scholarly journals A Family of Integer-Point Ternary Parametric Subdivision Schemes

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ghulam Mustafa ◽  
Muhammad Asghar ◽  
Shafqat Ali ◽  
Ayesha Afzal ◽  
Jia-Bao Liu
Keyword(s):  
General Formula ◽  
Integer Point ◽  
Laurent Polynomial ◽  
Polynomial Method ◽  
Subdivision Schemes ◽  
Smooth Curves ◽  
Curves And Surfaces

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.

scholarly journals A Novel Numerical Algorithm to Estimate the Subdivision Depth of Binary Subdivision Schemes

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 66 ◽  
Author(s):  
Aamir Shahzad ◽  
Faheem Khan ◽  
Abdul Ghaffar ◽  
Ghulam Mustafa ◽  
Kottakkaran Sooppy Nisar ◽  
...  
Keyword(s):  
Numerical Algorithm ◽  
Subdivision Schemes ◽  
Practical Applications ◽  
Curves And Surfaces ◽  
Geometrical Shapes ◽  
Subdivision Depth ◽  
Subdivision Curves

Subdivision schemes are extensively used in scientific and practical applications to produce continuous geometrical shapes in an iterative manner. We construct a numerical algorithm to estimate subdivision depth between the limit curves/surfaces and their control polygons after k-fold subdivisions. In this paper, the proposed numerical algorithm for subdivision depths of binary subdivision curves and surfaces are obtained after some modification of the results given by Mustafa et al in 2006. This algorithm is very useful for implementation of the parametrization.

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.

scholarly journals A Family of High Continuity Subdivision Schemes Based on Probability Distribution

2019 ◽  
Vol 38 (2) ◽  
pp. 389-398 ◽  
Author(s):  
Muhammad Asghar ◽  
Muhammad Javed Iqbal ◽  
Ghulam Mustafa
Keyword(s):  
Probability Distribution ◽  
Geometric Modeling ◽  
Probability Generating Function ◽  
Hölder Regularity ◽  
Binomial Probability ◽  
Shape Preserving ◽  
Subdivision Schemes ◽  
Smooth Curves ◽  
The Family ◽  
Holder Regularity

Subdivision schemes are famous for the generation of smooth curves and surfaces in CAGD (Computer Aided Geometric Design). The continuity is an important property of subdivision schemes. Subdivision schemes having high continuity are always required for geometric modeling. Probability distribution is the branch of statistics which is used to find the probability of an event. We use probability distribution in the field of subdivision schemes. In this paper, a simplest way is introduced to increase the continuity of subdivision schemes. A family of binary approximating subdivision schemes with probability parameter p is constructed by using binomial probability generating function. We have derived some family members and analyzed the important properties such as continuity, Holder regularity, degree of generation, degree of reproduction and limit stencils. It is observed that, when the probability parameter p = 1/2, the family of subdivision schemes have maximum continuity, generation degree and Holder regularity. Comparison shows that our proposed family has high continuity as compare to the existing subdivision schemes. The proposed family also preserves the shape preserving property such as convexity preservation. Subdivision schemes give negatively skewed, normal and positively skewed behavior on convex data due to the probability parameter. Visual performances of the family are also presented.

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

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Muhammad Asghar ◽  
Ghulam Mustafa
Keyword(s):  
Laurent Polynomial ◽  
Subdivision Schemes ◽  
Special Cases ◽  
The Family

The generalized symbols of the family of a-ary (a≥2) univariate stationary and nonstationary parametric subdivision schemes have been presented. These schemes are the new version of Lane-Riesenfeld algorithms. Comparison shows that our proposed family has higher continuity and generation degree comparative to the existing subdivision schemes. It is observed that many existing binary and ternary schemes are the special cases of our schemes. The analysis of proposed family of subdivision schemes is also presented in this paper.

scholarly journals Construction and Analysis of Binary Subdivision Schemes for Curves and Surfaces Originated from Chaikin Points

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Rabia Hameed ◽  
Ghulam Mustafa
Keyword(s):  
Shape Parameter ◽  
Weighted Average ◽  
Subdivision Schemes ◽  
Smoothing Operator ◽  
Curves And Surfaces ◽  
New Variant ◽  
Subdivision Operator ◽  
Cubic Polynomial ◽  
Smoothing Parameters ◽  
Two Parameters

We present a new variant of Lane-Riesenfeld algorithm for curves and surfaces both. Our refining operator is the modification of Chaikin/Doo-Sabin subdivision operator, while each smoothing operator is the weighted average of the four/sixteen adjacent points. Our refining operator depends on two parameters (shape and smoothing parameters). So we get new families of univariate and bivariate approximating subdivision schemes with two parameters. The bivariate schemes are the nontensor product schemes for quadrilateral meshes. Moreover, we also present analysis of our families of schemes. Furthermore, our schemes give cubic polynomial reproduction for a specific value of the shape parameter. The nonuniform setting of our univariate and bivariate schemes gives better performance than that of the uniform schemes.

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