Rotation–minimizing conformal 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 , fk are both parallel to fi. The Frenet frame (t,p,b) 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 conformal frames, that are rotation–minimizing with respect to the binormal b along a space curve. Such a frame (f ,g,b) incorporates osculating–plane vectors f ,g that have no rotation about b, and may be defined through a rotation of t,p by an amount equal to minus the integral of curvature with respect to arc length. In aeronautical terms, a rotation–minimizing conformal frame (RMCF) specifies “yaw–free” rigid–body motion on a curved path. The existence of rational RMCFs on polynomial space curves with rational Frenet frames is investigated, and it is shown that they must be degree 7 at least. The RMCF is also employed to construct a novel type of ruled surface, characterized by tangent planes that coincide with the osculating planes along a given space curve, and rulings that exhibit the least possible rate of rotation consistent with this constraint.