Rational Approximation of Offset Curves by Parametric Speed Approximation

The problem of parametric speed approximation of a curve is raised in this paper. The authors point out that the crux of offset curve approximation lies in the approximation of parametric speed, and two methods are provided. The parametric speed of the curve is firstly approximated by the Bernstein polynomial, which takes the lengths of control point vectors of the direction curve of normal as Bezier coordinates. Then the corresponding geometric offset approximation algorithm is given. Moreover, an offset approximation with high precision is obtained by degree elevation of the direction curve of normal. Based on the Legendre polynomials approximation and Jacobi polynomials approximation with endpoints interpolation of parametric speed of the curve, two algebraic rational approximation algorithms of offset curves, which preserve the direction of normal, are derived.