On optimal polynomial geometric interpolation of circular arcs according to the Hausdorff distance

Abstract

The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the Hausdorff distance and have not been studied yet in the literature. Parametric polynomial curves of low degree are used and a geometric continuity is prescribed at the boundary points of the circular arc. A general theory about the existence and the uniqueness of the optimal approximant is presented and a rigorous analysis is done for some special cases for which the degree of the polynomial curve and the order of the geometric smoothness differ by two. This includes practically interesting cases of parabolic G 0 , cubic G 1 , quartic G 2 and quintic G 3 interpolation. Several numerical examples are presented which confirm theoretical results.

Optimal parametric interpolants of circular arcs

“…In all cases the optimality order is measured by simplified radial distance. The only paper where the optimality of the best interpolant is proved according to the real radial distance is [9]. Let us also mention the G n interpolation of order n + 1.…”