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.
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[3] Sabuncuoğlu, A. Diferensiyel Geometri. 5. Basım. Ankara:
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timelike surfaces in the Lorentzian 3-space, International
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(2023).
[6] Ouarab, S., Smarandache Ruled Surfaces according to
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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
[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
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