A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves

A new approximation method for conic section by quartic Bézier curves is proposed. This method is based on the quartic Bézier approximation of circular arcs. We give the upper bound of Hausdorff distance between the conic section and the quartic Bézier curve, and also show that the approximation order is eight. And we prove that our approximation method has a smaller upper bound than previous quartic Bézier approximation methods. A quartic G-continuous spline approximation of conic sections is obtained by using the subdivision scheme at the shoulder point of the conic section.


Introduction
It is well-known that besides the straight line, the conic sections are the simplest geometric entity. Conic sections are widely used in the fields of CAGD or CAD/CAM. Since the most of conic sections cannot be accurately represented by polynomials in explicit form, the parameter polynomials are used to approximate the conic sections. Bézier curves and surfaces [1][2][3][4] are the modeling tools widely used in CAD/CAM systems. Most of the previous work on conic sections approximation is based on quartic Bézier curves.
In 1997, Ahn and Kim [5] presented the approximation of circular arcs by quartic and quintic Bézier curves with approximation orders eight and ten. The approximation of circular arcs by quartic Bézier curves with approximation order eight were represented in [6][7][8]. Fang [9] presented a method for approximating conic sections using quintic polynomial curves. The constructed quintic polynomial curve has G 3 -continuity with the conic section at the end points and G 1 -continuity at the parametric mid-point. Floater [10] found that the approximation of the conic section by Bézier curve of any odd degree n has optimal approximation order 2n. Ahn [11] presented two methods of the quartic Bézier approximation of the conic section. Hu [12] gave a method for approximating conic sections by constrained Bézier curves of arbitrary degree. In 2014, Hu [13] provided a new approximation method of conic sections by quartic Bézier curves, which has a smaller error bound than previous quartic Bézier approximations.
The outline of this paper is as follows: In section 2, we present a new approximation method for conic sections by quartic Bézier curves, and give an upper bound on the Hausdorff distance between the conic section and the quartic Bézier curve. It is shown that the approximation order is eight. And we prove that our approximation method has a smaller error bound than previous quartic Bézier approximations. Finally, we illustrate our results by some numerical examples.

Quartic Bézier Approximation of Conic Sections
In this section, we give a new highly accurate approximation method of conic section by quartic Bézier curves. The conic section can be represented as [14] ( ) It is also well known that ( ) t c is an ellipse when 1 ω < , a parabola when 1 ω = and a hyperbola when 1 ω > .
The quartic Bézier curve used to approximate the conic section ( ) t c is given by ( ) where 0 1 2 , , τ τ τ are the barycentric coordinates with respect to 0 1 2 ∆p p p . Any point 0 1 2 x ∈ ∆p p p can be expressed as The control points of the approximation curve ( ) can be expressed as is the midpoint of 0 p and 2 p . In order to ensure that the approximation curves ( ) is contained in 0 1 2 ∆p p p , α and β must satisfy 0 1 < α < and 0 1 < β < .
The point 1 2 lies on the line segment joining two points 1 p and m, and ( ) has the barycentric coordinates with respect to has zeros at 1 t 2 = .
In summary, we have 0 1 < α < , Theorem 2. For 1 0 ′ < ω < ω , the Hausdorff distance between the conic section (t) c and the approximation curve  Eq.(9), we can get the value of ( ) δ ω . The proof of Theorem 2 is completed. Floater [15] gave the result that 1 ω − and 0 , where h is the maximum length of the parametric interval under subdivision. So according to the error bound, the approximation order of the approximation curve 1 (t) b in Theorem 2 is eight. The error of Hu's approximation method [13] is smaller than that of other previous quartic Bézier curve approximation methods. Next, we will prove our error bound is smaller than that of Hu's method.
Hu showed in [13]    p , as shown in Fig 2(b)  (100, 0) = = b p , as shown in Fig. 3(b). The Hausdorff error bound is  In addition, if the bound on the Hausdorff error H 1 d ( , ) b c is larger than a user-specified error tolerance, we can consider the subdivision scheme for ( ) t c , consisting of alternately subdividing at the shoulder point ( ) 0.5 c and normalizing each subcurve, as stated in [15]. Using this subdivision scheme, the composite curve of the quartic Bézier approximation curve 1 (t) b is globally G 2 continuous.

Numerical Examples
Suppose the conic section ( )