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Darboux Frame of Pancake Curve in Euclidean 3-Space

Yıl 2024, , 571 - 577, 30.09.2024
https://doi.org/10.17776/csj.1444015

Öz

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.

Kaynakça

  • [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).
Yıl 2024, , 571 - 577, 30.09.2024
https://doi.org/10.17776/csj.1444015

Öz

Kaynakça

  • [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).
Toplam 11 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Diferansiyel ve İntegral Denklemlerin Sayısal Çözümü
Bölüm Natural Sciences
Yazarlar

Emre Öztürk 0000-0001-6638-3233

Yayımlanma Tarihi 30 Eylül 2024
Gönderilme Tarihi 28 Şubat 2024
Kabul Tarihi 12 Eylül 2024
Yayımlandığı Sayı Yıl 2024

Kaynak Göster

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