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Rotation-minimizing osculating frames

  • Author(s): Farouki, RT
  • Giannelli, C
  • Sampoli, ML
  • Sestini, A
  • et al.
Abstract

An orthonormal frame ( , , ) is rotation-minimizing with respect to if its angular velocity ω satisfies ω× ≡0 - or, equivalently, the derivatives of and are both parallel to . The Frenet frame (t,p,b) along a space curve is rotation-minimizing with respect to the principal normal p, and in recent years adapted frames that are rotation-minimizing with respect to the tangent t have attracted much interest. This study is concerned with rotation-minimizing osculating frames (f,g,b) incorporating the binormal b, and osculating-plane vectors f, g that have no rotation about b. These frame vectors may be defined through a rotation of t, p by an angle equal to minus the integral of curvature with respect to arc length. In aeronautical terms, the rotation-minimizing osculating frame (RMOF) specifies yaw-free rigid-body motion along a curved path. For polynomial space curves possessing rational Frenet frames, the existence of rational RMOFs is investigated, and it is found that they must be of degree 7 at least. The RMOF is also employed to construct a novel type of ruled surface, with the property that its tangent planes coincide with the osculating planes of a given space curve, and its rulings exhibit the least possible rate of rotation consistent with this constraint. © 2013 Elsevier B.V. f1 f2 f3 fi fi fj fk fi

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