Approximate multi-degree reduction of Q-B\xe9zier curves via generalized Bernstein polynomial functions

Xianzhi Hu,Gang Hu,Md Yushalify Misro,Muhammad Abbas

doi:10.1186/s13662-020-02871-y

Abstract

Quasi Bézier curves (or QB-curves, for short) possess the excellent geometric features of classical Bézier curves and also have good shape adjustability. In this paper, an algorithm for a multi-degree reduction of QB-curves based on L_{2} norm and by the analysis of geometric characteristics of QB-curves is constructed. The approximating approach for QB-curves of degree n+1 by degree m (mleq n) is also given. Secondly, by solving the linear equations under the constraints of C^{0} and C^{1} and without constraints, the explicit expression of the points of the approximating curve is obtained, which minimizes the error between the original curve and the approximating curve using the least square method. Some numerical examples of degree reduction under different constraints are given, and the corresponding errors are calculated as well. The results show that this method can be easily implemented, is highly precise and very effective.

Highlights

A parametric curve, such as Bézier curve, is a fundamental tool and research content in the field of Computer Aided Geometric Design (CAGD)/Computer Aided Manufacturing (CAM)
Problem 1 Let us consider that the n + 1th-degree QB curve determined by control points {P∗i }ni=+01 ∈ Rd, d = 2, 3, has the following form: n+1 r∗(θ ) = P∗i bi,n+1(θ ), (3)
6 Conclusions In this paper, the least square degree reduction approximation problem for QB-curves based on L2-norm without constrains and under the C0 and C1 constraints is studied

Summary

Introduction

A parametric curve, such as Bézier curve, is a fundamental tool and research content in the field of Computer Aided Geometric Design (CAGD)/Computer Aided Manufacturing (CAM). In [25], generalized Bernstein basis functions were used to construct a QB-curve with multiple shape parameters The curve of this kind possesses several significant geometric characteristics of classical Bézier curves, and it has bendable shape modification, i.e., the shape of a curve can be adjusted by altering the values of shape parameters to construct more complicate curves of the same degree. This makes QB-curves extensively used in different modelings and CAD/CAM systems and has certain application value in describing curves and surfaces. Li et al [39] constructed some geometric continuity conditions for the generalized cubic H-Bézier model for the purpose of constructing shape-controlled complex curves and surfaces in engineering.

Problem description

Degree reduction of QB-curves under unconstrained conditions

Conclusions