C-1 and C-2 interpolation of orientation data along spatial Pythagorean-hodograph curves using rational adapted spline frames

Abstract

The problem of constructing a rational adapted frame (f(1) (xi), f(2) (xi), f(3) (xi)) that interpolates a discrete set of orientations at specified nodes along a given spatial Pythagoreanhodograph (PH) curve r(xi) 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 normalplane 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 C-1 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 C-2 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. (C) 2018 Elsevier B.V. All rights reserved.