Journal of the Korean Society for Industrial and Applied Mathematics

  • 제15권2호
  • /
  • Pages.123-135
  • /
  • 2011
  • /
  • 1226-9433(pISSN)
  • /
  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
권호 표지
DOI QR코드

DOI QR Code

ARC-LENGTH ESTIMATIONS FOR QUADRATIC RATIONAL B$\acute{e}$zier CURVES COINCIDING WITH ARC-LENGTH OF SPECIAL SHAPES

  • 투고 : 2011.01.18
  • 심사 : 2011.04.05
  • 발행 : 2011.06.25
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we present arc-length estimations for quadratic rational B$\acute{e}$zier curves using the length of polygon and distance between both end points. Our arc-length estimations coincide with the arc-length of the quadratic rational B$\acute{e}$zier curve exactly when the weight ${\omega}$ is 0, 1 and ${\infty}$. We show that for all ${\omega}$ > 0 our estimations are strictly increasing with respect to ${\omega}$. Moreover, we find the parameter ${\mu}^*$ which makes our estimation coincide with the arc-length of the quadratic rational B$\acute{e}$zier curve when it is a circular arc too. We also show that ${\mu}^*$ has a special limit, which is used for optimal estimation. We present some numerical examples, and the numerical results illustrates that the estimation with the limit value of ${\mu}^*$ is an optimal estimation.

키워드

과제정보

연구 과제 주관 기관 : National Research Foundation of Korea(NRF)

참고문헌

  1. Y. J. Ahn. Conic approximation of planar curves. Comp. Aided Desi., 33:867-872, 2001. https://doi.org/10.1016/S0010-4485(00)00110-X
  2. Y. J. Ahn. Helix approximation with conic and qadratic Bezier curves. Comp. Aided Geom. Desi., 22:551-565, 2005. https://doi.org/10.1016/j.cagd.2005.02.003
  3. Y. J. Ahn. Approximation of conic sections by curvature continuous quartic Bezier curves. Comp. Math. Appl., 60:1986-1993, 2010. https://doi.org/10.1016/j.camwa.2010.07.032
  4. Y. J. Ahn and H. O. Kim. Curvatures of the quadratic rational Bezier curves. Comp. Math. Appl., 36:71-83, 1998.
  5. 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., 167:405-416, 2004. https://doi.org/10.1016/j.cam.2003.10.008
  6. E. Boebert. Computing the arc length of cubic Bezier curves, 1993. http://steve.hollasch.net/cgindex/curves/cbezarclen.html, accessed 2010.
  7. M. A. Earle. A vector solution for navigation on a great ellipse. J. Navi., 53:473-481, 2000. https://doi.org/10.1017/S0373463300008948
  8. G. Farin. Curves and Surfaces for Computer Aided Geometric Design. Morgan-Kaufmann, San Francisco, 2002.
  9. M. Floater. High-order approximation of conic sections by quadratic splines. Comp. Aided Geom. Desi., 12(6):617-637, 1995. https://doi.org/10.1016/0167-8396(94)00037-S
  10. M. Floater. An O($h^{2n}$) Hermite approximation for conic sectoins. Comp. Aided Geom. Desi., 14:135-151, 1997. https://doi.org/10.1016/S0167-8396(96)00025-8
  11. M. S. Floater. Arc length estimation and the convergence of polynomial curve interpolation. Journal BIT Numerical Mathematics, 45:679-694, 2005. https://doi.org/10.1007/s10543-005-0031-2
  12. M. S. Floater and A. F. Rasmussen. Point-based methods for estimating the length of a parametric curve. Journal of Computational and Applied Mathematics, 196:512-522, 2006. https://doi.org/10.1016/j.cam.2005.10.001
  13. M. S. Floater, A. F. Rasmussen, and U. Reif. Extrapolation methods for approximating arc length and surface area. Numer. Alg., 44:235-248, 2007. https://doi.org/10.1007/s11075-007-9095-1
  14. J. Gravesen. Adaptive subdivision and the length and energy of Bezier curves. Computational Geometry, 8:13-31, 1997. https://doi.org/10.1016/0925-7721(95)00054-2
  15. J. Gravesen. The arc-length and energy of rational Bezier curves. Mat-report 1997/26, Department of Mathematics, Technical University of Denmark, 1997.
  16. E. T. Lee. The rational Bezier representation for conics. In geometric modeling : Algorithms and new trends, pages 3-19. SIAM, Academic Press, 1987.
  17. J. Malczak. Quadratic Bezier curve length. Undated web page; http://segfaultlabs.com/docs/quadratic-bezier-curve-length, accessed 2010.
  18. A. Pallikaris and G. Latsas. New algorithm for great elliptic sailing (GES). J. Navi., 62:497-507, 2009.
  19. T. Pavlidis. Curve fitting with conic splines. ACM Trans. Graph., 2:1-31, 1983. https://doi.org/10.1145/357314.357315
  20. A. F. Rasmussen and M. S. Floater. A point-based method for estimating surface area. In Proceedings of the SIAM Conference on Geometric Design, 2005.
  21. J. A. Roulier. Specifying the arc length of Bezier curves. Comp. Aided Geom. Desi., 10:25-56, 1993. https://doi.org/10.1016/0167-8396(93)90050-D
  22. J. A. Roulier and B. Piper. Prescribing the length of parametric curves. Comp. Aided Geom. Desi., 13:3-22, 1996. https://doi.org/10.1016/0167-8396(95)00004-6
  23. J. A. Roulier and B. Piper. Prescribing the length of rational Bezier curves. Comp. Aided Geom. Desi., 13:23-42, 1996. https://doi.org/10.1016/0167-8396(95)00005-4
  24. M. Schechter. Which way is jerusalem? Navigating on a spheroid. The College Mathematics Journal, 38:96-105, 2007.
  25. W.-K. Tseng and H.-S. Lee. Navigation on a great ellipse. J. Mari. Scie. Tech., 18:369-375, 2010.
  26. H.Wang, J. Kearney, and K. Atkinson. Arc-length parametrized spline curves for real-time simulation. In Tom Lyche, Marie-Laurence Mazure, and Larry L. Schumaker, editors, Curve and Surface Design, pages 387-396, 2002.