The best quintic Chebyshev approximation of circular arcs of order ten

Abedallah M Rababah


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.


B\'ezier curves; quintic approximation; circular arc; high accuracy; approximation order; equioscillation; CAD.


Prof. Dr. Klaus Höllig

Fachbereich Mathematik, IMNG

Pfaffenwaldring 57, 70569 Stuttgart

Phone: (0711) 685-6 53 51

Fax: (0711) 685-6 53 75


Prof. Y. J. Ahn

Department of Mathematics Education, Chosun University, Seosuk-dong, Dong-gu, Gwangju 501-759,

Republic of Korea,


Professor Samer Alabed

Department of Electrical Engineering,

American University of the Middle East, Kuwait.


Professor Chatchai Khunboa

Department of Computer Engineering, Faculty of Engineering,

Khon Kaen University, 40002, Thailand


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