Partition of the space of planar quintic Pythagorean-hodograph curves

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 $r\left(t\right)$ may be constructed from a complex quadratic pre–image polynomial $w\left(t\right)$ by integration of $r^{\prime }\left(t\right)=w^2\left(t\right)$, and it thus incorporates (modulo translations) six real parameters — the real and imaginary parts of the coefficients of $w\left(t\right)$. 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 $w\left(t\right)$ passes through the origin of the complex plane. Correspondingly, extreme curvatures and tight loops on $r\left(t\right)$ are incurred by a close proximity of $w\left(t\right)$ 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.