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.
Highlights
This replacement makes sense because both E(t) and e(t) attain their roots and reach their extrema at the same parameters
The use of parametric representation of curves is convenient in the field of CAD
In [16], the idea that a parametric representation of a curve is not unique has been used to improve the order of approximation by polynomial curves of degree n from n + 1 to 2n
Summary
The notations (x(t), y(t)) and x(t) y(t) are used to represent parametric equations, and points will be used in this article. As the scheme in this paper is built on the idea of minimizing the error over all of the segment [0, 1], the right choice for the beginning control point p0 is as follows p0 = (−α0 cos(θ), −β0 sin(θ)), where values of α0 and β0 could but should not be the same. The right choice for the end control point p4 is as follows p4 = (−α0 cos(θ), β0 sin(θ)). In order to have the Bezier curve p begin in the third quadrant, go counter clockwise through fourth and first quadrants and end in the second quadrant as the circular arc c, the following conditions should be fulfilled α, β, γ, ζ > 0, ξ > 1. Thereafter, it is shown that these values satisfy the approximation conditions; this is carried out
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