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.

Review of Subdivision Schemes and their Applications

2021 ◽  
Vol 16 ◽  
Author(s):  
Yan Liu ◽  
Huahao Shou ◽  
Kangsong Ji
Keyword(s):  
Computer Aided Design ◽  
Geometric Modeling ◽  
Hot Spot ◽  
Geometric Constraints ◽  
Future Application ◽  
Subdivision Surfaces ◽  
Subdivision Schemes ◽  
Practical Applications ◽  
Curves And Surfaces ◽  
Subdivision Curves

Background: Subdivision surfaces modeling method and related technology research gradually become a hot spot in the field of computer-aided design(CAD) and computer graphics (CG). In the early stage, research on subdivision curves and surfaces mainly focused on the relationship between the points, thereby failing to satisfy the requirements of all geometric modeling. Considering many geometric constraints is necessary to construct subdivision curves and surfaces for achieving high-quality geometric modeling. Objective: This paper aims to summarize various subdivision schemes of subdivision curves and surfaces, particularly in geometric constraints, such as points and normals. The findings help scholars to grasp the current research status of subdivision curves and surfaces better and to explore their applications in geometric modeling. Methods: This paper reviews the theory and applications of subdivision schemes from four aspects. We first discuss the background and key concept of subdivision schemes. We then summarize the classification of classical subdivision schemes. Next, we show the subdivision surfaces fitting and summarize new subdivision schemes under geometric constraints. Applications of subdivision surfaces are also discussed. Finally, this paper gives a brief summary and future application prospects. Results: Many research papers and patents of subdivision schemes are classified in this review paper. Remarkable developments and improvements have been achieved in analytical computations and practical applications. Conclusion: Our review shows that subdivision curves and surfaces are widely used in geometric modeling. However, some topics need to be further studied. New subdivision schemes need to be presented to meet the requirements of new practical applications.

scholarly journals A Computational Method for Subdivision Depth of Ternary Schemes

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 817
Author(s):  
Faheem Khan ◽  
Ghulam Mustafa ◽  
Aamir Shahzad ◽  
Dumitru Baleanu ◽  
Maysaa M. Al-Qurashi
Keyword(s):  
Error Bounds ◽  
Numerical Experiments ◽  
Computational Method ◽  
Subdivision Schemes ◽  
Practical Applications ◽  
Refinement Procedure ◽  
Subdivision Algorithm ◽  
Subdivision Depth ◽  
Error Distance ◽  
New Framework

Subdivision schemes are extensively used in scientific and practical applications to produce continuous shapes in an iterative way. This paper introduces a framework to compute subdivision depths of ternary schemes. We first use subdivision algorithm in terms of convolution to compute the error bounds between two successive polygons produced by refinement procedure of subdivision schemes. Then, a formula for computing bound between the polygon at k-th stage and the limiting polygon is derived. After that, we predict numerically the number of subdivision steps (depths) required for smooth limiting shape based on the demand of user specified error (distance) tolerance. In addition, extensive numerical experiments were carried out to check the numerical outcomes of this new framework. The proposed methods are more efficient than the method proposed by Song et al.

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 Subdivision depth computation for n-ary subdivision curves/surfaces

2010 ◽  
Vol 26 (6-8) ◽  
pp. 841-851 ◽  
Author(s):  
Ghulam Mustafa ◽  
Muhammad Sadiq Hashmi
Keyword(s):  
Subdivision Depth ◽  
Subdivision Curves

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.

scholarly journals The Mask of Odd Pointsn-Ary Interpolating Subdivision Scheme

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Ghulam Mustafa ◽  
Jiansong Deng ◽  
Pakeeza Ashraf ◽  
Najma Abdul Rehman
Keyword(s):  
Explicit Formula ◽  
Error Bounds ◽  
Subdivision Scheme ◽  
Subdivision Schemes ◽  
Total Absolute Curvature ◽  
Absolute Curvature ◽  
Special Cases ◽  
Preservation Property ◽  
Subdivision Curves ◽  
And Control

We present an explicit formula for the mask of odd pointsn-ary, for any oddn⩾3, interpolating subdivision schemes. This formula provides the mask of lower and higher arity schemes. The 3-point and 5-pointa-ary schemes introduced by Lian, 2008, and (2m+1)-pointa-ary schemes introduced by, Lian, 2009, are special cases of our explicit formula. Moreover, other well-known existing odd pointn-ary schemes including the schemes introduced by Zheng et al., 2009, can easily be generated by our formula. In addition, error bounds between subdivision curves and control polygons of schemes are computed. It has been noticed that error bounds decrease when the complexity of the scheme decreases and vice versa. Also, as we increase arity of the schemes the error bounds decrease. Furthermore, we present brief comparison of total absolute curvature of subdivision schemes having different arity with different complexity. Convexity preservation property of scheme is also presented.

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