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

The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the Hausdorff distance and has 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. (C) 2021 Elsevier B.V. All rights reserved.

Automated summary

This article discusses the optimal approximation of circular arcs using parametric polynomial curves, with an optimization related to the Hausdorff distance that has not been studied before. A general theory is presented with a rigorous analysis for special cases, including practical interpolation scenarios. Several numerical examples are provided to confirm theoretical results.