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A Nonstationary Ternary 4-Point Shape-Preserving Subdivision Scheme

Jieqing Tan, Guangyue Tong, Ghulam Mustafa
2021 Journal of Mathematics  
cusps compared with a stationary 4-point ternary interpolatory subdivision scheme.  ...  Furthermore, the monotonicity preservation and convexity preservation are proved.  ...  Dyn et al. analyze the convexity preservation of the 4-point interpolatory scheme in [6] .  ... 
doi:10.1155/2021/6694241 fatcat:avkfz2xhdjeclgrtndg3a6hlyy

Convexity preserving interpolatory subdivision with conic precision

Gudrun Albrecht, Lucia Romani
2012 Applied Mathematics and Computation  
The paper is concerned with the problem of shape preserving interpolatory subdivision.  ...  Preserving convexity, while a curve is interpolating a given data set, is far from trivial.  ...  Definition of the scheme for globally convex data In this section we define a convexity preserving subdivision scheme for globally convex data which will then be the basis for the final shape preserving  ... 
doi:10.1016/j.amc.2012.10.048 fatcat:2b276np7zbas7pstqlztnk54ye

Ternary shape-preserving subdivision schemes

Francesca Pitolli
2014 Mathematics and Computers in Simulation  
We prove that any bell-shaped mask, satisfying the basic sum rules, gives rise to a convergent monotonicity preserving subdivision scheme, but convexity preservation is not guaranteed.  ...  We analyze the shape-preserving properties of ternary subdivision schemes generated by bell-shaped masks.  ...  A subdivision scheme is said to be convexity preserving if it preserves the convexity of the starting sequence.  ... 
doi:10.1016/j.matcom.2013.04.003 fatcat:i4zlrke4b5fkjnb77ovj7kgsw4

Ternary univariate curvature-preserving subdivision

Myungjin Jeon, Dongsoong Han, Kyeongsu Park, Gundon Choi
2005 Journal of Applied Mathematics and Computing  
We present an interpolating, univariate subdivision scheme which preserves the discrete curvature and tangent direction at each step of subdivision.  ...  We estimate the curvature bound of odd vertices and give an error estimate for restoring a curve from sampled vertices on curves.  ...  Among subdivision schemes, interpolatory subdivision is more intuitive for the construction of curves and surfaces.  ... 
doi:10.1007/bf02936568 fatcat:iwbjxsqbyjejpor3wmaprygbfu

A Variant Cubic Exponential B-Spline Scheme with Shape Control

Baoxing Zhang, Hongchan Zheng
2021 Mathematics  
For such a scheme, we show its C2-convergence and analyze the effect of the parameters on the shape of the generated curves and also discuss its convexity preservation.  ...  This paper presents a variant scheme of the cubic exponential B-spline scheme, which, with two parameters, can generate curves with different shapes.  ...  In fact, for the convexity preservation of non-stationary schemes, Akram et al. [16] analyzed the shape preservation of the non-stationary 4-pt interpolatory scheme [5] .  ... 
doi:10.3390/math9233116 fatcat:yiibsa5dkfgg3awghnyoe7tc4m

Convexity preserving interpolatory subdivision with conic precision [article]

Gudrun Albrecht, Lucia Romani
2010 arXiv   pre-print
The paper is concerned with the problem of shape preserving interpolatory subdivision.  ...  For arbitrarily spaced, planar input data an efficient non-linear subdivision algorithm is presented that results in G^1 limit curves, reproduces conic sections and respects the convexity properties of  ...  Definition of the scheme for globally convex data In this section we define a convexity preserving subdivision scheme for globally convex data which will then be the basis for the final shape preserving  ... 
arXiv:1004.1295v1 fatcat:v3sthkzajnhrjhkhhcvles5gti

Page 4166 of Mathematical Reviews Vol. , Issue 99f [page]

1999 Mathematical Reviews  
(NL-TWENA,; Enschede) Convexity preserving interpolatory subdivision schemes. (English summary) Constr. Approx. 14 (1998), no. 4, 609-630.  ...  The authors wish to characterize those subdivision interpolatory schemes on {x\”) := ih}, which are local (actually depend at most on four data pieces), preserve convexity, are invariant on affine functions  ... 

