Family of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>a</mml:mi></mml:mrow></mml:math>-Ary Univariate Subdivision Schemes Generated by Laurent Polynomial

Asghar, M.; Mustafa, Ghulam

doi:10.1155/2018/7824279

Cited by 7 publications

(11 citation statements)

References 29 publications

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“…In Figure 3, we present some visual performances of proposed 3-point ternary approximating subdivision scheme. In Figure 1, we present the comparison of the proposed 3-point scheme S 1 with 3-point scheme G 3 presented in [5] and 3-point scheme S 3 0 presented in [17]. Here, black dotted lines show the initial polygon, red solid lines are the limit curve of 3-point scheme S 3 0 presented in [17], and blue solid lines are the limit curve of 3-point scheme G 3 presented in [5].…”

Section: Comparison With Existing Schemesmentioning

confidence: 99%

“…In Figure 1, we present the comparison of the proposed 3-point scheme S 1 with 3-point scheme G 3 presented in [5] and 3-point scheme S 3 0 presented in [17]. Here, black dotted lines show the initial polygon, red solid lines are the limit curve of 3-point scheme S 3 0 presented in [17], and blue solid lines are the limit curve of 3-point scheme G 3 presented in [5]. Mathematically, the continuity of all 3-point schemes is the same, which is C 2 continuity.…”

Section: Comparison With Existing Schemesmentioning

confidence: 99%

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A Family of Integer-Point Ternary Parametric Subdivision Schemes

Mustafa

Asghar

Ali

et al. 2021

Journal of Mathematics

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New subdivision schemes are always required for the generation of smooth curves and surfaces. The purpose of this paper is to present a general formula for family of parametric ternary subdivision schemes based on the Laurent polynomial method. The different complexity subdivision schemes are obtained by substituting the different values of the parameter. The important properties of the proposed family of subdivision schemes are also presented. The continuity of the proposed family is C 2 m . Comparison shows that the proposed family of subdivision schemes has higher degree of polynomial generation, degree of polynomial reproduction, and continuity compared with the exiting subdivision schemes. Maple software is used for mathematical calculations and plotting of graphs.

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Section: Comparison With Existing Schemesmentioning

confidence: 99%

Section: Comparison With Existing Schemesmentioning

confidence: 99%

A Family of Integer-Point Ternary Parametric Subdivision Schemes

Mustafa

Asghar

Ali

et al. 2021

Journal of Mathematics

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“…Tan et al [14] presented a new 5-point binary approximating scheme with two parameters in 2017. In 2018, Asghar and Mustafa [15] presented a family of a-ary univariate subdivision schemes with single parameter.…”

Section: Introductionmentioning

confidence: 99%

A Class of Refinement Schemes With Two Shape Control Parameters

et al. 2020

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A subdivision scheme defines a smooth curve or surface as the limit of a sequence of successive refinements of given polygon or mesh. These schemes take polygons or meshes as inputs and produce smooth curves or surfaces as outputs. In this paper, a class of combine refinement schemes with two shape control parameters is presented. These even and odd rules of these schemes have complexity three and four respectively. The even rule is designed to modify the vertices of the given polygon, whereas the odd rule is designed to insert a new point between every edge of the given polygon. These schemes can produce high order of continuous shapes than existing combine binary and ternary family of schemes. It has been observed that the schemes have interpolating and approximating behaviors depending on the values of parameters. These schemes have an interproximate behavior in the case of non-uniform setting of the parameters. These schemes can be considered as the generalized version of some of the interpolating and B-spline schemes. The theoretical as well as the numerical and graphical analysis of the shapes produced by these schemes are also presented.

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“…Zheng & Zhang [5] applied the push-back operation in the nonstationary case and constructed a combined nonstationary scheme generating different exponential polynomials. Asghar & Mustafa [6] constructed -ary nonstationary schemes which are new versions of the Lane-Riesenfeld algorithms. For other nonstationary schemes generating exponential polynomials, see [7][8][9] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

A Generalized Cubic Exponential B‐Spline Scheme with Shape Control

Zhang

Zheng

Pan

2019

Mathematical Problems in Engineering

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In this paper, a generalized cubic exponential B-spline scheme is presented, which can generate different kinds of curves, including the conics. Such a scheme is obtained by generalizing the cubic exponential B-spline scheme based on an iteration from the generation of exponential polynomials and a suitable function with two parameters μ and ν. By changing the values of μ and ν, the sensitivity of the shape of the subdivision curve to the initial control value v0 can be changed and different kinds of curves can then be obtained by adjusting the value of v0. For this new scheme, we show that, with any admissible choice of μ and ν, it owns the same smoothness order and support as the cubic exponential B-spline scheme. Besides, based on a different iteration and another suitable function, we construct a similar nonstationary scheme to generate more curves with different shapes and show the role of iterations and suitably chosen functions in the construction and analysis of such schemes. Several examples are given to illustrate the performance of our new schemes.

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Family of a-Ary Univariate Subdivision Schemes Generated by Laurent Polynomial

Cited by 7 publications

References 29 publications

A Family of Integer-Point Ternary Parametric Subdivision Schemes

A Family of Integer-Point Ternary Parametric Subdivision Schemes

A Class of Refinement Schemes With Two Shape Control Parameters

A Generalized Cubic Exponential B‐Spline Scheme with Shape Control

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