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On new types of rational rotation-minimizing frame space curves. (English) Zbl 1332.53016

A rotation-minimizing frame on a space curve is an adapted frame whose normal plane vectors exhibit no instantaneous rotation about the curve tangent. In general, it is not rational, not even for Pythagorean hodograph curves (PH curves) which have a polynomial parametric equation with a rational arc-length function. PH curves with rational rotation-minimizing frames are characterized by a certain identity between rational functions. An inequality between the degrees of the involved functions was wrongly claimed in [R. T. Farouki and T. Sakkalis, ibid. 45, No. 8, 844–856 (2010; Zbl 1204.53009)] and in a later corrigendum re-formulated as a conjecture [ibid. 58, 99–102 (2013; Zbl 1283.53010)]. In this paper, the authors demonstrate that this conjecture is actually false. They construct a wealth of counterexamples and, as a consequence, obtain new types of PH curves with rational rotation minimizing frame.

MSC:

53A17 Differential geometric aspects in kinematics
53A04 Curves in Euclidean and related spaces
68U07 Computer science aspects of computer-aided design
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[1] Farouki, R. T., Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable (2008), Springer: Springer Berlin · Zbl 1144.51004
[2] Farouki, R. T.; Giannelli, C.; Manni, C.; Sestini, A., Quintic space curves with rational rotation-minimizing frames, Comput. Aided Geom. Des., 26, 580-592 (2009) · Zbl 1205.65079
[3] Farouki, R. T., Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves, Adv. Comput. Math., 33, 331-348 (2010) · Zbl 1202.53004
[4] Farouki, R. T.; Sakkalis, T., Rational rotation-minimizing frames on polynomial space curves of arbitrary degree, J. Symb. Comput., 45, 844-856 (2010) · Zbl 1204.53009
[5] Farouki, R. T.; Sakkalis, T., A complete classification of quintic space curves with rational rotation-minimizing frames, J. Symb. Comput., 47, 214-226 (2012) · Zbl 1232.53004
[6] Farouki, R. T.; Sakkalis, T., Corrigendum to “Rational rotation-minimizing frames on polynomial space curves of arbitrary degree” [J. Symbolic Comput. 45 (8) (2010) 844-856], J. Symb. Comput., 59, 99-102 (2013) · Zbl 1283.53010
[7] Gentili, G.; Struppa, D. C., On the multiplicity of zeros of polynomials with quaternionic coefficients, Milan J. Math., 76, 15-25 (2008) · Zbl 1194.30054
[8] Gordon, B.; Motzkin, T. S., On the zeros of polynomials over division rings, Trans. Am. Math. Soc., 116, 218-226 (1965) · Zbl 0141.03002
[9] Han, C. Y., Nonexistence of rational rotation-minimizing frames on cubic curves, Comput. Aided Geom. Des., 25, 298-304 (2008) · Zbl 1172.65327
[10] Klok, F., Two moving coordinate frames for sweeping along a 3D trajectory, Comput. Aided Geom. Des., 3, 217-229 (1986) · Zbl 0631.65145
[11] Topuridze, N., On roots of quaternion polynomials, J. Math. Sci., 160, 6, 843-855 (2009) · Zbl 1243.16022
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