Nonexistence of rational rotation-minimizing frames on cubic curves

We prove there is no rational rotation-minimizing frame (RMF) along any non-planar regular cubic polynomial curve. Although several schemes have been proposed to generate rational frames that approximate RMF's, exact rational RMF's have been only observed on certain Pythagorean-hodograph curves of degree seven. Using the Euler–Rodrigues frames naturally defined on Pythagorean-hodograph curves, we characterize the condition for the given curve to allow a rational RMF and rigorously prove its nonexistence in the case of cubic curves.     

Rotation-minimizing osculating frames

An orthonormal frame (_f1,_f2,_f3)$({\mathbf{f}}_1,{\mathbf{f}}_2,{\mathbf{f}}_3)$ is rotation-minimizing   with respect to _fi${\mathbf{f}}_i$ if its angular velocity ω   satisfies ω⋅_fi≡0$\mathit{\omega }\cdot {\mathbf{f}}_i\equiv 0$ — or, equivalently, the derivatives of _fj${\mathbf{f}}_j$ and _fk${\mathbf{f}}_k$ are both parallel to _fi${\mathbf{f}}_i$. The Frenet frame (t,p,b)$(\mathbf{t},\mathbf{p},\mathbf{b})$ 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)$(\mathbf{f},\mathbf{g},\mathbf{b})$ 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.     

Dual representation of spatial rational Pythagorean-hodograph curves

In this paper, the dual representation of spatial parametric curves and its properties are studied. In particular, rational curves have a polynomial dual representation, which turns out to be both theoretically and computationally appropriate to tackle the main goal of the paper: spatial rational Pythagorean-hodograph curves (PH curves). The dual representation of a rational PH curve is generated here by a quaternion polynomial which defines the Euler–Rodrigues frame of a curve. Conditions which imply low degree dual form representation are considered in detail. In particular, a linear quaternion polynomial leads to cubic or reparameterized cubic polynomial PH curves. A quadratic quaternion polynomial generates a wider class of rational PH curves, and perhaps the most useful is the ten-parameter family of cubic rational PH curves, determined here in the closed form.