document: Hermite interpolation by rotation-invariant spatial Pythagorean-hodograph curves 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. -LRB- 1993 -RRB- , that admit compact descriptions in terms of quaternions - an instance of the `` PH representation map '' proposed by Choi et al. -LRB- 2002 -RRB- . 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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