Motion interpolation with Euler–Rodrigues frames on extremal Pythagorean-hodograph curves

Abstract

We introduce a novel subset of spatial Pythagorean-hodograph (PH) quintic curves characterized by a unique extremal configuration in the quaternion space. For each generic set of $C^1$ Hermite motion data, there exist exactly four interpolants of these extremal PH curves, each of them matching the specified frames by its Euler–Rodrigues frame (ERF). The four extremal interpolants can be distinguished by the signs that are extracted from their generating quaternion polynomials, and are invariant under orthogonal transformations. Remarkably, not only are the extremal interpolants planar when applied to planar motion data, but they also demonstrate superior geometric properties in comparison to other PH quintic motion interpolants, particularly in terms of their bending energy and the angular variation of their ERF.