Mapping rational rotation-minimizing frames from polynomial curves on to rational curves

Abstract Given a polynomial space curve r ( ξ ) that has a rational rotation–minimizing frame (an RRMF curve), a methodology is developed to construct families of rational space curves r ˜ ( ξ ) with the same rotation–minimizing frame as r ( ξ ) at corresponding points. The construction employs the dual form of a rational space curve, interpreted as the edge of regression of the envelope of a family of osculating planes, having normals in the direction u ( ξ ) = r ′ ( ξ ) × r ″ ( ξ ) and distances from the origin specified in terms of a rational function f ( ξ ) as f ( ξ ) / ‖ u ( ξ ) ‖ . An explicit characterization of the rational curves r ˜ ( ξ ) generated by a given RRMF curve r ( ξ ) in this manner is developed, and the problem of matching initial and final points and frames is shown to impose only linear conditions on the coefficients of f ( ξ ) , obviating the non–linear equations (and existence questions) that arise in addressing this problem with the RRMF curve r ( ξ ) . Criteria for identifying low–degree instances of the curves r ˜ ( ξ ) are identified, by a cancellation of factors common to their numerators and denominators, and the methodology is illustrated by a number of computed examples.

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