A new selection scheme for spatial Pythagorean hodograph quintic Hermite interpolants

For given C¹ Hermite data, there exists a two-parameter family of Pythagorean hodograph (PH) quintic curves which interpolate the data (two end-points and end-derivatives) as observed by Farouki et al. (2002b). As “good” candidate curves for a selection problem, we propose a special type of PH quintic interpolating curves called extremal interpolants and prove that the extremal interpolants preserve planarity, i.e., they are planar curves if the data are planar. Since there are only four distinct extremal interpolants, the selection problem, when only considering extremal interpolants as possible candidates, is reduced to picking one curve from finite candidates. Due to the preservation of planarity, extremal interpolants coincide with one of p_0,0(t),p_0,π(t),p_π,0(t),p_π,π(t) if the data are planar, where p_ϕ0,ϕ2 (t) denotes the parametrization proposed by Šír and Jüttler (2005). However, any of the four extremal interpolants is generically not identical to the interpolants for non-planar data, and empirical results suggest that being compared with the unique cubic interpolant, the best curve is one of extremal interpolants among all extremal interpolants and p_0,0(t),p_0,π(t),p_π,0(t),p_π,π(t).