The best uniform quadratic approximation of circular arcs with high accuracy

Abstract

Abstract:In this article, the issue of the best uniform approximation of circular arcs with parametrically defined polynomial curves is considered. The best uniform approximation of degree 2 to a circular arc is given in explicit form. The approximation is constructed so that the error function is the Chebyshev polynomial of degree 4; the error function equioscillates five times; the approximation order is four. For  D =4 arcs (quarter of a circle), the uniform error is 5:5 10 3 . The numerical examples demonstrate the efficiency and simplicity of the approximation method as well as satisfy the properties of the best uniform approximation and yield 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]. .…”

Quartic approximation of circular arcs using equioscillating error function

“…There are methods in the literature that are G 1 − and G 2 −continuous, see for example [6], [9], [10], [13], [14], [16], [17], [18], [19], [22].…”

“…To approximate the circle c, there is a need to find a parametrically defined polynomial curve p : t → (x(t), y(t)) , 0 ≤ t ≤ 1, where x(t), y(t) are polynomials of degree 4, that approximates c with "minimum" error. Many researchers have tackled this issue using different norms and methods, see [2], [3], [4], [5], [6], [9], [10], [14], [16], [18]. For details and numerical comparisons with these works, see section 6.…”

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.…”