Rotation–minimizing osculating frames

An orthonormal frame (f1,f2,f3) is rotation–minimizing with respect to fi if its angular velocity ω satisfies ω⋅fi≡0 — or, equivalently, the derivatives of fj and fk are both parallel to fi. 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.