Approximation of circular arcs using quartic bezier curves with barycentric coordinates satisfying G2 data

Azhar Ahmad

Abstract

This paper proposes four methods to approximate circular arcs using quartic Bezier curves. Barycentric coordinates of two/three combination of control points are used to obtain an optimal approximation. Interior control points of quartic Bezier curves are found by satisfying G 2 data from given circular arcs. The maximum errors between circular arcs and approximated curves are calculated using Hausdorff distance approach. Four different approaches are discussed which varies in terms of accuracy, computation cost and approximation order.

References

[1] A. Ahmad, R. Masri, N.M. Idrus and J.M. Ali, New Approach to Approximate Circular Arc by Quartic Bezier Curve, Open Journal of Applied Sciences, 2 (2012), 132-737. https://doi.org/10.4236/ojapps.2012.24b032 [2] Y.J. Ahn and H.O. Kim, Approximation of circular arcs by Bezier curves, Journal of Computational and Applied Mathematics, 81 (1997), 145-163 https://doi.org/10.1016/s0377-0427(97)00037-x [3] C. de Boor, K. Hollig and M. Sabin, High accuracy geometric Hermite interpolation, Computer Aided Geometric Design, 4 (1997), 269–278. https://doi.org/10.1016/0167-8396(87)90002-1 [4] T. Dokken, M. Dehlen, T. Lyche and K. Morken, Good approximation of circles by curvature-continuous Bezier curves, Computer Aided Geometric Design, 7 (1990), 33-41. https://doi.org/10.1016/0167-8396(90)90019-n [5] L. Fang, Circular arc approximation by quintic polynomial curves, Computer Aided Geometric Design, 15 (1998), 843-861. https://doi.org/10.1016/s0167-8396(98)00019-3 [6] G. Farin, Curves and Surfaces for Computer Aided Geometric Design; A practical guide, 4th Edition, Academic Press, New York, 1997. [7] M. Goldapp, Approximation of circular arcs by cubic polynomial, Computer Aided Geometric Design, 8 (1991), 227-238. https://doi.org/10.1016/0167-8396(91)90007-x [8] S.H. Kim and Y.J. Ahn, An approximation of circular arcs by quartic Bezier curves, Computer Aided Geometric Design, 39 (2007), 490-493 https://doi.org/10.1016/j.cad.2007.01.004