An Approximation Method of Bézier Curve

Zhi Wu,Chu Yi Song,De Xi Bao

doi:10.5539/cis.v10n4p67

Abstract

It is proved that the linear space constructed by power base is a banach space under 2-norm by using approximation method. For the Bézier curve--the elements in banach space, the linear combination of the low-order S power base is used to approximate optimal the high-order Bernstein base function. The original Bézier curve is instituted by the linear combination of low-order S power base and the optimal approximation element of the original Bézier curve is obtained.

Highlights

Approximating the high-order polynomial curves and curved surfaces with the low-order ones plays an important role in data compression, data transmission, and data exchange, etc., in geometric modeling tasks
The first category is based on the base function conversion
The second category can be summarized as the geometric approximation algorithms with the control vertexes

Summary

Introduction

Approximating the high-order polynomial curves and curved surfaces with the low-order ones plays an important role in data compression, data transmission, and data exchange, etc., in geometric modeling tasks. Current algorithms for Bézier Curve degree reduction can be summarized into two categories. Farouki proposes the degree reduction algorithm based on the Legendre Polynomials in reference (Rida T Farould 2000). The second category can be summarized as the geometric approximation algorithms with the control vertexes. In this category, reference (Forrest A R 1972) is based on the interpolation algorithm; S M. Based on the Sánchez-Reyes’s work, we further propose a degree reduction method with the optimal approximation. This method is built on the Bernstein base function and the properties of S power base

The Theoretical Problem

Approximation Error Estimation and Numerical Examples

Conclution