A New Approach to Increase the Flexibility of Curves and Regular Surfaces Produced by 4-Point Ternary Subdivision Scheme

Rabia Hameed, Ghulam Mustafa, Amina Liaqat, Dumitru Baleanu, Faheem Khan, Maysaa M. Al-Qurashi, Yu-Ming Chu, Eric Florentin
2020 Mathematical Problems in Engineering  
In this article, we present a new subdivision scheme by using an interpolatory subdivision scheme and an approximating subdivision scheme.  ...  The construction of the subdivision scheme is based on translation of points of the 4-point interpolatory subdivision scheme to the new position according to three displacement vectors containing two shape  ...  We use the following result of [12] to analyze convexity-preserving property for our subdivision scheme. Theorem 6 (see [12] ).  ... 
doi:10.1155/2020/6096545 fatcat:z4jean4uabfe5bqhpupmwjxemm

Error bounds for a convexity-preserving interpolation and its limit function

S. Amat, K. Dadourian, R. Donat, J. Liandrat, J.C. Trillo
2008 Journal of Computational and Applied Mathematics  
We show that this interpolation is convexity preserving, as its associated subdivision scheme. Finally, some numerical experiments are presented.  ...  Error bounds between a nonlinear interpolation and the limit function of its associated subdivision scheme are estimated. The bounds can be evaluated without recursive subdivision.  ...  From this step we either consider straight lines between the control points, building a polygonal approximation to the limit function of the subdivision scheme, or build the approximation to the limit  ... 
doi:10.1016/ fatcat:dri2b6kicrhctn2bud52g5w2nu

Non-uniform Interpolatory Subdivision Based on Local Interpolants of Minimal Degree [chapter]

Kęstutis Karčiauskas, Jörg Peters
2014 Lecture Notes in Computer Science  
We briefly compare these solutions to recent non-linear interpolatory subdivision schemes.  ...  This paper presents new univariate linear non-uniform interpolatory subdivision constructions that yield high smoothness, C 3 and C 4 , and are based on least-degree spline interpolants.  ...  One referee pointed out an alternative construction for interpolants [BCR13] that appeared after our submission.  ... 
doi:10.1007/978-3-642-54382-1_15 fatcat:q3hzcaweybgcvnhgkavkmbldau

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

Ghazala Akram, Khalida Bibi, Kashif Rehan, Shahid S. Siddiqi
2017 Journal of Computational and Applied Mathematics  
We will analyzed what conditions should be introduced on the initial control points so that the limit curve achieved by the subdivision schemes presented in [9] are both monotonicity and convexity preserving  ...  In this paper, the shape preserving properties of the stationary 4-point quaternary approximating and interpolating subdivision schemes of Ko [9] are fully investigated.  ...  [2] introduce a new approach towards proving convexity preserving properties for interpolatory subdivision scheme. In 2014, Tan et al.  ... 
doi:10.1016/ fatcat:duldcd2ovzgnbd3q52oaoexygu

Shape Preserving Interpolatory Subdivision Schemes for Nonuniform Data

Frans Kuijt, Ruud van Damme
2002 Journal of Approximation Theory  
This article is concerned with constructing suitable subdivision methods for the f-data which preserve convexity; i.e., the data at the kth level, {x (k) i , f i (k)} is a convex data set for all k provided  ...  This leads to explicit rational convexity preserving subdivision schemes which generate continuously differentiable limit functions from initial convex data.  ...  Convexity preserving interpolatory subdivision algorithms have also been discussed in [DLL92, LU94] .  ... 
doi:10.1006/jath.2001.3628 fatcat:qmbh5jznyvdjtdftufx3oldxou


Yan Wang, Zhiming Li
2018 Journal of Pure and Applied Mathematics: Advances and Applications  
It is shown that the resulting curves are 2 C continuous for the certain range of the parameter.  ...  A new five-point binary subdivision scheme with a parameter is proposed.  ...  Acknowledgements This work is supported by the Special Funds for Young Scholars of Hefei Normal University under Grant No. 2017QN42. Special thanks also go to the referees.  ... 
doi:10.18642/jmsaa_7100121984 fatcat:l736c4svlfesbcjju47d3ekylq

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

Pakeeza Ashraf, Mehak Sabir, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Ilyas Khan
2020 Frontiers in Physics  
Convexity-preservation of the limit curve of the four-point scheme is also analyzed. The shape-preserving behavior of the four-point scheme is also shown through several numerical examples.  ...  In addition, the outcomes are generalized to determine conditions for positivity-and monotonicity-preservation of the limit curves.  ...  Basically, a subdivision scheme satisfies the convexity-preserving property if, for an initial convex control polygon, the limit curves generated by the scheme preserve the convexity of the initial data  ... 
doi:10.3389/fphy.2019.00241 fatcat:dz4n6olymngnpoegfn7tpap5by

Page 8718 of Mathematical Reviews Vol. , Issue 99m [page]

1999 Mathematical Reviews  
interpolatory subdivision schemes.  ...  Summary: “A scheme is described for interactively modifying the shape of convexity-preserving planar interpolating curves.  ... 
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