Equivalence of distinct characterizations for rational rotation-minimizing frames on quintic space curves

A rotation-minimizing frame on a space curve r(t) is an orthonormal basis (f₁,f₂,f₃) for R³, where f₁=r^′/|r^′| is the curve tangent, and the normal-plane vectors f₂,f₃ exhibit no instantaneous rotation about f₁. Such frames are useful in spatial path planning, swept surface design, computer animation, robotics, and related applications. The simplest curves that have rational rotation-minimizing frames (RRMF curves) comprise a subset of the quintic Pythagorean-hodograph (PH) curves, and two quite different characterizations of them are currently known: (a) through constraints on the PH curve coefficients; and (b) through a certain polynomial divisibility condition. Although (a) is better suited to the formulation of constructive algorithms, (b) has the advantage of remaining valid for curves of any degree. A proof of the equivalence of these two different criteria is presented for PH quintics, together with comments on the generalization to higher-order curves. Although (a) and (b) are both sufficient and necessary criteria for a PH quintic to be an RRMF curve, the (non-obvious) proof presented here helps to clarify the subtle relationships between them.