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

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A rotation-minimizing adapted frame on a space curve r ( t ) is an orthonormal basis ( f 1 , f 2 , f 3 ) for R 3 such that f 1 is coincident with the curve tangent t = r ′ / | r ′ | at each point and the normal-plane vectors f 2 , f 3 exhibit no instantaneous rotation about f 1 . 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 u 2 ( t ) + v 2 ( t ) + p 2 ( t ) + q 2 ( t ) = a 2 ( t ) + b 2 ( 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.