G 1 interpolation by rational cubic PH curves in R3

In this paper the G1 interpolation of two data points and two tangent directions with spatial cubic rational PH curves is considered. It is shown that interpolants exist for any true spatial data configuration. The equations that determine the interpolants are derived by combining a closed form representation of a ten parametric family of rational PH cubics given in Kozak et al. (2014), and the Gram matrix approach. The existence of a solution is proven by using a homotopy analysis, and numerical method to compute solutions is proposed. In contrast to polynomial PH cubics for which the range of G1 data admitting the existence of interpolants is limited, a switch to rationals provides an interpolation scheme with no restrictions.