Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves
A rotation minimizing frame (RMF) of a space curve is an adapted frame whose angular velocity about the curve tangent vanishes. It is used in motion design and for the generation of swept surfaces. For general curves, RMFs cannot be computed in closed form. An important exception in this respect are Pythagorean hodograph curves (PH curves) whose transcendental RMFs are obtained by integrating a rational function. The article under review presents two new criteria for the existence of \textit{rational} rotation minimizing frames on quintic PH curves. One of the criteria pertains to the Hopf map description of PH curves, the other one to the description via quaternions. In contrast to the earlier criteria of \textit{R.~T.~Farouki, C.~Giannelli, C.~Manni,} and \textit{A.~Sestini} [Comput. Aided Geom. Design 26, No.~5 580--592 (2009; Zbl 1205.65079)] they are of low algebraic degree and exhibit the expected symmetry in the coefficients of the defining polynomials. The quaternion characterization is especially nice and concise.
scientific article; zbMATH DE number 5799383
scientific article