Abstract
The interpolation of first-order Hermite data by spatial Pythagorean-hodograph curves that exhibit closure under arbitrary 3-dimensional rotations is addressed. The hodographs of such curves correspond to certain combinations of four polynomials, given by Dietz et al. [4], that admit compact descriptions in terms of quaternions – an instance of the “PH representation map” proposed by Choi et al. [2]. The lowest-order PH curves that interpolate arbitrary first-order spatial Hermite data are quintics. It is shown that, with PH quintics, the quaternion representation yields a reduction of the Hermite interpolation problem to three “simple” quadratic equations in three quaternion unknowns. This system admits a closed-form solution, expressing all PH quintic interpolants to given spatial Hermite data as a two-parameter family. An integral shape measure is invoked to fix these two free parameters.
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Farouki, R.T., al-Kandari, M. & Sakkalis, T. Hermite Interpolation by Rotation-Invariant Spatial Pythagorean-Hodograph Curves. Advances in Computational Mathematics 17, 369–383 (2002). https://doi.org/10.1023/A:1016280811626
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DOI: https://doi.org/10.1023/A:1016280811626