Rational rotation-minimizing frames on polynomial space curves of arbitrary degree

AbstractA rotation-minimizing adapted frame on a space curve r(t) is an orthonormal basis (f1,f2,f3) for R3 such that f1 is coincident with the curve tangent t=r′/|r′| at each point and the normal-plane vectors f2, f3 exhibit no instantaneous rotation about f1. Such frames are of interest in applications such as spatial path planning, computer animation, robotics, and swept surface constructions. Polynomial curves with rational rotation-minimizing frames (RRMF curves) are necessarily Pythagorean-hodograph (PH) curves–since only the PH curves possess rational unit tangents–and they may be characterized by the fact that a rational expression in the four polynomials u(t), v(t), p(t), q(t) that define the quaternion or Hopf map form of spatial PH curves can be written in terms of just two polynomials a(t), b(t). As a generalization of prior characterizations for RRMF cubics and quintics, a sufficient and necessary condition for a spatial PH curve of arbitrary degree to be an RRMF curve is derived herein for the generic case satisfying u2(t)+v2(t)+p2(t)+q2(t)=a2(t)+b2(t). This RRMF condition amounts to a divisibility property for certain polynomials defined in terms of u(t), v(t), p(t), q(t) and their derivatives