scholarly journals A New 4-PointC3Quaternary Approximating Subdivision Scheme

2009 ◽  
Vol 2009 ◽  
pp. 1-14 ◽  
Author(s):  
Ghulam Mustafa ◽  
Faheem Khan
Keyword(s):  
Shape Parameter ◽  
Approximation Order ◽  
Subdivision Scheme ◽  
Subdivision Schemes ◽  
Approximating Subdivision Scheme

A new 4-pointC3quaternary approximating subdivision scheme with one shape parameter is proposed and analyzed. Its smoothness and approximation order are higher but support is smaller in comparison with the existing binary and ternary 4-point subdivision schemes.

scholarly journals Generalized 5-Point Approximating Subdivision Scheme of Varying Arity

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 474 ◽  
Author(s):  
Sardar Muhammad Hussain ◽  
Aziz Ur Rehman ◽  
Dumitru Baleanu ◽  
Kottakkaran Sooppy Nisar ◽  
Abdul Ghaffar ◽  
...  
Keyword(s):  
Error Bound ◽  
Shape Parameter ◽  
Geometric Design ◽  
Hölder Regularity ◽  
Subdivision Scheme ◽  
Subdivision Schemes ◽  
Computer Aided Geometric Design ◽  
Approximating Subdivision Scheme ◽  
Computer Aided ◽  
Holder Regularity

The Subdivision Schemes (SSs) have been the heart of Computer Aided Geometric Design (CAGD) almost from its origin, and various analyses of SSs have been conducted. SSs are commonly used in CAGD and several methods have been invented to design curves/surfaces produced by SSs to applied geometry. In this article, we consider an algorithm that generates the 5-point approximating subdivision scheme with varying arity. By applying the algorithm, we further discuss several properties: continuity, Hölder regularity, limit stencils, error bound, and shape of limit curves. The efficiency of the scheme is also depicted with assuming different values of shape parameter along with its application.

scholarly journals A Family of Even-Point Ternary Approximating Schemes

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Abdul Ghaffar ◽  
Ghulam Mustafa
Keyword(s):  
General Formula ◽  
Shape Parameter ◽  
Sufficient Conditions ◽  
Approximation Order ◽  
Subdivision Schemes ◽  
The Family ◽  
Curvature And Torsion

We presented a general formula to generate the family of even-point ternary approximating subdivision schemes with a shape parameter for describing curves. Some sufficient conditions for C0 to C7 continuity and approximation order for certain ranges of parameter are discussed. The proposed even-point ternary schemes compare remarkably with existing even-point ternary schemes because they are able to generate limit functions with higher smoothness and approximation order. In addition, we measured curvature and torsion that assist the quality of subdivided curves.

Designing General p-Ary n-Point Smooth Subdivision Schemes

2014 ◽  
Vol 472 ◽  
pp. 510-515 ◽  
Author(s):  
Hong Chan Zheng ◽  
Qian Song
Keyword(s):  
Smooth Curve ◽  
Approximation Order ◽  
Subdivision Scheme ◽  
Subdivision Schemes ◽  
Special Cases ◽  
Two Parameters ◽  
Interpolatory Subdivision

In this paper, in order to produce smooth curve, we design a class of n-point p-ary smooth interpolatory subdivision schemes that can reproduce polynomials of degree n-1 with approximation order n. Many classical interpolatory subdivision schemes are special cases of this kind of subdivision. We illustrate the approach with a new 5-point quaternary interpolatory subdivision scheme with two parameters, which reproduces polynomial of degree 4 with approximation order of 5 and can generate new interpolatory curves.

scholarly journals Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials

2021 ◽  
Vol 19 (1) ◽  
pp. 909-926
Author(s):  
Zeze Zhang ◽  
Hongchan Zheng ◽  
Lulu Pan
Keyword(s):  
Approximation Order ◽  
High Order ◽  
Subdivision Scheme ◽  
Exponential Polynomial ◽  
Subdivision Schemes ◽  
Exponential Polynomials ◽  
Variable Parameters ◽  
Numerical Examples

Abstract In this paper, we propose a family of non-stationary combined ternary ( 2 m + 3 ) \left(2m+3) -point subdivision schemes, which possesses the property of generating/reproducing high-order exponential polynomials. This scheme is obtained by adding variable parameters on the generalized ternary subdivision scheme of order 4. For such a scheme, we investigate its support and exponential polynomial generation/reproduction and get that it can generate/reproduce certain exponential polynomials with suitable choices of the parameters and reach 2 m + 3 2m+3 approximation order. Moreover, we discuss its smoothness and show that it can produce C 2 m + 2 {C}^{2m+2} limit curves. Several numerical examples are given to show the performance of the schemes.

A ternary 4-point approximating subdivision scheme

2007 ◽  
Vol 190 (2) ◽  
pp. 1563-1573 ◽  
Author(s):  
Kwan Pyo Ko ◽  
Byung-Gook Lee ◽  
Gang Joon Yoon
Keyword(s):  
Subdivision Scheme ◽  
Approximating Subdivision Scheme

A unified three point approximating subdivision scheme

2011 ◽  
Vol 27 (1) ◽  
pp. 10-20 ◽  
Author(s):  
Ghulam Mustafa ◽  
Faheem Khan ◽  
Muhammad Sadia Hashmi ◽  
Muhammad Zeshan Afzal
Keyword(s):  
Subdivision Scheme ◽  
Approximating Subdivision Scheme

scholarly journals Recursive Process for Constructing the Refinement Rules of New Combined Subdivision Schemes and Its Extended Form

2021 ◽  
Vol 2021 ◽  
pp. 1-23
Author(s):  
Rabia Hameed ◽  
Ghulam Mustafa ◽  
Jiansong Deng ◽  
Shafqat Ali
Keyword(s):  
New Method ◽  
Subdivision Scheme ◽  
Control Points ◽  
Local Translation ◽  
Extended Form ◽  
Subdivision Schemes ◽  
Tension Parameter ◽  
Displacement Vectors ◽  
The Family ◽  
Initial Control

In this article, we present a new method to construct a family of 2 N + 2 -point binary subdivision schemes with one tension parameter. The construction of the family of schemes is based on repeated local translation of points by certain displacement vectors. Therefore, refinement rules of the 2 N + 2 -point schemes are recursively obtained from refinement rules of the 2 N -point schemes. Thus, we get a new subdivision scheme at each iteration. Moreover, the complexity, polynomial reproduction, and polynomial generation of the schemes are increased by two at each iteration. Furthermore, a family of interproximate subdivision schemes with tension parameters is also introduced which is the extended form of the proposed family of schemes. This family of schemes allows a different tension value for each edge and vertex of the initial control polygon. These schemes generate curves and surfaces such that some initial control points are interpolated and others are approximated.

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