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The flow-curvature of plane parametrized curves

Yıl 2023, Cilt: 72 Sayı: 2, 417 - 428, 23.06.2023

Mircea Crasmareanu

https://doi.org/10.31801/cfsuasmas.1165123
Cited By: 1

Öz

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.

Anahtar Kelimeler

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

Teşekkür

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.

Kaynakça

  • 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
  • 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
  • Chou, K.-S., Zhu, X.-P., The Curve Shortening Problem, Boca Raton, FL: Chapman & Hall/CRC, 2001. Zbl 1061.53045
  • 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
  • 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
  • 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
  • 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
  • Jensen, G. R., Musso, E., Nicolodi, L., Surfaces in Classical Geometries. A Treatment by Moving Frames, Universitext, Springer, 2016. Zbl 1347.53001
  • 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
  • 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
  • Miron, R., The geometry of Myller configurations. Applications to theory of surfaces and nonholonomic manifolds, Bucharest: Editura Academiei Romˆane, 2010. Zbl 1206.53003
  • 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
  • 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
  • Younes, L., Shapes and Diffeomorphisms, 2nd Updated Edition, Applied Mathematical Sciences 171, Berlin, Springer, 2019. Zbl 1423.53002
  • 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

Yıl 2023, Cilt: 72 Sayı: 2, 417 - 428, 23.06.2023

Mircea Crasmareanu

https://doi.org/10.31801/cfsuasmas.1165123
Cited By: 1

Öz

Kaynakça

  • 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
  • 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
  • Chou, K.-S., Zhu, X.-P., The Curve Shortening Problem, Boca Raton, FL: Chapman & Hall/CRC, 2001. Zbl 1061.53045
  • 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
  • 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
  • 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
  • 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
  • Jensen, G. R., Musso, E., Nicolodi, L., Surfaces in Classical Geometries. A Treatment by Moving Frames, Universitext, Springer, 2016. Zbl 1347.53001
  • 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
  • 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
  • Miron, R., The geometry of Myller configurations. Applications to theory of surfaces and nonholonomic manifolds, Bucharest: Editura Academiei Romˆane, 2010. Zbl 1206.53003
  • 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
  • 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
  • Younes, L., Shapes and Diffeomorphisms, 2nd Updated Edition, Applied Mathematical Sciences 171, Berlin, Springer, 2019. Zbl 1423.53002
  • 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
Toplam 15 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Research Article
Yazarlar

Mircea Crasmareanu 0000-0002-5230-2751

Yayımlanma Tarihi 23 Haziran 2023
Gönderilme Tarihi 22 Ağustos 2022
Kabul Tarihi 30 Aralık 2022
Yayımlandığı Sayı Yıl 2023 Cilt: 72 Sayı: 2

Kaynak Göster

APA Crasmareanu, M. (2023). The flow-curvature of plane parametrized curves. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 72(2), 417-428. https://doi.org/10.31801/cfsuasmas.1165123
AMA Crasmareanu M. The flow-curvature of plane parametrized curves. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Haziran 2023;72(2):417-428. doi:10.31801/cfsuasmas.1165123
Chicago Crasmareanu, Mircea. “The Flow-Curvature of Plane Parametrized Curves”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72, sy. 2 (Haziran 2023): 417-28. https://doi.org/10.31801/cfsuasmas.1165123.
EndNote Crasmareanu M (01 Haziran 2023) The flow-curvature of plane parametrized curves. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72 2 417–428.
IEEE M. Crasmareanu, “The flow-curvature of plane parametrized curves”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., c. 72, sy. 2, ss. 417–428, 2023, doi: 10.31801/cfsuasmas.1165123.
ISNAD Crasmareanu, Mircea. “The Flow-Curvature of Plane Parametrized Curves”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72/2 (Haziran 2023), 417-428. https://doi.org/10.31801/cfsuasmas.1165123.
JAMA Crasmareanu M. The flow-curvature of plane parametrized curves. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72:417–428.
MLA Crasmareanu, Mircea. “The Flow-Curvature of Plane Parametrized Curves”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, c. 72, sy. 2, 2023, ss. 417-28, doi:10.31801/cfsuasmas.1165123.
Vancouver Crasmareanu M. The flow-curvature of plane parametrized curves. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72(2):417-28.

Cited By

The Flow-geodesic Curvature and the Flow-evolute of Hyperbolic Plane Curves
International Electronic Journal of Geometry
https://doi.org/10.36890/iejg.1229215

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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