On G2 planar Hermite interpolants with quintic PH curves

This paper addresses planar G2 Hermite interpolation by means of a single quintic Pythagorean–Hodograph (PH) curve. Using the complex representation of planar PH curves, the interpolation conditions are reduced from a nonlinear system in four scalar unknowns to a system in two real variables. This reduction reveals the intrinsic algebraic structure of the problem and shows that, for prescribed G2 data, the interpolant is fully determined, with no remaining free shape parameters, although multiple isolated solutions may exist. As a consequence, additional geometric constraints, such as prescribed arc length, cannot in general be imposed. The reduced system naturally splits into two branches corresponding to a sign choice; numerical evidence indicates that only the branch with positive sign yields the relevant solution in most cases. The singular configuration corresponding to parallel endpoint tangents is also analyzed, leading to a compatibility condition on the endpoint curvatures. The symmetric case reduces to a single cubic equation, that can be solved in a closed form, revealing geometric insights for the existence of multiple solutions. Several numerical examples illustrate the behavior, robustness, and solution structure of the proposed approach.