A Nonstationary Ternary 4-Point Shape-Preserving Subdivision Scheme

Jieqing Tan,Guangyue Tong,Ghulam Mustafa

doi:10.1155/2021/6694241

Abstract

This paper uses the continued fraction technique to construct a nonstationary 4-point ternary interpolatory subdivision scheme, which provides the user with a tension parameter that effectively handles cusps compared with a stationary 4-point ternary interpolatory subdivision scheme. Then, the continuous nonstationary 4-point ternary scheme is analyzed, and the limit curve is at least C 2 -continuous. Furthermore, the monotonicity preservation and convexity preservation are proved.

Highlights

Subdivision schemes are wildly used in many areas, including CAGD, CG, and related areas
Hassan et al present the 4-point ternary subdivision scheme in [3], in which the limit curves by the 4-point ternary subdivision scheme are continuous
Cai presents binary and ternary 4-point interpolatory subdivision schemes that are continuous in nonuniform control points and discusses the limit curve’s convexity preservation [7, 8]

Summary

Introduction

Subdivision schemes are wildly used in many areas, including CAGD, CG, and related areas. Beccari et al present a nonstationary subdivision scheme in [4, 5], which generates continuous limit curves. Cai presents binary and ternary 4-point interpolatory subdivision schemes that are continuous in nonuniform control points and discusses the limit curve’s convexity preservation [7, 8]. Tan et al discuss the monotonicity preservation and convexity preservation of the binary subdivision scheme [11, 12]. Nonstationary subdivision schemes have been studied [5, 22], but the limit curves are not monotonicity preserving and convexity preserving. Our paper aims to construct a nonstationary 4-point ternary interpolatory subdivision scheme, which is shape-preserving using a continued fraction.

A Stationary Subdivision

Convexity Preservation

Improved Subdivision Interpolation Scheme at Endpoints