C¹ and C² interpolation of orientation data along spatial Pythagorean-hodograph curves using rational adapted spline frames

The problem of constructing a rational adapted frame (f₁(ξ),f₂(ξ),f₃(ξ)) that interpolates a discrete set of orientations at specified nodes along a given spatial Pythagorean-hodograph (PH) curve r(ξ) is addressed. PH curves are the only polynomial space curves that admit rational adapted frames, and the Euler–Rodrigues frame (ERF) is a fundamental instance of such frames. The ERF can be transformed into other rational adapted frame by applying a rationally-parametrized rotation to the normal-plane vectors. When orientation and angular velocity data at curve end points are given, a Hermite frame interpolant can be constructed using a complex quadratic polynomial that parametrizes the normal-plane rotation, by an extension of the method recently introduced to construct a rational minimal twist frame (MTF). To construct a rational adapted spline frame, a representation that resolves potential ambiguities in the orientation data is introduced. Based on this representation, a C¹ rational adapted spline frame is constructed through local Hermite interpolation on each segment, using angular velocities estimated from a cubic spline that interpolates the frame phase angle relative to the ERF. To construct a C² rational adapted spline frame, which ensures continuity of the angular acceleration, a complex-valued cubic spline is used to directly interpolate the complex exponentials of the phase angles at the nodal points.