title: Hermite interpolation by rotation-invariant spatial Pythagorean-hodograph curves
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. (1993), that admit compact descriptions in terms of quaternions - an instance of the "PH representation map" proposed by Choi et al. (2002). 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
keyphrases: interpolation first-order Hermite data polynomials quaternions closed-form solution integral shape measure spatial Pythagorean hodograph curves