Chebyshev Approximation of Plane Curves by Splines

Abstract

Given a parametric plane curve p and any Bezier curve q of degree n such that p and q have contact of order k at the common end points, we use the normal vector field of p to measure the distance of corresponding points of p and q. Applying the theory of nonlinear Chebyshev approximation, we show that the maximum norm of this distance (or error) function ρ q is locally minimal for q if and only if ρ q is an alternant with 2 · ( n − k − 1) + 1 extreme points. Finally, a Remes type algorithm is presented for the numerical computation of a locally best approximation to p.