Geometric Hermite interpolation by monotone helical quintics

Monotone helical curves are polynomial helices whose unit tangent maintains a fixed sense of rotation about an axis. We investigate the interpolation of geometric Hermite data consisting of end points, tangents, and curvatures by monotone helical quintics. Based on the Hopf map model for spatial Pythagorean-hodograph curves, which subsume polynomial helices, we show that the geometric Hermite interpolation can be determined by solving a certain univariate polynomial equation of degree twelve.

Research highlights
► Spatial PH interpolants to Hermite data including curvature are found.
► For monotone helical quintics it amounts to solving an equation of degree 12.
► Empirical results suggest five distinct solutions at most.