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 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.

解决了构造一条曲线的问题，该曲线可以与给定的初始/最终位置以及定向框架进行插值。更详细地，所得的内插曲线是7级的PH曲线，在可以与空间PH曲线相关联的自适应帧中，我们考虑了Euler-Rodrigues（ER）帧。而且$G^{\mathrm{1个}}$施加了框架之间的连续性，并研究了实现一般几何连续性的条件。它也表明我们的构造$G^{ķ}$ ER框架的连续性意味着 $G^{ķ+\mathrm{1个}}$相应的PH曲线的连续性，因此该方法对于定义样条运动可能很有用。利用单位球面上旋转矩阵和四元数之间的关系，通过相应四元数多项式上的条件来表示框架上的几何连续性条件。这导致了一个非线性方程组的研究，该方程组的可解性得到了研究，并得出了从光滑参数曲线及其总体适应框架中采样数据时解的渐近分析。结果表明，除了Frenet帧的情况外，PH插值的最佳逼近阶数为6，其中逼近阶数最大为4。