A well–known feature of the Pythagorean–hodograph (PH) curves is the multiplicity of solutions arising from their construction through the interpolation of Hermite data. In general, there are four distinct planar quintic PH curves that match first–order Hermite data, and a two–parameter family of spatial quintic PH curves compatible with such data. Under certain special circumstances, however, the number of distinct solutions is reduced. The present study characterizes these singular cases, and analyzes the properties of the resulting quintic PH curves. Specifically, in the planar case it is shown that there may be only three (but not less) distinct Hermite interpolants, of which one is a “double” solution. In the spatial case, a constant difference between the two free parameters reduces the dimension of the solution set from two to one, resulting in a family of quintic PH space curves of different shape but identical arc lengths. The values of the free parameters that result in formal specialization of the (quaternion) spatial problem to the (complex) planar problem are also identified, demonstrating that the planar PH quintics, including their degenerate cases, are subsumed as a proper subset of the spatial PH quintics.

勾股勾线图（PH）曲线的一个众所周知的特征是通过对Hermite数据进行插值而构造出的多个解。通常，有四个与一阶Hermite数据匹配的截然不同的平面五次PH曲线，以及一个与此类数据兼容的两参数系列空间五次PH曲线。但是，在某些特殊情况下，减少了不同解决方案的数量。本研究表征了这些奇异情况，并分析了所得五次PH曲线的性质。具体而言，在平面情况下，表明可能只有三个（但不少于其他）不同的Hermite插值，其中一个是“双重”解决方案。在空间情况下，两个自由参数之间的恒定差将解集的维数从2减小为1，产生了一系列形状各异但弧长相同的五次PH空间曲线。还确定了导致（四元数）空间问题正式特殊化为（复杂）平面问题的自由参数的值，表明平面PH五元组（包括退化的情况）被归为空间的适当子集PH五分法。