Applied and Computational Mathematics
Volume 5, Issue 2, April 2016, Pages: 40-45

A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves

Zhi Liu1, *, Na Wei1, Jieqing Tan1, Xiaoyan Liu2

1School of Mathematics, Hefei University of Technology, Hefei, China

2Department of Mathematics, University of La Verne, La Verne, USA

(Zhi Liu)
(Na Wei)
(Jieqing Tan)
(Xiaoyan Liu)

Zhi Liu, Na Wei, Jieqing Tan, Xiaoyan Liu. A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves. Applied and Computational Mathematics. Vol. 5, No. 2, 2016, pp. 40-45. doi: 10.11648/j.acm.20160502.11

Abstract: 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 G2-continuous spline approximation of conic sections is obtained by using the subdivision scheme at the shoulder point of the conic section.

Keywords: Conic Section, Quartic Bézier Curve, Hausdorff Distance, Approximation, G2-Continuous, Subdivision Scheme

Contents

1. 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 [1-4] 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 [6-8]. Fang [9] presented a method for approximating conic sections using quintic polynomial curves. The constructed quintic polynomial curve has G3-continuity with the conic section at the end points and G1-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.

2. 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]

,

where , ,  are the control points,  is the weight associated with ,  is the quadratic Bernstein polynomial given by

It is also well known that  is an ellipse when , a parabola when  and a hyperbola when .

The quartic Bézier curve used to approximate the conic section  is given by

where  are the control points,  are the quartic Bernstein polynomials defined by

Lemma 1. [15] Suppose that  is any continuous curve which lies entirely inside the (closed) triangle  such that  and . Then

(1)

where  is the Hausdorff distance [12] defined by

,

and f:  is a function defined by

(2)

where  are the barycentric coordinates with respect to . Any point  can be expressed as . The curve  satisfies the equation  for .

The control points of the approximation curve  can be expressed as

, , , , ,

where  is the midpoint of  and . In order to ensure that the approximation curves  is contained in ,  and  must satisfy  and .

The point  lies on the line segment joining two points  and m, and  has the barycentric coordinates with respect to

(3)

Obviously  satisfy. Substituting Eq.(3) into Eq.(2), we can get

(4)

where

Suppose  has zeros at . Then from  it follows . But  will tend to infinity as tends to 1, which does not meet our requirement. So we choose

(5)

Substituting Eq.(5) into Eq.(4), we can get

(6)

where

(7)

The approximation curve  contacts with the conic  at  and 1 with multiplicity at least two respectively. If  and  are the roots of , we can get

(8)

By Eqs. (6), (7) and (8) we can get the leading coefficient  of  as follows

Since the approximation curve  is chosen to contact with the conic  at  with multiplicity 2,1,2,1 and 2, we have

From , it follows

(9)

The value of  is only determined by . If we want to determine an upper bound on the Hausdorff distance between the approximation curves and the conics, we need to obtain the range of  to ensure that the approximation curve  lies inside .

Theorem 1. If the weight satisfies , then the curve  lies inside , where  and .

Proof. According to the convex hull property of Bézier curves, the quartic Bézier curve  lies inside  when  and .

Substituting  into Eq. (5), we can get

.

Differentiating  with respect to  gives

Since the equation  has no real roots, there are only two possibilities, either , or  for all .

From , it follows  for all . Therefore  is a monotonically increasing function with respect to  for . It is easy to get , . Let . Then

.

Similarly, differentiating  with respect to  gives

The equation  has a unique zero . Since  , we have  for any .

In summary, we have ,  for .

Theorem 2. For , the Hausdorff distance between the conic section  and the approximation curve  is bounded as

(10)

where

Proof. The polynomial  has the maximum  at  in the interval [0, 1]. By Eq.(1) and Eq.(9), we can get the value of . The proof of Theorem 2 is completed.

Floater [15] gave the result that and  are , where  is the maximum length of the parametric interval under subdivision. So according to the error bound, the approximation order of the approximation curve  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] that for

(11)

where

.

Theorem 3. For , the upper bound on the Hausdorff error (10) by our method is smaller than that by Hu’s method, i.e., .

Proof. Comparing  in (10) with  in (11) reveals that to show  is equivalent to proving the following inequality

It is obvious that  for  according to Fig 1. Therefore

as asserted.

Fig. 1. The graph of .

3. Numerical Examples

Example 1. Let the conic section  be given with the control points , ,  and the weight , as shown in Fig 2(a). The quartic Bézier  has the control points , , , , , as shown in Fig 2(b). The Hausdorff error bound is

by Theorem 2 in our method. Obviously, this error bound is smaller than that with Hu’ s method, which is

obtained by (11).

(a) the conic section

(b) the quartic Bézier approximation

Fig. 2. The quartic Bézier approximation of conic section with .

Example 2. Let the conic section  be given with the control points , ,  and the weight , as shown in Fig 3(a). The quartic Bézier  has the control points , ,   , as shown in Fig. 3(b). The Hausdorff error bound is

by Theorem 2 in our method. Obviously, this error bound is smaller than that with Hu’ s method, which is

obtained by (11).

(a) the conic section ;

(b) the quartic Bézier approximation

Fig. 3. The quartic Bézier approximation of conic section with .

In addition, if the bound on the Hausdorff error  is larger than a user-specified error tolerance, we can consider the subdivision scheme for , consisting of alternately subdividing at the shoulder point  and normalizing each subcurve, as stated in [15]. Using this subdivision scheme, the composite curve of the quartic Bézier approximation curve  is globally G2 continuous. Suppose the conic section  is subdivided at  into two parts  and . Then  and  have control points as

and the weight  associated with the control points  and .

In Example 2, If the error tolerance is , then we should split the conic  at  into two segments  and , as shown in Fig 4(a), at the shoulder point by the subdivision scheme proposed in [16].

Using our method, the quartic Bézier approximations  and  have the Hausdorff error bounds

The composition curve of  and  yields the quartic G2continuous spline approximation  of the conic section , as shown in Fig. 4(b).

(a)The conic section  and .

(b) The composite curve of  and .

Fig. 4. The G2 continuous quartic Bézier approximation curve.

4. Conclusion

We give a new approximation method for conic section by quartic Bézier curves, and prove that our approximation method has a smaller error bound than previous quartic Bézier approximations. Although the approximations are not optimal, but the result is high accuracy. The next question considered is to find a better zeros sequence in order to have smaller error bound.

Acknowledgements

The authors would like to thank the referees for their valuable comments which greatly help improve the clarity and quality of the paper.

This work was supported in part by the National Natural Science Foundation of China (Grant No. 61472466 and 11471093), Key Project of Scientific Research, Education Department of Anhui Province of China under Grant No. KJ2014ZD30. The Fundamental Research Funds for the Central Universities under Grant No. JZ2015HGXJ0175.

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