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

Muhammad Asghar,Ghulam Mustafa

doi:10.1155/2018/7824279

Muhammad Asghar, Ghulam Mustafa

Open Access

https://doi.org/10.1155/2018/7824279

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Abstract

The generalized symbols of the family of a-ary (a≥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.

Highlights

Subdivision schemes are powerful tools in CAGD for generation of smooth curves and surfaces
Lane-Riesenfeld algorithm has been used for the generation of a family of binary parametric subdivision schemes [17, 18]
The lower bound of Holder regularity of a family of a-ary subdivision schemes corresponding to the symbol Sna,e(z) is computed as r ≥ n + 2 − loga(μ), where μ is defined as μ = a − a2w and μ = aw

Summary

Introduction

Subdivision schemes are powerful tools in CAGD for generation of smooth curves and surfaces. Presented a new family of subdivision schemes by using sixpoint variant on the Lane-Riesenfeld algorithm. Siddiqi and Rehan [11] modified 3-point binary and ternary subdivision schemes with a tension parameter which generate a family of C1 limiting curves for certain range of tension parameter. Lane-Riesenfeld algorithm has been used for the generation of a family of binary parametric subdivision schemes [17, 18]. (i) The generalization of Lane-Riesenfeld algorithm to generate high arity schemes (ii) A unified way to present families of univariate stationary and nonstationary high arity parametric subdivision schemes (iii) To propose families of schemes with higher continuity and generation degree comparative to the existing schemes [7, 17,18,19] (see Tables 3 and 4).

Construction of Algorithm

Nonstationary Algorithm

Conclusion