Rotation-minimizing osculating frames

• The rotation-minimizing osculating frame (RMOF) on space curves is introduced.
• RMOFs define “yaw-free” rigid-body motion along a space curve.
• The RMOF can be used to construct a novel type of ruled surface.
• Polynomial space curves with rational RMOFs must be of degree 7 at least.

An orthonormal frame (f1,f2,f3)(f1,f2,f3) is rotation-minimizing   with respect to fifi if its angular velocity ω   satisfies ω⋅fi≡0ω⋅fi≡0 — or, equivalently, the derivatives of fjfj and fkfk are both parallel to fifi. The Frenet frame (t,p,b)(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)(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.