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 (f"1,f"2,f"3) for R^3, where f"1=r^'/|r^'| is the curve tangent, and the normal-plane vectors f"2,f"3 exhibit no instantaneous rotation about f"1. 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.