Rotation-minimizing osculating frames

Abstract

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.

Multilevel Editing of B-Spline Curves with Robust Orientation of Details

“…RMFs are widely used in many applications in computer graphics, sweep surface modeling [19], generalized cylinder modeling [20,21] and motion design and control [22]. However, exact RMF (i.e., closed form) computation is non-trivial and thus a number of approximation techniques [5,[23][24][25] have been proposed. We employ RMFs as a reference coordinate system and represent the edited details in RMFs.…”

“…We choose rotation minimizing frames (RMFs) [5,[23][24][25] for the requirement above. However, finding the explicit form of exact RMFs on general B-spline curves is known to be a non-trivial problem and thus many approximation techniques have been proposed in practice.…”

Rational rotation-minimizing frames—Recent advances and open problems

“…As previously noted, the most commonly-studied RMF is the adapted frame, which is rotation-minimizing with respect to the tangent t. The Frenet frame is itself rotation-minimizing with respect to the principal normal p. A third possibility, a frame that is rotation-minimizing with respect to the binormal b, was studied in [20]. In the terminology of aerodynamics, these three frames define roll-free, pitch-free, and yaw-free motions.…”

“…It was shown in [20] that no DPH curves of degree less than 7 with rational osculating RMFs exist. The study of curves with such frames was extended in [42] to the rational case, based on the dual representation [30,41], and it was shown that rational curves with rational osculating RMFs of degree ≥ 6 exist, and the minimum degree of polynomial curves with this property is 7.…”

“…When the normal-plane vectors p, b are replaced by vectors u, v in that plane having no instantaneous rotation about t, we obtain a rotation-minimizing adapted frame (t, u, v) -also known as a Bishop frame [3]. A frame (f, g, b) retaining the binormal b but replacing the vectors t, p in the osculating plane (i.e., the plane that most nearly contains the curve at each point) by vectors f, g in that plane having no instantaneous about b, is a rotation-minimizing osculating frame [20]. Other types of rotation-minimizing frame, not based on modifying the Frenet frame, are also possible.…”