Quartic approximation of circular arcs using equioscillating error function

Abstract

Abstract-A high accuracy quartic approximation for circular arc is given in this article. The approximation is constructed so that the error function is of degree 8 with the least deviation from the x-axis; the error function equioscillates 9 times; the approximation order is 8. The numerical examples demonstrate the efficiency and simplicity of the approximation method as well as satisfying the properties of the approximation method and yielding the highest possible accuracy.

Approximating offset Curves using Bezier curves with high accuracy

“…By the symmetry of the approximation of the circular arc, in equation 8, x a ( 1 a ) and y a ( 1 a ) equal to zero at the same parameters, and (9), x a ( 1 b ) and y a ( 1 b ) equal to one at the same parameters, then a equals the parameter in (8) and b equals the parameter in (9).…”

“…The offset approximation in this paper is based on the best uniform approximation of the circular arc and yields a polynomial offset approximation curve. The best uniform approximation of the circular arc of degree 3 presented in [7] where the error function is the Chebyshev polynomial of degree 6, see also [8][9][10][11][12][13][14][15][16]. .…”

High order approximation of degree nine and order eighteen

“…The polynomial of best approximation of degree nine is found in this paper; it has nineteen equioscillations rather than eleven equioscillations theoretically guaranteed by the theorems of Chebyshev and Borel but cannot be found. The cases of n = 2, 3, 4 are considered in [12][13][14]; i.e., the nonic piecewise approximation for planar curves α 1 = α 2 = 9 is studied. The approximation order is raised to eighteen rather than ten.…”