# APPROXIMATION ORDER OF C3 QUARTIC B-SPLINE APPROXIMATION OF CIRCULAR ARC

• BAE, SUNG CHUL (DEPARTMENT OF MATHEMATICS EDUCATION, KOREA UNIVERSITY) ;
• AHN, YOUNG JOON (DEPARTMENT OF MATHEMATICS EDUCATION, CHOSUN UNIVERSITY)
• Accepted : 2016.06.14
• Published : 2016.06.25

#### Abstract

In this paper, we present a $C^3$ quartic B-spline approximation of circular arcs. The Hausdorff distance between the $C^3$ quartic B-spline curve and the circular arc is obtained in closed form. Using this error analysis, we show that the approximation order of our approximation method is six. For a given circular arc and error tolerance we find the $C^3$ quartic B-spline curve having the minimum number of control points within the tolerance. The algorithm yielding the $C^3$ quartic B-spline approximation of a circular arc is also presented.

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

#### References

1. C. de Boor and M. Sabin, High accuracy geometric Hermite interpolation, Comp. Aided Geom. Desi., 4 (1987), 269-278. https://doi.org/10.1016/0167-8396(87)90002-1
2. K. Morken, Best approximation of circle segments by quadratic Bezier curves, Curves and Surfaces, Academic Press, (1990), 387-396.
3. T. Dokken, M. Daehlen, T. Lyche, and K. Morken, Good approximation of circles by curvature-continuous Bezier curves, Comp. Aided Geom. Desi., 7 (1990), 33-41. https://doi.org/10.1016/0167-8396(90)90019-N
4. M. Goldapp, Approximation of circular arcs by cubic polynomials, Comp. Aided Geom. Desi., 8 (1991), 227-238. https://doi.org/10.1016/0167-8396(91)90007-X
5. T. Dokken, Controlling the shape of the error in cubic ellipse approximation, Curve and Surface Design, Saint-Malo, Nashboro Press (2002), 113-122.
6. M. Floater, High-order approximation of conic sections by quadratic splines, Comp. Aided Geom. Desi., 12(6) (1995) 617-637. https://doi.org/10.1016/0167-8396(94)00037-S
7. M. Floater. An O($h^{2n}$) Hermite approximation for conic sections, Comp. Aided Geom. Desi., 14 (1997) 135-151. https://doi.org/10.1016/S0167-8396(96)00025-8
8. T. Dokken. Aspects of intersection algorithms and approximation, PhD thesis, University of Oslo, (1997).
9. Y. J. Ahn and H. O. Kim. Approximation of circular arcs by Bezier curves, J. Comp. Appl. Math., 81 (1997) 145-163. https://doi.org/10.1016/S0377-0427(97)00037-X
10. L. Fang. Circular arc approximation by quintic polynomial curves, Comp. Aided Geom. Desi., 15 (1998), 843-861. https://doi.org/10.1016/S0167-8396(98)00019-3
11. S. H. Kim and Y. J. Ahn. Approximation of circular arcs by quartic Bezier curves, Comp. Aided Desi., 39(6) (2007), 490-493. https://doi.org/10.1016/j.cad.2007.01.004
12. S. Hur and T. Kim. The best $G^1$ cubic and $G^2$ quartic Bezier approximations of circular arcs, J. Comp. Appl. Math., 236 (2011), 1183-1192. https://doi.org/10.1016/j.cam.2011.08.002
13. Z. Liu, J. Tan, X. Chen, and L. Zhang. An approximation method to circular arcs, Appl. Math. Comp., 15 (2012), 1306-1311.
14. S. W. Kim and Y. J. Ahn. Circle approximation by quartic $G^2$ spline using alternation of error function, J. KSIAM, 17 (2013), 171-179.
15. B. Kovac and E. Zagar. Some new $G^1$ quartic parametric approximants of circular arcs, Appl. Math. Comp., 239 (2014), 254-264. https://doi.org/10.1016/j.amc.2014.04.100
16. I.-K. Lee, M.-S. Kim, and G. Elber. Planar curve offset based on circle approximation, Comp. Aided Desi., 28 (1996), 617-630. https://doi.org/10.1016/0010-4485(95)00078-X
17. G. Elber, I.-K. Lee, and M.-S. Kim. Comparing offset curve approximation methods, IEEE Comp. Grap. Appl., 17(3) (1997), 62-71. https://doi.org/10.1109/38.586019
18. Y. J. Ahn, Y. S. Kim, and Y. Shin. Approximation of circular arcs and offset curves by Bezier curves of high degree, J. Comp. Appl. Math., 167 (2004), 405-416. https://doi.org/10.1016/j.cam.2003.10.008
19. Y. J. Ahn, C. M. Hoffmann, and Y. S. Kim. Curvature-continuous offset approximation based on circle approximation using quadratic Bezier biarcs, Comp. Aided Desi., 43 (2011), 1011-1017. https://doi.org/10.1016/j.cad.2011.04.005
20. Y. J. Ahn and C. M. Hoffmann. Circle approximation using LN Bezier curves of even degree and its application, J. Math. Anal. Appl., 40 (2014), 257-266.
21. W. Yang and X. Ye. Approximation of circular arcs by $C^2$ cubic polynomial B-splines, 10th IEEE Intern. Conf. Comput.-Aided Des. Comput. Graph., (2007), 417-420.
22. B.-G. Lee, Y. Park, and J. Yoo. Application of Legendre-Bernstein basis transformations to degree elevation and degree reduction, Comp. Aided Geom. Desi., 19 (2002), 709-718. https://doi.org/10.1016/S0167-8396(02)00164-4
23. H. Sunwoo. Matrix representation for multi-degree reduction of Bezier curves, Comp. Aided Geom. Desi., 22 (2005), 261-273. https://doi.org/10.1016/j.cagd.2004.12.002
24. L. Piegl and W. Tiller. The NURBS book. Springer Science & Business Media, (2012).
25. G. Farin. Curves and Surfaces for CAGD. Morgan-Kaufmann, San Francisco, (2002).
26. W. Bohm. Inserting new knots into B-spline curves, Comp. Aided Desi., 4 (1980), 199-201.
27. E. Cohen, T. Lyche, and R. Riesenfeld. Discrete B-splines and subdivision techniques in Computer-Aided Geometric Design and Computer Graphics, Comp. Grap. Image Proc., 14 (1980), 87-111. https://doi.org/10.1016/0146-664X(80)90040-4
28. W. Bohm, G. Farin, and J. Kahmann. A survey of curve and surface methods in CAGD, Comp. Aided Geom. Desi., 1 (1984), 1-60. https://doi.org/10.1016/0167-8396(84)90003-7
29. K. Morken, M. Reomers, and C. Schulz. Computing intersections of planar spline curves using knot insertion, Comp. Aided Geom. Desi., 26 (2009), 351-366. https://doi.org/10.1016/j.cagd.2008.07.005