The flow-curvature of plane parametrized curves

Year 2023, Volume: 72 Issue: 2, 417 - 428, 23.06.2023

Mircea Crasmareanu

https://doi.org/10.31801/cfsuasmas.1165123

Cited By: 1

Abstract

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.

Keywords

Plane parametrized curve, angular vector field, flow-frame, flow-curvature

Thanks

I am grateful to Professor Dr. Vladimir Balan for several corrections to an initial version of this work. Also, I am extremely indebted to two anonymous referees for their remarks concerning my paper.

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