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

Humaira Mustanira Tariq,Rabia Hameed,Ghulam Mustafa,Jia-Bao Liu

doi:10.1155/2021/6679201

Humaira Mustanira Tariq, Rabia Hameed + Show 2 more

Open Access

https://doi.org/10.1155/2021/6679201

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Abstract

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.

Highlights

Nowadays, subdivision schemes have got great importance in the field of geometric modeling
New points are added at each refinement level in order to get a smooth final shape. e number of points which are inserted at each refinement level is known as the arity of the curve subdivision scheme
If we move the single point of the control polygon, the shape of the curve changes over the specific region. is region is known as the support of the scheme

Summary

Introduction

Subdivision schemes have got great importance in the field of geometric modeling. Ternary schemes give us small support as compared to the binary subdivision scheme and give better local control on the shapes. Is type of subdivision schemes is known as approximating subdivision schemes and was introduced by [7,8,9,10]. In [13,14,15,16], the subdivision schemes are introduced with parameters which can produce both the interpolatory and approximating shapes. We present a class of ternary combined schemes which is obtained from the well-known classes of interpolatory and approximating subdivision schemes. We unify interpolatory and approximating classes of subdivision schemes into a single class of combined schemes by applying certain mathematical operations on their refinement rules.

Preliminaries

Formulation of the Class of Combined Schemes

Conclusion