The quintics are the lowest–order planar Pythagorean–hodograph (PH) curves suitable for free–form design, since they can exhibit inflections. A quintic PH curve $\mathbf{r}(t)$ may be constructed from a complex quadratic pre–image polynomial $\mathbf{w}(t)$ by integration of ${\mathbf{r}}^{\prime }(t)={\mathbf{w}}^2(t)$, and it thus incorporates (modulo translations) six real parameters — the real and imaginary parts of the coefficients of $\mathbf{w}(t)$. Within this 6–dimensional space of planar PH quintics, a 5–dimensional hypersurface separates the inflectional and non–inflectional curves. Points of the hypersurface identify exceptional curves that possess a tangent–continuous point of infinite curvature, corresponding to the fact that the parabolic locus specified by the quadratic pre–image polynomial $\mathbf{w}(t)$ passes through the origin of the complex plane. Correspondingly, extreme curvatures and tight loops on $\mathbf{r}(t)$ are incurred by a close proximity of $\mathbf{w}(t)$ to the origin of the complex plane. These observations provide useful insight into the disparate shapes of the four distinct PH quintic solutions to the first–order Hermite interpolation problem.

五次曲线是适用于自由形式设计的最低阶平面勾股图 (PH) 曲线，因为它们可以表现出拐点。五次PH曲线$\mathbf{r}（t）$可以由复杂的二次原像多项式构造$\mathbf{w}（t）$通过整合${\mathbf{r}}^{\prime }（t）={\mathbf{w}}^2（t）$，因此它包含（模平移）六个实数参数 - 系数的实部和虚部$\mathbf{w}（t）$。在平面 PH 五次方程的 6 维空间内，5 维超曲面将拐点曲线和非拐点曲线分开。超曲面的点识别具有无限曲率的切连续点的异常曲线，对应于由二次原像多项式指定的抛物线轨迹的事实$\mathbf{w}（t）$经过复平面的原点。相应地，极端曲率和紧密环$\mathbf{r}（t）$是由于非常接近而引起的$\mathbf{w}（t）$到复平面的原点。这些观察结果为了解一阶 Hermite 插值问题的四种不同 PH 五次解的不同形状提供了有用的见解。