Absolute hodograph winding number and planar PH quintic splines

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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 C2 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 C2 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.

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Cited by (11)

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    The polynomial speed functions furnish the PH curves with many nice properties such as exact arc length evaluation, rational offset curves, rational unit tangent vectors. To utilize these properties, many algorithms for PH curve construction on various conditions have been developed (Choi et al., 2008; Choi and Kwon, 2008; Farouki et al., 2002, 2008; Farouki and Neff, 1995; Huard et al., 2014; Jüttler, 2001; Moon et al., 2001; Šír and Jüttler, 2007). Especially, the PH curve construction problems under the arc length constraint have been addressed recently (Farouki, 2016, 2019).

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    Because of the square map in the PH condition, the Hermite interpolation problems are expressed as systems of quadratic equations, so the solution might not be unique. The Hermite interpolation problems for planar PH curves usually have multiple solutions, and several selection schemes of the best solution have been reported (Choi et al., 2008; Choi and Kwon, 2008; Farouki and Neff, 1995; Moon et al., 2001). On the other hand, the Hermite interpolation problems for spatial PH curves have infinitely many solutions (Farouki et al., 2002, 2008; Kwon, 2010), which also require the selection schemes for the optimal solution.

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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.

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