The best quintic Chebyshev approximation of circular arcs of order ten

Abstract

Mathematically, circles are represented by trigonometric parametric equations and implicit equations. Both forms are not proper for computer applications and CAD systems. In this paper, a quintic polynomial approximation for a circular arc is presented. This approximation is set so that the error function is  of degree $10$ rather than $6$; the Chebyshev error function equioscillates $11$ times rather than $7$; the approximation order is $10$ rather than $6$. The method approximates more than the full circle with Chebyshev   uniform error  of  $1/2^{9}$. The examples show the competence and simplicity of the proposed approximation, and that it can not be improved.