A new method approximating offset curve by Bézier curve using parallel derivative curves

Abstract

In this paper, we present a new method for the offset approximation of an n-th degree Bezier curve by an \(n+1\)-th degree Bezier curve using parallel derivative curves. Our method is based on the Gauss map approximation of the n-th degree Bezier curve by the \(n+1\)-th degree Bezier curve whose derivative curve is parallel to that of the n-th degree Bezier curve. Thus, the convolution of the Gauss map approximation curve and n-th Bezier curve is a polynomial of degree \(n+1\), which is our offset approximation of the n-th degree Bezier curve. Our approximation method has two advantages. One is that the offset approximation method yields a non-rational (polynomial) curve which is obtained by the simple sum of two Bezier curves. The other is that the offset approximation curve satisfying the endpoint interpolation of the given offset curve automatically has the same tangent lines as the offset curve at both endpoints. We illustrate our offset approximation method with some numerical examples.