We present a scheme to find spatial quintic Pythagorean-hodograph (PH) curves that interpolate given first-order Hermite data and Frenet frames. The two free parameters that appear in general quintic PH interpolants are determined to adjust the orientation of binormal vectors. Using the unique cubic interpolant to the given set of Hermite data as a reference, we produce a quintic PH interpolant that shares not only the first-order Hermite data but also the Frenet frames at the endpoints with the cubic counterpart. This approach can be readily applied to a sequence of points to generate a quintic PH $C^1$ spline curve with Frenet-frame continuity. We also prove that for any nonlinear analytic curve, such PH Frenet interpolants uniquely exist if the Hermite data are extracted from sufficiently close points on the curve.

我们提出了一种方案来寻找空间五次勾股图 (PH) 曲线，该曲线对给定的一阶 Hermite 数据和 Frenet 框架进行插值。确定出现在一般五次 PH 插值中的两个自由参数来调整副法向量的方向。使用给定 Hermite 数据集的唯一三次插值作为参考，我们生成了一个五次 PH 插值，它不仅共享一阶 Hermite 数据，而且与三次对应的端点处的 Frenet 框架共享。这种方法可以很容易地应用于一系列点以生成五次 PH$C^1$具有 Frenet 框架连续性的样条曲线。我们还证明，对于任何非线性解析曲线，如果 Hermite 数据是从曲线上足够接近的点提取的，则这种 PH Frenet 插值是唯一存在的。