scholarly journals Analysis of Geometric Properties of Ternary Four-Point Rational Interpolating Subdivision Scheme

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 338 ◽  
Author(s):  
Pakeeza Ashraf ◽  
Bushra Nawaz ◽  
Dumitru Baleanu ◽  
Kottakkaran Sooppy Nisar ◽  
Abdul Ghaffar ◽  
...  
Keyword(s):  
Graphical Representation ◽  
Necessary Conditions ◽  
Shape Preservation ◽  
Subdivision Scheme ◽  
Control Points ◽  
Limit Curve ◽  
Subdivision Schemes ◽  
Geometric Properties ◽  
Smooth Curves ◽  
Initial Control

Shape preservation has been the heart of subdivision schemes (SSs) almost from its origin, and several analyses of SSs have been established. Shape preservation properties are commonly used in SSs and various ways have been discovered to connect smooth curves/surfaces generated by SSs to applied geometry. With an eye on connecting the link between SSs and applied geometry, this paper analyzes the geometric properties of a ternary four-point rational interpolating subdivision scheme. These geometric properties include monotonicity-preservation, convexity-preservation, and curvature of the limit curve. Necessary conditions are derived on parameter and initial control points to ensure monotonicity and convexity preservation of the limit curve of the scheme. Furthermore, we analyze the curvature of the limit curve of the scheme for various choices of the parameter. To support our findings, we also present some examples and their graphical representation.

scholarly journals Shape Preservation of Tension Controlled Ternary 4-point Non-Stationary Interpolating Subdivision Scheme

2018 ◽  
Author(s):  
Khurram Pervez ◽  
Syed Hussain Shah
Keyword(s):  
Shape Preservation ◽  
Subdivision Scheme ◽  
Control Points ◽  
Shape Preserving ◽  
Subdivision Schemes ◽  
Tension Parameter ◽  
Initial Control ◽  
Shape Preserving Properties ◽  
Limiting Case

The aim of this work is to analyze and investigate the shape preserving properties of ternary 4-point non-stationary interpolating subdivision schemes constructed by Beccari et al. [1] with a tension parameter !k+1 which can reproducing exponential. Moreover, the conditions on the initial control points are developed that allow user to generate shape preserving limit curves after a nite number of subdivision steps and generalize these results in limiting case. Signicance of derived conditions are illustrated through graphs and the whole discussion is followed by examples.

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.

scholarly journals Shape-Preserving Properties of a Relaxed Four-Point Interpolating Subdivision Scheme

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 806 ◽  
Author(s):  
Pakeeza Ashraf ◽  
Abdul Ghaffar ◽  
Dumitru Baleanu ◽  
Irem Sehar ◽  
Kottakkaran Sooppy Nisar ◽  
...  
Keyword(s):  
Graphical Representation ◽  
Subdivision Scheme ◽  
Control Points ◽  
Positivity Preserving ◽  
Shape Preserving ◽  
Numerical Examples ◽  
Initial Control ◽  
Point Scheme ◽  
Shape Preserving Properties ◽  
Monotonicity Preserving

In this paper, we analyze shape-preserving behavior of a relaxed four-point binary interpolating subdivision scheme. These shape-preserving properties include positivity-preserving, monotonicity-preserving and convexity-preserving. We establish the conditions on the initial control points that allow the generation of shape-preserving limit curves by the four-point scheme. Some numerical examples are given to illustrate the graphical representation of shape-preserving properties of the relaxed scheme.

scholarly journals Odd-Ary Approximating Subdivision Schemes and RS Strategy for Irregular Dense Initial Data

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Muhammad Aslam ◽  
W. P. Abeysinghe
Keyword(s):  
Initial Data ◽  
Regularization Method ◽  
Noise Removal ◽  
Limit Curve ◽  
Control Data ◽  
Subdivision Schemes ◽  
Initial Control ◽  
Data Points ◽  
Variational Regularization

We investigate the implementation of approximating subdivision schemes on noisy or irregular initial control data. Presence of noise in the initial data generates oscillatory curves by subdivision schemes. To reduce or completely eliminate these oscillations, we combine subdivision schemes with other noise removal schemes such as variational regularization method. This setup will allow us to produce the limit curve with less oscillations and still stay as close as possible to the initial data points.

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 Univariate approximating schemes and their non-tensor product generalization

2018 ◽  
Vol 16 (1) ◽  
pp. 1501-1518 ◽  
Author(s):  
Ghulam Mustafa ◽  
Robina Bashir
Keyword(s):  
Initial Data ◽  
Tensor Product ◽  
Limit Curve ◽  
Subdivision Schemes ◽  
Tension Parameter ◽  
The Family

AbstractThis article deals with univariate binary approximating subdivision schemes and their generalization to non-tensor product bivariate subdivision schemes. The two algorithms are presented with one tension and two integer parameters which generate families of univariate and bivariate schemes. The tension parameter controls the shape of the limit curve and surface while integer parameters identify the members of the family. It is demonstrated that the proposed schemes preserve monotonicity of initial data. Moreover, continuity, polynomial reproduction and generation of the schemes are also discussed. Comparison with existing schemes is also given.

scholarly journals Shape-Preservation of the Four-Point Ternary Interpolating Non-stationary Subdivision Scheme

2020 ◽  
Vol 7 ◽  
Author(s):  
Pakeeza Ashraf ◽  
Mehak Sabir ◽  
Abdul Ghaffar ◽  
Kottakkaran Sooppy Nisar ◽  
Ilyas Khan
Keyword(s):  
Shape Preservation ◽  
Subdivision Scheme ◽  
Stationary Subdivision

scholarly journals Shape preservation of 4-point interpolating non-stationary subdivision scheme

2017 ◽  
Vol 319 ◽  
pp. 480-492 ◽  
Author(s):  
Ghazala Akram ◽  
Khalida Bibi ◽  
Kashif Rehan ◽  
Shahid S. Siddiqi
Keyword(s):  
Shape Preservation ◽  
Subdivision Scheme ◽  
Stationary Subdivision

scholarly journals A New Class of 2q-Point Nonstationary Subdivision Schemes and Their Applications

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 639 ◽  
Author(s):  
Abdul Ghaffar ◽  
Mehwish Bari ◽  
Zafar Ullah ◽  
Mudassar Iqbal ◽  
Kottakkaran Sooppy Nisar ◽  
...  
Keyword(s):  
Curve Surface ◽  
Asymptotic Equivalence ◽  
Shape Parameters ◽  
Limit Curve ◽  
Subdivision Schemes ◽  
Local Shape ◽  
New Class ◽  
Stationary Subdivision

The main objective of this study is to introduce a new class of 2 q -point approximating nonstationary subdivision schemes (ANSSs) by applying Lagrange-like interpolant. The theory of asymptotic equivalence is applied to find the continuity of the ANSSs. These schemes can be nicely generalized to contain local shape parameters that allow the user to locally adjust the shape of the limit curve/surface. Moreover, many existing approximating stationary subdivision schemes (ASSSs) can be obtained as nonstationary counterparts of the proposed ANSSs.

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.

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