CIRCLE APPROXIMATION BY QUARTIC G²SPLINE USING ALTERNATION OF ERROR FUNCTION

Abstract

ABSTRACT. In this paper we present a method of circular arc approximation by quartic Bézier curve. Our quartic approximation method has a smaller error than previous quartic approximation methods due to the alternation of the error function of our quartic approximation. Our method yields a closed form of error so that subdivision algorithm is available, and curvaturecontinuous quartic spline under the subdivision of circular arc with equal-length until error is less than tolerance. We illustrate our method by some numerical examples.

The best quintic Chebyshev approximation of circular arcs of order ten

“…The circle c is approximated in this paper using a quintic parametrically defined polynomial curve p : t → (x(t), y(t)) , 0 ≤ t ≤ 1, where x(t), y(t) are polynomials of degree 5, that approximates c with least deviation. Many researchers have tackled this issue using different degrees, norms, and methods, see for example [2,3,4,5,6,7,8,9] and the references therein.…”

The best sextic approximation of hyperbola with order twelve

“…This solution is presented in Figure 2 and Figure 3; the corresponding error is shown in Figure 4. More related results can be found in [13][14][15] and the references therein.…”

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.…”