IFSs are, in general, incapable of reproducing important shape properties inherent in a given data set. Abandoning the polynomiality of the functions defining the IFS, we introduce a new class of rational IFS that generates fractal functions (self-referential functions) for solving constrained interpolation problems. Suitable values of the rational IFS parameters are identified so that: (i) the corresponding FIF inherits positivity and/or monotonicity properties present in the data set, and (ii) the attractor of the IFS lies within an axis-aligned rectangle. The proposed IFS schemes for the shape preserving interpolation generalize some of the classical non-recursive interpolation methods, and expand the interpolation/approximation, including approximants for which functions themselves or the first derivatives can even be non-differentiable in a dense set of points of the domain. For appropriate values of the IFS parameters, the resulting rational quadratic FIF converges uniformly to the original function Φ ∈ C 3 [x 1 , x n ] with h 3 order of convergence, where h denotes the norm of the partition. We also provide a number of examples intended to demonstrate the proposed schemes and to suggest how these schemes outperform their classical counterparts.

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Local shape control of the rational interpolation curves with quadratic denominator

Cited by 6 publications

References 20 publications

Two Methods of Object Designing by Rational Splines

Two Methods of Object Designing by Rational Splines

Local Control of the Curves Using Rational Cubic Spline

A rational iterated function system for resolution of univariate constrained interpolation

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