Arc length preserving approximation of circular arcs by Pythagorean-hodograph curves of degree seven

TL;DR

This paper develops a method for approximating circular arcs with degree seven Pythagorean-hodograph curves that preserve arc length, providing a detailed analysis, solution criteria, and numerical validation.

Contribution

It introduces a complex representation approach for arc length preserving PH curve approximation of circular arcs, including solution selection criteria and asymptotic analysis.

Findings

Numerical examples confirm theoretical results.

Multiple solutions exist with criteria for optimal selection.

The approach effectively approximates circular arcs with degree seven PH curves.

Abstract

In this paper interpolation of two planar points, corresponding tangent directions and curvatures with Pythagorean-hodograph (PH) curves of degree seven preserving an arc length is considered. A general approach using complex representation of PH curves is presented and a detailed analysis of the problem for data arising from a circular arc is provided. In the case of several solutions some criteria for the selection of the most appropriate one are described and an asymptotic analysis is given. Several numerical examples are included which confirm theoretical results.