A Highly Accurate Approximation of Conic Sections by Quartic Bézier Curves

Zhi Liu

doi:10.11648/j.acm.20160502.11

Abstract

A new approximation method for conic section by quartic Bezier curves is proposed. This method is based on the quartic Bezier approximation of circular arcs. We give the upper bound of Hausdorff distance between the conic section and the quartic Bezier 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 Bezier 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.

Highlights

It is well-known that besides the straight line, the conic sections are the simplest geometric entity
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

Summary

Introduction

It is well-known that besides the straight line, the conic sections are the simplest geometric entity. 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. Fang [9] presented a method for approximating conic sections using quintic polynomial curves. 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. 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.

Quartic Bézier Approximation of Conic Sections

Numerical Examples

Conclusion