Absolute hodograph winding number and planar PH quintic splines

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.