Families of univariate and bivariate subdivision schemes originated from quartic B-spline

Mathematical Problems in Engineering

2018

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The generalized symbols of the family of -ary ( ≥ 2) univariate stationary and nonstationary parametric subdivision schemes have been presented. These schemes are the new version of Lane-Riesenfeld algorithms. Comparison shows that our proposed family has higher continuity and generation degree comparative to the existing subdivision schemes. It is observed that many existing binary and ternary schemes are the special cases of our schemes. The analysis of proposed family of subdivision schemes is also presented in this paper.

Section: Application and Comparisonmentioning

confidence: 87%

Family of a-Ary Univariate Subdivision Schemes Generated by Laurent Polynomial

Asghar

Mathematical Problems in Engineering

2018

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“…is type of subdivision schemes is known as approximating subdivision schemes and was introduced by [7][8][9][10]. e analysis of both types of schemes is done by [11,12].…”

Section: Introductionmentioning

confidence: 99%

A New Paradigm to Design a Class of Combined Ternary Subdivision Schemes

Tariq

Hameed

2021

Journal of Mathematics

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Subdivision schemes play a vital role in curve modeling. The curves produced by the class of 2 n + 2 -point ternary scheme (Deslauriers and Dubuc (1989)) interpolate the given data while the curves produced by a class of 2 n + 2 -point ternary B-spline schemes approximate the given data. In this research, we merge these two classes to introduce a consolidated and unified class of combined subdivision schemes with two shape control parameters in order to grow versatility for overseeing valuable necessities. However, the proposed class of subdivision schemes gives optimal smoothness in the final shapes, yet we can increase its smoothness by using a proposed general formula in form of its Laurent polynomial. The theoretical analysis of the class of subdivision schemes is done by using various mathematical tools and using their coding in the Maple environment. The graphical analysis of the class of schemes is done in the Maple environment by writing the codes based on the recursive mathematical expressions of the class of subdivision schemes.

“…e families of such schemes were introduced in [5][6][7][8]. Furthermore, a parametric subdivision scheme can be converted to the nonuniform subdivision scheme by defining the local tension parameter.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, a parametric subdivision scheme can be converted to the nonuniform subdivision scheme by defining the local tension parameter. Mustafa and Hameed [6,7] converted their families of univariate and bivariate subdivision schemes to nonuniform approximating subdivision schemes by defining local tension parameters. A parametric subdivision scheme which can produce both interpolatory and approximating curves is called the combined subdivision scheme.…”

Section: Introductionmentioning

confidence: 99%

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

Hameed

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.