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.


53A17 Differential geometric aspects in kinematics
53A04 Curves in Euclidean and related spaces
68U07 Computer science aspects of computer-aided design
Full Text: DOI


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