Geometric interpolation of ER frames with G2 Pythagorean-hodograph curves of degree 7

The problem of constructing a curve that interpolates given initial/final positions along with orientational frames is addressed. In more detail, the resulting interpolating curve is a PH curve of degree 7 and among the adaptive frames that can be associated to a spatial extcolor{blue}{PH} curve, we consider the Euler-Rodrigues (ER) frame. Moreover G^1 continuity between frames is imposed and conditions for achieving general geometric continuity are investigated. It is also shown that our construction of G^k continuity of ER frames implies G^{k+1} continuity of the corresponding PH curves, and hence this approach can be useful to define spline motions. Exploiting the relation between rotational matrices and quaternions on the unit sphere, geometric continuity conditions on the frames are expressed through conditions on the corresponding quaternion polynomials. This leads to a nonlinear system of equations whose solvability is investigated, and asymptotic analysis of the solutions in the case of data sampled from a smooth parametric curve and its general adapted frame is derived. It is shown that there exist PH interpolants with optimal approximation order 6, except for the case of the Frenet frame, where the approximation order is at most 4. Several numerical examples are presented, which confirm the theoretical results.