Rotation-minimizing osculating frames

摘要

An orthonormal frame (f_1,f_2,f_3) is rotation-minimizing with respect to f_i if its angular velocity ω satisfies ω · f_i = 0 - or, equivalently, the derivatives of f_j and f_k are both parallel to f_i. 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.