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

Rabia Hameed,Shafqat Ali,Jiansong Deng,Ghulam Mustafa,Efthymios G Tsionas

doi:10.1155/2021/6639706

Abstract

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.

Highlights

Subdivision schemes are efficient tools for generating smooth curves/surfaces as the limit of an iterative process based on simple refinement rules starting from certain control points defining a control polygon/mesh
Approximating schemes yield smoother curves with higher order continuity, interpolating schemes are more useful for engineering applications as they preserve the shape of the coarse mesh. e special family of interpolatory schemes consists of the schemes with refinement rules that preserve the points associated with the coarse mesh and only generate new points related to the additional vertices of the refined mesh
An important family of interpolatory schemes was introduced by Deslauriers and Dubuc [1], and latest tools for its analysis were introduced by Amat et al [2] whereas an important family of approximating subdivision schemes that is the dual counterparts of the schemes of Deslauriers and Dubuc [1] was proposed by Dyn et al [3]

Summary

Introduction

Subdivision schemes are efficient tools for generating smooth curves/surfaces as the limit of an iterative process based on simple refinement rules starting from certain control points defining a control polygon/mesh. There exist the parametric subdivision schemes, which can produce family of smooth approximating curves for special choices of the tension parameters. Pan et al [9] and Novara and Romani [10] presented the combined ternary subdivision schemes to fit interpolatory and approximating curves. We present a recursive method to construct the (2N + 2)-point combined subdivision schemes with one tension parameter to control the given points of the initial polygon. We present an extended form of this family of combined schemes by defining another tension parameter to control the insertion of new point between the given points in order to smooth the given polygon. We convert our schemes to interproximate schemes that generate smooth and oscillation-free curves and surfaces such that some initial control points are interpolated and others are approximated.

Preliminaries

Construction of the Families of Subdivision Schemes

Analysis of the Families of Schemes

Interproximate Subdivision Schemes

Conclusion