The flow-curvature of plane parametrized curves

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.

References

1. Bates, L. M., Melko, O. M., On curves of constant torsion I, J. Geom., 104 (2) (2013), 213–227. https://doi.org/10.1007/s00022-013-0166-2
2. Bishop, R. L., There is more than one way to frame a curve, Am. Math. Mon., 82 (1975), 246–251. https://doi.org/10.2307/2319846
3. Chou, K.-S., Zhu, X.-P., The Curve Shortening Problem, Boca Raton, FL: Chapman & Hall/CRC, 2001. Zbl 1061.53045
4. Crasmareanu, M., The flow-curvature of spacelike parametrized curves in the Lorentz plane, Proceedings of the International Geometry Center, 15 (2) (2022), 100–108. https://doi.org/10.15673/tmgc.v15i2.2281
5. Crasmareanu, M., The flow-geodesic curvature and the flow-evolute of hyperbolic plane curves, Int. Electron. J. Geom., 16 (2023), no. 1, 225—231. https://doi.org/10.36890/iejg.1229215
6. Crasmareanu, M., Frigioiu, C. Unitary vector fields are Fermi-Walker transported along Rytov-Legendre curves, Int. J. Geom. Methods Mod. Phys., 12 (10) (2015), , Article ID 1550111. https://doi.org/10.1142/S021988781550111X
7. Gozdz, S., Curvature type functions for plane curves, An. S¸tiint¸. Univ. Al. I. Cuza Ia¸si Mat., 39 (3) (1993), 295–303. Zbl 0851.53001
8. Jensen, G. R., Musso, E., Nicolodi, L., Surfaces in Classical Geometries. A Treatment by Moving Frames, Universitext, Springer, 2016. Zbl 1347.53001

9. Mazur, B., Perturbations, deformations, and variations (and “near-misses”) in geometry, physics, and number theory, Bull. Am. Math. Soc., 41 (3) (2004), 307–336. https://doi.org/10.1090/S0273-0979-04-01024-9
10. Miron, R., Une generalisation de la notion de courbure de parallelisme, Gaz. Mat. Fiz., Bucure¸sti, Ser. A 10 (63) (1958), 705–708. Zbl 0087.36101
11. Miron, R., The geometry of Myller configurations. Applications to theory of surfaces and nonholonomic manifolds, Bucharest: Editura Academiei Romˆane, 2010. Zbl 1206.53003
12. Özen, K. E., Tosun, M., A new moving frame for trajectories with non-vanishing angular momentum, J. of Mathematical Sciences and Modelling, 4 (1) (2021), 7–18. https://doi.org/10.33187/jmsm.869698
13. Soliman, M. A., Nassar, H.A.-A., Hussien, R. A., Youssef, T., Evolutions of the ruled surfaces via the evolution of their directrix using quasi frame along a space curve, J. of Applied Mathematics and Physics, 6 (2018), 1748–1756. https://doi.org/10.4236/jamp.2018.68149
14. Younes, L., Shapes and Diffeomorphisms, 2nd Updated Edition, Applied Mathematical Sciences 171, Berlin, Springer, 2019. Zbl 1423.53002
15. Zhu, X.-P., Lectures on Mean Curvature Flows, AMS/IP Studies in Advanced Mathematics vol. 32, Providence, RI: American Mathematical Society, 2002. Zbl 1197.53087