A simple and efficient approximation of a Bézier piece by its cutdown polygon

Abstract

A new concept named the cutdown polygon is introduced for a Bézier piece approximation. The cutdown polygon is constructed by directly connecting some selected control points of the Bézier piece. Based on the proposed approximation, we obtained two explicit quantitative bounds. Apart from the degree of the polynomial, the two bounds depend only on the maximal absolute second differences or the sum of absolute second differences of the sequence of control points, respectively. Compared with control polygon approximation, the key advantage of using the cutdown polygon is that the edge number can be dramatically reduced to about one half of that using control polygon approximation or less while maintaining the same estimate bound.