The best quintic Chebyshev approximation of circular arcs of order ten

Abedallah M Rababah

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.

Keywords


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

References


Prof. Dr. Klaus Höllig

Fachbereich Mathematik, IMNG

Pfaffenwaldring 57, 70569 Stuttgart

Phone: (0711) 685-6 53 51

Fax: (0711) 685-6 53 75

Email: klaus.hoellig@gmail.com

hoellig@mathematik.uni-stuttgart.de

Prof. Y. J. Ahn

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

Republic of Korea,

Email: ahn@chosun.ac.kr

Professor Samer Alabed

Department of Electrical Engineering,

American University of the Middle East, Kuwait.

Email: samer.al-abed@aum.edu.kw

samer.alabed@nt.tu-darmstadt.de

Professor Chatchai Khunboa

Department of Computer Engineering, Faculty of Engineering,

Khon Kaen University, 40002, Thailand

Email: chatchai@kku.ac.th




DOI: http://doi.org/10.11591/ijece.v9i5.pp%25p
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