Browse > Article

http://dx.doi.org/10.14477/jhm.2014.27.5.329
###

A Brief History of Study on the Bound for Derivative of Rational Curves in CAGD |

Park, Yunbeom (Dept. of Math. Edu., Seowon Univ.) |

Publication Information

Abstract

CAGD(Computer Aided Geometric Design) is a branch of applied mathematics concerned with algorithms for the design of smooth curves and surfaces and for their efficient mathematical representation. The representation is used for the computation of the curves and surfaces, as well as geometrical quantities of importance such as curvatures, intersection curves between two surfaces and offset surfaces. The curves, B-spline, rational curves and NURBS(Non-Uniform Rational B-Spline) are basically and widely used in CAGD. The definitions and properties of these curves are presented in this paper. And a brief history of study on the bound for derivative of rational curves in CAGD is also presented.

Keywords

CAGD; Rational curves; Derivative of rational curves;

Citations & Related Records

- Reference

1 | C. Deng, Improved bounds on the magnitude of the derivative of rational Bezier curves, Applied Mathematics and Computation 218 (2011), 204-206. DOI |

2 | H. E. Bez, N. Bez, On derivative bounds for the rational quadratic Bezier paths, Computer Aided Geometric Design 30 (2013), 254-261. DOI |

3 | H. E. Bez, N. Bez, New minimal bounds for the derivatives of rational Bezier paths and rational rectangular Bezier surfaces, Applied Mathematics and Computation 225 (2013), 475-479. DOI |

4 | C. de Boor, A Practical Guide to Splines, Springer-Verlag, 2001. |

5 | C. Deng, Y. Li, A new bound on the magnitude of the derivation of rational Bezier curve, Computer Aided Geometric Design 30 (2013), 175-180. DOI |

6 | E. Dimas, D. Briassoulis, 3D geometric modelling based on NURBS: a review, Advances in Engineering Software 30 (1999), 741-751. DOI |

7 | G. Farin, Algorithms for rational Bezier curves, Computer Aided Design 15 (1983), 73-77. DOI |

8 | G. Farin, Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, Academic Press, 1996. |

9 | G. Farin, J. Hoschek, M.-S. Kim, Handbook of Computer Aided Geometric Design, Elsevier Science, 2002. |

10 | Y. Huang, H. Su, The bound on derivatives of rational Bezier curves, Computer Aided Geometric Design 23 (2006), 698-702. DOI |

11 | M. Floater, Evaluation and properties of the derivative of a NURBS curve, Mathematical Methods in CAGD, T. Lyche Ed., 1992. |

12 | M. Floater, Derivative of rational Bezier curves, Computer Aided Geometric Design 9 (1992), 161-174. DOI |

13 | T. Hermann, On the derivatives of second and third degree rational Bezier curves, Computer Aided Geometric Design 16 (1999), 157-173. DOI |

14 | Y. Li, C. Deng, W. Jin, N. Zhao, On the bounds of the derivative of rational Bezier curves, Applied Mathematics and Computation 219 (2013), 10425-10433. DOI |

15 | H. Lin, On the derivative formula of a rational Bezier curve at a corner, Applied Mathematics and Computation 210 (2009), 197-201. DOI |

16 | V. P. Nguyena et al, An introduction to Isogeometric Analysis with Matlab implementation: FEM and XFEM formulations, preprint, 2012. |

17 | L. Piegl, W. Ma, W. Tiller, An alternative method of curve interpolation, Visual Computer 21 (2005), 104-117. DOI |

18 | L. Piegl, W. Tiller, The NURBS Book, Springer-Verlag, 1995. |

19 | J. Prochazkova, D. Prochazka, Implementation of NURBS curve derivatives in engineering practice, WSCG'2007 Posters Proceedings, V. Skala Ed., 2007. |

20 | I. Selimovic, On NURBS algorithms using tangent cones, Computer Aided Geometric Design 26 (2009), 772-778. DOI |

21 | Z. Wu, F. Lin, H. S. Seah, K. Y. Chan, Evaluation of difference bounds for computing rational Bezier curves and surfaces, Computers & Graphics 28 (2004), 551-558. DOI |

22 | R. J. Zhang, W. Ma, Some improvements on the derivative bounds of rational Bezier curves and surfaces, Computer Aided Geometric Design 23 (2006), 563-572. DOI |