## Abstract

We present a new semi-topological quantity, called the absolute hodograph winding number, that measures how close the quintic PH spline interpolating a given sequence of points is to the cubic spline interpolating the same sequence. This quantity then naturally leads into a new criterion of determining the best quintic PH spline interpolant. This seems to work favorably compared with the elastic bending energy criterion developed by Farouki [Farouki, R.T., 1996. The elastic bending energy of Pythagorean-hodograph curves. Comput. Aided Geom. Design 13 (3), 227-241]. We also present a fast method that is a modification of the method of Albrecht, Farouki, Kuspa, Manni, and Sestini [Albrecht, G., Farouki, R.T., 1996. Construction of C^{2} Pythagorean-hodograph interpolating splines by the homotopy method. Adv. Comput. Math. 5 (4), 417-442; Farouki, R.T., Kuspa, B.K., Manni, C., Sestini, A., 2001. Efficient solution of the complex quadratic tridiagonal system for C^{2} PH quintic splines. Numer. Algorithms 27 (1), 35-60]. While the basic scheme of our approach is essentially the same as theirs, ours differs in that the underlying space in which the Newton-Raphson method is applied is the double covering space of the hodograph space, whereas theirs is the hodograph space itself. This difference, however, seems to produce more favorable results, when viewed from the above mentioned semi-topological criterion.

Original language | English |
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Pages (from-to) | 230-246 |

Number of pages | 17 |

Journal | Computer Aided Geometric Design |

Volume | 25 |

Issue number | 4-5 |

DOIs | |

State | Published - May 2008 |

### Bibliographical note

Funding Information:* Corresponding author at: Department of Mathematical Sciences, Seoul National University, 599 Gwanangno, Gwanak-gu, Seoul, 151-742, South Korea. E-mail address: [email protected] (S.-H. Kwon). 1 The first author also holds joint appointment in the Research Institute of Mathematics, Seoul National University. 2 This work was supported by the BK21 project of the Ministry of Education, Korea.