Research Article
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Darboux Frame of Pancake Curve in Euclidean 3-Space

Year 2024, Volume: 45 Issue: 3, 571 - 577, 30.09.2024

Emre Öztürk

https://doi.org/10.17776/csj.1444015
https://izlik.org/JA78UY64HK

Abstract

In this paper, we focus on a rare curve, “The pancake curve” which was named by Robert Ferréol. We investigate the Darboux frame of the curve, and we calculate the curvatures: geodesic curvature, normal curvature, and geodesic torsion. We discuss two cases, for instance, pancake curve as an intersection of Plücker’s conoid and unit cylinder, and the intersection of a hyperbolic paraboloid and unit cylinder. Moreover, we give the relations between the Frenet and Darboux curvatures. Finally, we give some examples related to the Frenet and Darboux frame and curvatures.

Keywords

Plücker's conoid Hyperbolic paraboloid Darboux frame Unit cylinder Pancake curve

References

  • [1] Encyclopédie des formes mathématiques remarquables. Available at: https://mathcurve.com/courbes3d.gb/crepe/crepe.shtml. Retrieved: December 20, 2023.
  • [2] O’Neill B. Elementary Differential Geometry Revised Second Edition. USA: Academic Press Elsevier, (2006) p. 248.
  • [3] Sabuncuoğlu, A. Diferensiyel Geometri. 5. Basım. Ankara: Nobel Yayınevi, (2014) p. 306.
  • [4] Kühnel, W. “Differential Geometry Curves-SurfacesManifolds”. USA: American Mathematical Society, (2006).
  • [5] Li, Y., Gür Mazlum, S., Şenyurt, S., The Darboux trihedrons of timelike surfaces in the Lorentzian 3-space, International Journal of Geometric Methods in Modern Physics, 20 (2) (2023).
  • [6] Ouarab, S., Smarandache Ruled Surfaces according to Darboux Frame in , Journal of Mathematics, Article ID 9912624, (1) (2021) 1-10.
  • [7] Aksan, B., Gür Mazlum, S., On Darboux Frames of Indicatrices of Spacelike Salkowski Curve with Spacelike Binormal in , Black Sea, Journal of Engineering and Science, 6 (4) (2023) 401-413.
  • [8] Ünlütürk, Y., Cimdiker, M., Ekici, C., Characteristic properties of the parallel ruled surfaces with Darboux frame in Euclidean 3- space., Communication in Mathematical Modeling and Applications, 1 (1) (2016) 26-43.
  • [9] Gür Mazlum, S., Geometric properties of timelike surfaces in Lorentz-Minkowski 3-space, Filomat, 37 (17) (2023) 5735– 5749.
  • [10] Altın, M., Kazan, A., Tubular Hypersurfaces According to Extended Darboux Frame Field of First Kind in , Turkish Journal of Science, 7 (2) (2022) 75-84.
  • [11] Gray, A., Abbena, E., Salamon, S. Differential Geometry of Curves and Surfaces with Mathematica. 3rd edition. USA: Chapman & Hall/CRC, (2006).

Year 2024, Volume: 45 Issue: 3, 571 - 577, 30.09.2024

Emre Öztürk

https://doi.org/10.17776/csj.1444015
https://izlik.org/JA78UY64HK

Abstract

References

  • [1] Encyclopédie des formes mathématiques remarquables. Available at: https://mathcurve.com/courbes3d.gb/crepe/crepe.shtml. Retrieved: December 20, 2023.
  • [2] O’Neill B. Elementary Differential Geometry Revised Second Edition. USA: Academic Press Elsevier, (2006) p. 248.
  • [3] Sabuncuoğlu, A. Diferensiyel Geometri. 5. Basım. Ankara: Nobel Yayınevi, (2014) p. 306.
  • [4] Kühnel, W. “Differential Geometry Curves-SurfacesManifolds”. USA: American Mathematical Society, (2006).
  • [5] Li, Y., Gür Mazlum, S., Şenyurt, S., The Darboux trihedrons of timelike surfaces in the Lorentzian 3-space, International Journal of Geometric Methods in Modern Physics, 20 (2) (2023).
  • [6] Ouarab, S., Smarandache Ruled Surfaces according to Darboux Frame in , Journal of Mathematics, Article ID 9912624, (1) (2021) 1-10.
  • [7] Aksan, B., Gür Mazlum, S., On Darboux Frames of Indicatrices of Spacelike Salkowski Curve with Spacelike Binormal in , Black Sea, Journal of Engineering and Science, 6 (4) (2023) 401-413.
  • [8] Ünlütürk, Y., Cimdiker, M., Ekici, C., Characteristic properties of the parallel ruled surfaces with Darboux frame in Euclidean 3- space., Communication in Mathematical Modeling and Applications, 1 (1) (2016) 26-43.
  • [9] Gür Mazlum, S., Geometric properties of timelike surfaces in Lorentz-Minkowski 3-space, Filomat, 37 (17) (2023) 5735– 5749.
  • [10] Altın, M., Kazan, A., Tubular Hypersurfaces According to Extended Darboux Frame Field of First Kind in , Turkish Journal of Science, 7 (2) (2022) 75-84.
  • [11] Gray, A., Abbena, E., Salamon, S. Differential Geometry of Curves and Surfaces with Mathematica. 3rd edition. USA: Chapman & Hall/CRC, (2006).
There are 11 citations in total.

Details

Primary Language English
Subjects Numerical Solution of Differential and Integral Equations
Journal Section Research Article
Authors

Emre Öztürk 0000-0001-6638-3233

Submission Date February 28, 2024
Acceptance Date September 12, 2024
Publication Date September 30, 2024
DOI https://doi.org/10.17776/csj.1444015
IZ https://izlik.org/JA78UY64HK
Published in Issue Year 2024 Volume: 45 Issue: 3

Cite

APA Öztürk, E. (2024). Darboux Frame of Pancake Curve in Euclidean 3-Space. Cumhuriyet Science Journal, 45(3), 571-577. https://doi.org/10.17776/csj.1444015

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