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 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 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 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 forq-Bernstein-Stancu Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Yali Wang ◽  
Yinying Zhou
Keyword(s):  
Bernstein Polynomials ◽  
Shape Preserving ◽  
Positive Basis ◽  
Stancu Operators ◽  
Variation Diminishing ◽  
Totally Positive ◽  
Shape Preserving Properties ◽  
Monotonicity Preserving

We investigate shape preserving forq-Bernstein-Stancu polynomialsBnq,α(f;x)introduced by Nowak in 2009. Whenα=0,Bnq,α(f;x)reduces to the well-knownq-Bernstein polynomials introduced by Phillips in 1997; whenq=1,Bnq,α(f;x)reduces to Bernstein-Stancu polynomials introduced by Stancu in 1968; whenq=1,α=0, we obtain classical Bernstein polynomials. We prove that basicBnq,α(f;x)basis is a normalized totally positive basis on[0,1]andq-Bernstein-Stancu operators are variation-diminishing, monotonicity preserving and convexity preserving on[0,1].

scholarly journals A Reverse Non-Stationary Generalized B-Splines Subdivision Scheme

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2628
Author(s):  
Abdellah Lamnii ◽  
Mohamed Yassir Nour ◽  
Ahmed Zidna
Keyword(s):  
Optimization Problem ◽  
Linear Optimization ◽  
Subdivision Scheme ◽  
Third Order ◽  
Linear Optimization Problem ◽  
Numerical Examples ◽  
B Splines ◽  
Initial Control ◽  
Order Scheme ◽  
Different Shapes

In this paper, two new families of non-stationary subdivision schemes are introduced. The schemes are constructed from uniform generalized B-splines with multiple knots of orders 3 and 4, respectively. Then, we construct a third-order reverse subdivision framework. For that, we derive a generalized multi-resolution mask based on their third-order subdivision filters. For the reverse of the fourth-order scheme, two methods are used; the first one is based on least-squares formulation and the second one is based on solving a linear optimization problem. Numerical examples are given to show the performance of the new schemes in reproducing different shapes of initial control polygons.

Shape Preserving Properties with Constraints on the Tension Parameter of Binary Three-Point Approximating Subdivision Scheme

2020 ◽  
Vol 20 (01) ◽  
pp. 2050005
Author(s):  
Khalida Bibi ◽  
Ghazala Akram ◽  
Kashif Rehan
Keyword(s):  
Initial Data ◽  
Subdivision Scheme ◽  
Shape Preserving ◽  
Tension Parameter ◽  
Approximating Subdivision Scheme ◽  
Shape Preserving Properties

The paper analyzes conditions for preserving the shape properties from the initial data to the limit curves of the binary three-point approximating subdivision scheme. We provide suitable conditions on the initial data utilizing the tension parameter [Formula: see text], thus the scheme can maintain three important shape properties, namely positivity, monotonicity and convexity in the limit curves. The use of derived conditions is illustrated in few examples, which offers more flexibility in the generation of smooth limit curves endowed with shape preserving properties.

scholarly journals On the Numerical Solution of Fractional Boundary Value Problems by a Spline Quasi-Interpolant Operator

Axioms ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 61
Author(s):  
Francesca Pitolli
Keyword(s):  
Fractional Derivative ◽  
Boundary Value Problems ◽  
Numerical Experiments ◽  
Boundary Value ◽  
Bernstein Operator ◽  
Explicit Formulas ◽  
Shape Preserving ◽  
Engineering Control ◽  
Shape Preserving Properties ◽  
Fractional Boundary Value Problems

Boundary value problems having fractional derivative in space are used in several fields, like biology, mechanical engineering, control theory, just to cite a few. In this paper we present a new numerical method for the solution of boundary value problems having Caputo derivative in space. We approximate the solution by the Schoenberg-Bernstein operator, which is a spline positive operator having shape-preserving properties. The unknown coefficients of the approximating operator are determined by a collocation method whose collocation matrices can be constructed efficiently by explicit formulas. The numerical experiments we conducted show that the proposed method is efficient and accurate.

An Improved Four Point Interpolatory Subdivision Scheme

2013 ◽  
Vol 380-384 ◽  
pp. 1555-1557
Author(s):  
Xin Fen Zhang ◽  
Yu Zhen Liu
Keyword(s):  
Chord Length ◽  
Subdivision Scheme ◽  
Shape Preserving ◽  
Curve Interpolation ◽  
Interpolatory Polynomial ◽  
Interpolatory Subdivision

In this paper we propose a new kind of geometry driven subdivision scheme for curve interpolation. We use cubic Lagrange interpolatory polynomial to construct a new point, selecting parameters by accumulated chord length method. The new scheme is shape preserving. It can overcome the shortcoming of the initial four point subdivision scheme proposed.

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