Four point interpolatory-corner cutting subdivision

Tan, Jieqing; Tong, Guangyue; Zhang, Li; Xie, Junshan

doi:10.1016/j.amc.2015.05.107

Cited by 4 publications

(5 citation statements)

References 13 publications

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“…Here, 3 , and f 1 i,4 are the initial points of the 4-point, 6-point, and 8-point relaxed subdivision schemes obtained by substituting the values of N equal to 1, 2, and 3, respectively, in (13) and (14). Since the proposed subdivision schemes are stationary, the refinement rules are same at each level of subdivision.…”

Section: Framework For the Construction Of A Family Ofmentioning

confidence: 99%

“…Since the proposed subdivision schemes are stationary, the refinement rules are same at each level of subdivision. erefore, for other subdivision levels, we apply (11) while the coefficients of points f k i,N+1 remain same as the coefficients of points f 0 i,N+1 � f 0 i obtained from (13). Also, 3 , and f k+1 i,4 are the control points at (k + 1)-th subdivision level obtained by applying the 2-point, 4-point, 6-point, and 8-point relaxed subdivision schemes on the k-th level points 3 , and f k i,4 , respectively.…”

Section: Framework For the Construction Of A Family Ofmentioning

confidence: 99%

“…Li and Zheng [11] combined the 4-point scheme of Dyn et al [12] and the cubic B-spline binary refinement scheme to construct an interproximate subdivision scheme. Tan et al [13] combined the 4-point scheme of Dyn et al [12] and a 2point corner cutting scheme to construct another interproximate subdivision scheme. However, these schemes give interproximate behavior but are not easy to implement and analyze.…”

Section: Introductionmentioning

confidence: 99%

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Recursive Process for Constructing the Refinement Rules of New Combined Subdivision Schemes and Its Extended Form

Hameed

Mustafa

Deng

et al. 2021

Journal of Mathematics

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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.

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Section: Framework For the Construction Of A Family Ofmentioning

confidence: 99%

Section: Framework For the Construction Of A Family Ofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Recursive Process for Constructing the Refinement Rules of New Combined Subdivision Schemes and Its Extended Form

Hameed

Mustafa

Deng

et al. 2021

Journal of Mathematics

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“…Li and Zheng [14] combined the 4-point scheme of Dyn et al [9] and the cubic B-spline binary refinement scheme to construct a shape controlled subdivision scheme. Tan et al [19] combined the 4-point scheme of Dyn et al [9] and a 2-point corner cutting scheme to construct another shape controlled subdivision scheme. In this paper, we present a recursive method to construct the (2N + 2)-point combined subdivision schemes which is a new trend in the construction of subdivision schemes.…”

Section: Introductionmentioning

confidence: 99%

Recursive process for constructing the refinement rules of new combined subdivision schemes and its extended form

Hameed¹,

Mustafa²

2018

Preprint

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In this article, we present a new method to construct a family of (2N + 2)-point binary subdivision schemes with one tension parameter where N is a non-negative integer. The construction of the family of schemes is based on repeated local translation of points by certain displacement vectors. Therefore, the refinement rules of a (2N + 2)-point scheme for N = M are recursively obtained from the refinement rules of the (2N + 2)-point schemes for N = 0, 1, 2, . . . , M − 1. The complexity, polynomial reproduction and polynomial generation of these schemes are increased by two for the successive values of N . Furthermore, we modify this family of schemes to a family of (2N + 3)-point schemes with two tension parameters. Moreover, a family of interproximate subdivision schemes with tension parameters is also introduced, which allows a different tension value for each edge and vertex of the initial control polygon. Interproximate schemes generate curves and surfaces such that some initial control points are interpolated and others are approximated.

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“…Tan et al [14] studied the monotonicity preserving of a binary scheme based on [13]. Tan et al [15] presented a more practical algorithm to generate curves, which can interpolate some initial vertices and approximate the other vertices. Rehan and Siddiqi [16] introduced a combined six-point subdivision scheme with tension parameters that can generate C 1 , C 2 continuous interpolating limit curves and C 1 , C 2 , C 3 continuous approximating limit curves.…”

Section: Introductionmentioning

confidence: 99%

A Five-Point Subdivision Scheme with Two Parameters and a Four-Point Shape-Preserving Scheme

Tan

Wang

Shi

2017

MCA

Self Cite

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Abstract:In order to improve the flexibility of curves, a new five-point binary approximating subdivision scheme with two parameters is presented. The generating polynomial method is used to investigate the uniform convergence and C k -continuity of this scheme. In a special case, the five-point scheme changes into a four-point scheme, which can generate C 3 limit curves. The shape-preserving properties of the four-point scheme are analyzed, and a few examples are given to illustrate the efficiency and the shape-preserving effect of this special case.

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Four point interpolatory-corner cutting subdivision

Cited by 4 publications

References 13 publications

Recursive Process for Constructing the Refinement Rules of New Combined Subdivision Schemes and Its Extended Form

Recursive Process for Constructing the Refinement Rules of New Combined Subdivision Schemes and Its Extended Form

Recursive process for constructing the refinement rules of new combined subdivision schemes and its extended form

A Five-Point Subdivision Scheme with Two Parameters and a Four-Point Shape-Preserving Scheme

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