The Best Uniform Cubic Approximation of Circular Arcs with High Accuracy

Abdalah Rababah


In this article, the issue of the best uniform cubic approximation of circular arcs with parametrically defined polynomial curves is considered. By a proper choice of the Bézier points, the best uniform approximation of degree 3 to a circular arc is given in explicit form. The approximation is constructed so that the error function is the Chebyshev polynomial of degree 6; the error function equioscillates 7 times; the approximation order is 6. The numerical examples demonstrate the efficiency and simplicity of the approximation method as well as satisfying the properties of the best uniform approximation and yielding the best approximation of least deviation and the highest possible accuracy.


Bezier curves; Best uniform approximation; Circular arc; high accuracy; approximation order; Equioscillation

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P. Bézier, The Mathematical Basis of the UNISURF CAD System, Butterworth-Heinemann Newton, MA, USA (1986).

J. Blinn, How many ways can you draw a circle?, Computer Graphics and Applications, IEEE 7 (8) (1987), 39–44. Boor, K. Höllig and M. Sabin, High accuracy geometric Hermite interpolation, Comput. Aided Geom. Design 4 (1987), 269–278.

T. Dokken, M. Dæhlen, T. Lyche and K. Mørken, Good approximation of circles by curvaturecontinuous Bézier curves, Comput. Aided Geom. Design 7 (1990), 33–41.

M. Floater, High order approximation of conic sections by quadratic splines, Comput. Aided Geom. Design 12 (1995), 617–637.

M. Goldapp, Approximation of circular arcs by cubic polynomials, Comput. Aided Geom. Design 8 (1991), 227–238.

K. Höllig and J. Hörner, Approximation and Modeling with B-Splines, SIAM, Titles in Applied Mathematics 132 (2013).

A. Makarov, Splines with minimal defect and decomposition matrices, Communications in Mathematics and Applications 3 (3) (2012), 355–367.

A. Rababah, Taylor theorem for planar curves, Proc. Amer. Math. Soc. 119 (3) (1993), 803–810.

A. Rababah, High order approximation method for curves, Comput. Aided Geom. Design 12 (1995), 89–102.

A. Rababah, High accuracy Hermite approximation for space curves in (Re^d), Journal of Mathematical Analysis and Applications 325 (2) (2007), 920–931.

A. Rababah and S. Mann, Iterative process for (G^2)-multi degree reduction of Bézier curves, Applied Mathematics and Computation 217 (20) (2011), 8126–8133.

A. Rababah and S. Mann, Linear methods for (G^1), (G^2), and (G^3)-multi-degree reduction of Bézier curves, Computer-Aided Design 45 (2) (2013), 405–414.

A. Rababah, The best uniform quadratic approximation of circular arcs with high accuracy, Open Mathematics 14 (1) (2016), 118–127.

J. Rice, The approximation of functions, Vol. 1: linear theory. Addison-Wesley, 1964.

R. Shamoyan and W. Xu, A note on sharp embedding theorems for holomorphic classes based on Lorentz spaces on a unit circle, Communications in Mathematics and Applications 1 (1) (2010), 47–51.


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