Nonexistence of rational rotation-minimizing frames on cubic curves

Quintic space curves with rational rotation-minimizing frames

“…A sufficient and necessary condition for the existence of a rational RMF on a spatial PH curve has been derived by Han, in terms of the quaternion representation: see Theorem 5 and related discussion in Han (2008). In terms of the Hopf map representation, this condition can be phrased as follows.…”

“…Using the quaternion representation of spatial PH curves, Han (2008) has shown that only degenerate (linear or planar) cubics have rational RMFs. Prior to analysing quintic RRMF curves, it is instructive to deduce this result from the Hopf map condition (8), using the form of w(t) defined in Lemma 1.…”

“…These facts have motivated recent interest in two special classes of PH curves -the double Pythagorean-hodograph (DPH) curves (Beltran and Monterde, 2007;Farouki et al, 2009aFarouki et al, , 2009b which have rational Frenet frames (and thus might also be called RFF curves), and the set of PH curves with rational RMFs (Choi and Han, 2002;Han, 2008). For brevity, we shall call the latter RRMF curves -bearing in mind that they are necessarily PH curves.…”

“…Choi and Han (2002) studied conditions under which the ERF of a PH curve coincides with an RMF, and showed that, for PH cubics, the ERF and Frenet frame are equivalent; for PH quintics, the ERF can be rotation-minimizing only in the degenerate case of planar curves; and the simplest non-planar PH curves for which the ERF can be an RMF are of degree 7. More recently, Han (2008) presented an algebraic criterion characterizing RRMFs of any (odd) degree, and showed that RRMF cubics are degenerate -i.e., they are either planar PH curves, or PH curves with non-primitive hodographs.…”

Corrigendum to “Rational rotation-minimizing frames on polynomial space curves of arbitrary degree” [J. Symbolic Comput. 45 (8) (2010) 844–856]

“…An adapted orthonormal frame (f 1 , f 2 , f 3 ) on r(t), where f 1 is the curve tangent, is a rotation-minimizing frame (RMF) if its angular velocity ω satisfies ω · f 1 ≡ 0 [1]. For a rational RMF, it is sufficient and necessary [7] that the condition…”