commun. fac. sci. univ. ank. ser. a1 math. stat. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 1303-5991 2618-6470 Ankara University 10.31801/cfsuasmas.1165123 Mathematical Sciences Matematik The flow-curvature of plane parametrized curves https://orcid.org/0000-0002-5230-2751 Crasmareanu Mircea University Al. I. Cuza, Iasi 06 23 2023 72 2 417 428 08 22 2022 12 30 2022 Copyright © 1948, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 1948 Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics

We introduce and study a new frame and a new curvature function for a fixed parametrization of a plane curve. This new frame is called flow since it involves the time-dependent rotation of the usual Frenet flow; the angle of rotation is exactly the current parameter. The flow-curvature is calculated for several examples obtaining the logarithmic spirals (and the circle as limit case) and the Grim Reaper as flat-flow curves. A main result is that the scaling with$\frac{1}{\sqrt{2}}$ of both Frenet and flow-frame belong to the same fiber of the Hopf bundle. Moreover, the flow-Fermi-Walker derivative is defined and studied.

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