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Farouki, R.T., Giannelli, C., Manni, C. and Sestini, A. (2009) Quintic Space Curves with Rational Rotation-Minimizing Frames. Computer Aided Geometric Design, 26, 580-592.
https://doi.org/10.1016/j.cagd.2009.01.005

has been cited by the following article:

• TITLE: Global Optimization Method to Comprise Rotation-Minimizing Euler-Rodrigues Frames of Pythagorean-Hodograph Curve

  AUTHORS: Fengfu Peng, Jinrong Pang, Yuting Pan

  KEYWORDS: Rotation-Minimizing, Euler-Rodrigues Frames, Pythagorean-Hodograph Curves

  JOURNAL NAME: Journal of Applied Mathematics and Physics, Vol.11 No.5, May 10, 2023

  ABSTRACT: The rotation-minimizing frame is the main research object for a spatial curve. Although the mathematical description is not complicated, it is not easy to directly make an exact minimizing-rotation frame for the Euler-Rodrigues frame. The condition for the non-normalized Euler-Rodrigues frame of the Pythagorean-Hodograph curve to become the rotation-minimizing frame is given in this article, which is an ordinary differential equation with rational form, the analytical solution that does not always exist. To avoid calculating the solution of ordinary differential equations, a global optimization algorithm for the conditions is proposed, that has a weight function in the objective function. The quintic Pythagorean-Hodograph curve is analyzed concretely with the method, and its objective function and constraint conditions of optimization are clarified. The example is analyzed by using this method with different weight functions and contrasting that approach with its exact value.