Geometric modeling and applications of generalized blended trigonometric B\xe9zier curves with shape parameters

Sidra Maqsood,Abdul Majeed,Azhar Iqbal,Muhammad Abbas,Kenjiro T Miura

doi:10.1186/s13662-020-03001-4

Abstract

A Bézier model with shape parameters is one of the momentous research topics in geometric modeling and computer-aided geometric design. In this study, a new recursive formula in explicit expression is constructed that produces the generalized blended trigonometric Bernstein (or GBT-Bernstein, for short) polynomial functions of degree m. Using these basis functions, generalized blended trigonometric Bézier (or GBT-Bézier, for short) curves with two shape parameters are also constructed, and their geometric features and applications to curve modeling are discussed. The newly created curves share all geometric properties of Bézier curves except the shape modification property, which is superior to the classical Bézier. The C^{3} and G^{2} continuity conditions of two pieces of GBT-Bézier curves are also part of this study. Moreover, in contrast with Bézier curves, our generalization gives more shape adjustability in curve designing. Several examples are presented to show that the proposed method has high applied values in geometric modeling.

Highlights

1 Introduction Bézier curves are the powerful mechanism for modeling in computer-aided geometric design (CAGD) and computer graphics (CG)
5 Conclusions In this study, we proposed the GBT-Bézier curves of degree m associated with two shape parameters and studied their characteristics
These proposed GBT-Bézier curves have almost all characteristics of the classical Bézier curves but the shape-adjustable quality is an additional quality if compared to the classical Bézier curves

Summary

Introduction

Bézier curves are the powerful mechanism for modeling in computer-aided geometric design (CAGD) and computer graphics (CG). Hu et al [14] constructed geometric continuity conditions for the construction of free-form generalized Bézier curves with n shape parameters These free-form complex shape-adjustable generalized Bézier curves can be modeled by using shape-adjustable generalized Bernstein basis functions. Maqsood et al [16] constructed the generalized trigonometric Bézier (GT-Bézier) curves via GT-basis functions with shape parameters They modeled some complex curves and surfaces using C3 and G2 continuity conditions. The quadratic trigonometric basis functions were constructed by Bashir et al [20] using two shape parameters They modeled a rational quadratic trigonometric Bézier curve using these trigonometric basis functions as well as two curve segments connected by using C2 and G2 continuity conditions. GBT-Bézier curves with two shape parameters are constructed based on GBT-Bernstein basis functions of degree m. Z, the GBT-Bernstein basis functions of degree m become the same as classical Bernstein basis functions of degree m

Derivative at the end points:

Geometric invariance

Variation diminishing property

Shape control property

Conclusions