Journal of the Korean Society for Industrial and Applied Mathematics

  • 제28권2호
  • /
  • Pages.59-70
  • /
  • 2024
  • /
  • 1226-9433(pISSN)
  • /
  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
권호 표지
DOI QR코드

DOI QR Code

APPROXIMATION OF HELIX BY G2 CUBIC POLYNOMIAL CURVES

  • YOUNG JOON AHN (DEPARTMENT OF MATHEMATICS EDUCATION, CHOSUN UNIVERSITY)
  • 투고 : 2023.11.26
  • 심사 : 2024.03.24
  • 발행 : 2024.06.25
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper we present the approximation method of the circular helix by G2 cubic polynomial curves. The approximants are G1 Hermite interpolation of the circular helix and their approximation order is four. We obtain numerical examples to illustrate the geometric continuity and the approximation order of the approximants. The method presented in this paper can be extended to approximating the elliptical helix. Using the property of affine transformation invariance we show that the approximant has G2 continuity and the approximation order four. The numerical examples are also presented to illustrate our assertions.

키워드

과제정보

This study was supported by research funds from Chosun University, 2023. The author is very grateful to two anonymous reviewers for their valuable comments and constructive suggestions.

참고문헌

  1. T. Dokken, M. Daehlen, T. Lyche, and K. Morken. Good approximation of circles by curvature-continuous Bezier curves. Comput. Aided Geom. Design, 7:33-41, 1990. 
  2. M. Goldapp. Approximation of circular arcs by cubic polynomials. Comput. Aided Geom. Design, 8:227-238, 1991. 
  3. M. Knez and E. Zagar. Interpolation of circular arcs by parametric polynomials of maximal geometric smoothness. Comput. Aided Geom. Design, 63:66-77, 2018. 
  4. E. Zagar. Arc length preserving G2 Hermite interpolation of circular arcs. J. Comput. Appl. Math., 424:115008, 2023. 
  5. H. M. Yoon and Y. J. Ahn. Circular arc approximation by hexic polynomial curves. Comput. Appl. Math., 42:256, 2023. 
  6. K. Hollig and J. Koch. Geometric Hermite interpolation. Comput. Aided Geom. Design, 12(6):567-580, 1995. 
  7. K. Hollig and J. Koch. Geometric Hermite interpolation with maximal order and smoothness. Comput. Aided Geom. Design, 13(8):681-695, 1996. 
  8. L. Xu and J. Shi. Geometric Hermite interpolation for space curves. Comput. Aided Geom. Design, 18:817-829, 2001. 
  9. M. Krajnc. Geometric Hermite interpolation by cubic G1 splines. Nonlinear Analysis: Theory, Methods & Applications, 70:2614-2626, 2009. 
  10. G. Jaklic and E. Zagar. Curvature variation minimizing cubic Hermite interpolants. Appl. Math. Comput., 218:3918-3924, 2011. 
  11. C. de Boor, K. Hollig, and M. Sabin. High accuracy geometric Hermite interpolation. Comput. Aided Geom. Design, 4:269-278, 1987. 
  12. M. Floater. An O(h2n) Hermite approximation for conic sections. Comput. Aided Geom. Design, 14:135-151, 1997. 
  13. G. Jaklic, J. Kozak, M. Krajnc, V. Vitrih, and E. Zagar. High-order parametric polynomial approximation of conic sections. Constr. Approx., 38:1-18, 2013. 
  14. G. Jaklic. Uniform approximation of a circle by a parametric polynomial curve. Comput. Aided Geom. Design, 41:36-46, 2016. 
  15. G. Jaklic and J. Kozak. On parametric polynomial circle approximation. Numerical Algorithms, 77:433-450, 2018. 
  16. M. Floater. High-order approximation of conic sections by quadratic splines. Comput. Aided Geom. Design, 12(6):617-637, 1995. 
  17. Y. J. Ahn and H. O. Kim. Approximation of circular arcs by Bezier curves. J. Comput. Appl. Math., 81:145-163, 1997. 
  18. L. Fang. Circular arc approximation by quintic polynomial curves. Comput. Aided Geom. Design, 15:843-861, 1998. 
  19. L. Fang. G3 approximation of conic sections by quintic polynomial. Comput. Aided Geom. Design, 16:755-766, 1999. 
  20. S. Mick and O. Roschel. Interpolation of helical patches by kinematic rational Bezier patches. Computers and Graphics, 14:275-280, 1990. 
  21. I. Juhasz. Approximating the helix with rational cubic Bezier curves. Computer-Aided Design, 27:587-593, 1995. 
  22. G. Seemann. Approximating a helix segment with a rational Bezier curve. Comput. Aided Geom. Design, 14:475-490, 1997. 
  23. X. Yang. High accuracy approximation of helices by quintic curves. Comput. Aided Geom. Design, 20:303- 317, 2003. 
  24. Y. J. Ahn. Helix approximation with conic and qadratic Bezier curves. Comput. Aided Geom. Design, 22:551-565, 2005. 
  25. L. Lu. On polynomial approximation of circular arcs and helices. Comput. Math. Appl., 63:1192-1196, 2012. 
  26. S. Kilicoglu. On approximation of helix by 3rd, 5th and 7th order Bezier curves in E3. Thermal Science, 26:525-538, 2022. 
  27. G. Jaklic, B. Juttler, M. Krajnc, V. Vitrih, and E. Zagar. Hermite interpolation by rational Gk motions of low degree. J. Comput. Appl. Math., 240:20-30, 2013. 
  28. M. Knez and M. L. Sampoli. Geometric interpolation of ER frames with G2 Pythagorean-hodograph curves of degree 7. Comput. Aided Geom. Design, 88:102001, 2021. 
  29. T. N. Goodman. Properties of β-splines. Journal of Approximation Theory, 44:132-153, 1985. 
  30. T. D. DeRose and B. A. Barsky. Geometric continuity, shape parameters, and geometric constructions for Catmull-Rom splines. ACM Transactions on Graphics (TOG), 7:1-41, 1988. 
  31. A. Vavpetic and E. Zagar. A general framework for the optimal approximation of circular arcs by parametric polynomial curves. J. Comput. Appl. Math., 345:146-158, 2019. 
  32. R. Herzog and P. Blanc. Optimal G2 Hermite interpolation for 3D curves. Comput.-Aided Design, 117:102752, 2019. 
  33. Y. A. H. Louie, R. L. Somorjai, and A. Klug. Differential geometry of proteins: helical approximations. Journal of Molecular Biology, 168(1):143-162, 1983. 
  34. D. Fuchs, I. Izmestiev, M. Raffaelli, G. Szewieczek, and S. Tabachnikov. Differential geometry of space curves: forgotten chapters. The Mathematical Intelligencer, 120:1-13, 2023.