A New Class of 2q-Point Nonstationary Subdivision Schemes and Their Applications

Abdul Ghaffar,Dumitru Baleanu,Mudassar Iqbal,Mehwish Bari,Kottakkaran Sooppy Nisar,Zafar Ullah

doi:10.3390/math7070639

Abstract

The main objective of this study is to introduce a new class of 2 q -point approximating nonstationary subdivision schemes (ANSSs) by applying Lagrange-like interpolant. The theory of asymptotic equivalence is applied to find the continuity of the ANSSs. These schemes can be nicely generalized to contain local shape parameters that allow the user to locally adjust the shape of the limit curve/surface. Moreover, many existing approximating stationary subdivision schemes (ASSSs) can be obtained as nonstationary counterparts of the proposed ANSSs.

Highlights

The importance of subdivision schemes (SSs) cannot be denied because it plays an important role in computer aided geometric design (CAGD), geometric modeling, computer graphics, image processing, visualization and engineering, etc
We prove that the SSs (21) and (25) are asymptotically equivalent
In order to present the achievements of the nonstationary subdivision schemes (NSSs) (19) and (21), we discuss continuity, shape of limit curves, curvature, and torsion

Summary

Introduction

The importance of subdivision schemes (SSs) cannot be denied because it plays an important role in computer aided geometric design (CAGD), geometric modeling, computer graphics, image processing, visualization and engineering, etc. SSs can be distinguished in various types: Range from uniform to non-uniform, binary to an arbitrary arity, interpolatory to approximating, and stationary to nonstationary. It seems that stationary subdivision schemes (SSSs) have interesting features, but reconstruction of special types of limit curves of various shapes—including polynomial functions and conic sections such as circles, ellipses, and spiral curves—could not be accomplished without the nonstationary subdivision schemes (NSSs). De Rham [1] constructed approximating SSS and, later on, Chaikin [2] introduced the corners cutting SSS. In 2003, Jena et al [5] constructed a four-point interpolating

Objectives

Results

Conclusion