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 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 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-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.

Maintaining and Updating the Storing Data by the Control Points

2013 ◽  
Vol 457-458 ◽  
pp. 793-796
Author(s):  
I. Mimorov ◽  
I. Livshits ◽  
V. Vasilev
Keyword(s):  
Good Practice ◽  
New Method ◽  
Control Points ◽  
Competitive Advantages ◽  
Distance Teaching ◽  
Time Cost ◽  
Teaching System ◽  
System Usage ◽  
Potential Risks ◽  
To Receive

This paper describes the new method that improves the processing and storing of data, which was used during the development of distance teaching system. Usage of a modern methodologies and good practice has reduced the time cost for working with information, helps to identify the out of day information, operate potential risks and shows how to receive competitive advantages.

Sutton’s Swan Song

2020 ◽  
pp. 112-120
Author(s):  
Gavin Weightman
Keyword(s):  
New Method ◽  
The Family

This chapter recounts how, once he had moved out of Sutton House, Daniel Sutton became itinerant, moving from one West End street to another in quick succession. In 1779, he announced that he had been 'engaged by the Governors of the General Inoculation Dispensary' and he had moved nearby to Southampton Street in Bloomsbury. Although he was still inoculating on his own account on his usual terms of 10 guineas, to have any kind of official post was out of character. Times had changed and he made it clear in yet another newspaper advertisement that he was well aware of the waning of his celebrity. Announcing his appointment to the dispensary, he felt it necessary to plead that he was the 'identical person who, in 1767 (by royal approbation) was complimented with a grant of the following honorary Patent for his singular and new method of inoculation'. This method, he claimed, was now 'very materially improved'. Once again the family coat of arms awarded to himself and his family was evoked. The chapter then looks at the publication in 1796 of Daniel's account of his discoveries as an inoculator.

Rapid Mapping Method Based on Free Blocks of Surveys

2016 ◽  
Vol 10 (2) ◽  
Author(s):  
Xianwen Yu ◽  
Huiqing Wang ◽  
Jinling Wang
Keyword(s):  
Coordinate System ◽  
Large Scale ◽  
Implementation Process ◽  
Mapping Method ◽  
New Method ◽  
Control Points ◽  
Coordinate Systems ◽  
Free Block ◽  
Transformation Parameters ◽  
Geodetic Coordinate System

AbstractWhile producing large-scale larger than 1:2000 maps in cities or towns, the obstruction from buildings leads to difficult and heavy tasks of measuring mapping control points. In order to avoid measuring the mapping control points and shorten the time of fieldwork, in this paper, a quick mapping method is proposed. This method adjusts many free blocks of surveys together, and transforms the points from all free blocks of surveys into the same coordinate system. The entire surveying area is divided into many free blocks, and connection points are set on the boundaries between free blocks. An independent coordinate system of every free block is established via completely free station technology, and the coordinates of the connection points, detail points and control points in every free block in the corresponding independent coordinate systems are obtained based on poly-directional open traverses. Error equations are established based on connection points, which are determined together to obtain the transformation parameters. All points are transformed from the independent coordinate systems to a transitional coordinate system via the transformation parameters. Several control points are then measured by GPS in a geodetic coordinate system. All the points can then be transformed from the transitional coordinate system to the geodetic coordinate system. In this paper, the implementation process and mathematical formulas of the new method are presented in detail, and the formula to estimate the precision of surveys is given. An example has demonstrated that the precision of using the new method could meet large-scale mapping needs.

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