The quintics are the lowestâorder planar Pythagoreanâhodograph (PH) curves suitable for freeâform design, since they can exhibit inflections. A quintic PH curve r(t) may be constructed from a complex quadratic preâimage polynomial w(t) by integration of râČ(t)=w2(t), and it thus incorporates (modulo translations) six real parameters â the real and imaginary parts of the coefficients of w(t). Within this 6âdimensional space of planar PH quintics, a 5âdimensional hypersurface separates the inflectional and nonâinflectional curves. Points of the hypersurface identify exceptional curves that possess a tangentâcontinuous point of infinite curvature, corresponding to the fact that the parabolic locus specified by the quadratic preâimage polynomial w(t) passes through the origin of the complex plane. Correspondingly, extreme curvatures and tight loops on r(t) are incurred by a close proximity of 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 firstâorder Hermite interpolation problem.