References

  1. Balgetir H., Bektaş M., Ergüt M. Bertrand curves for non-null curves in 3-dimensional Lorentzian space. Hadronic Journal 2004, 27 (2), 229–236.
  2. Balgetir H., Bektaş M., Inoguchi J. Null Bertrand curves in Minkowski 3-space and their characterizations. Note Mat. 2004, 23 (1), 7–13. doi:10.1285/i15900932v23n1p7
  3. Bertrand J.M. Mémoire sur la théorie des courbes á double courbure. J. Math. Pures Appl. (9) 1850, 15, 332–350.
  4. Bonnor W.B. Null curves in a Minkowski space-time. Tensor (N.S.) 1969, 20, 229–242.
  5. Camcı Ç., Uçum A., İlarslan K. A New Approach to Bertrand Curves in Euclidean 3-Space. J. Geom. 2020, 111, article number 49. doi:10.1007/s00022-020-00560-5
  6. Choi J.H., Kang T.H., Kim Y.H. Bertrand curves in 3-dimensional space forms. Appl. Math. Comput. 2012, 219 (3), 1040–1046. doi:10.1016/j.amc.2012.07.008
  7. Ekmekci N., İlarslan K. On Bertrand curves and their characterization. Differ. Geom. Dyn. Syst. 2001, 3 (2), 17–24.
  8. Güler F., Bayram E., Kasap E. Offset surface pencil with a common asymptotic curve. Int. J. Geom. Methods Mod. Phys. 2018, 15 (11), 1850195. doi:10.1142/S0219887818501955
  9. Honda S., Takahashi M. Bertrand and Mannheim curves of framed curves in the 3-dimensional Euclidean space. Turkish J. Math. 2020, 44 (3), 883–899. doi:10.3906/mat-1905-63
  10. Inoguchi J.I., Lee S. Null curves in Minkowski 3-space. Int. Electron. J. Geom. 2008, 1 (2), 40–83.
  11. Jin D.H. Null Bertrand curves in a Lorentz manifold. J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 2008, 15 (3), 209–215.
  12. Jie H., Pei D. Singular Special Curves in 3-Space Forms. Mathematics 2020, 8 (5), 846. doi:10.3390/math8050846
  13. Kuhnel W. Differential geometry: curves-surfaces-manifolds. In: Student mathematical library, 16. American Mathematical Society. Braunschweig, Wiesbaden, 1999.
  14. Lopez R. Differential geometry of curves and surfaces in Lorentz-Minkowski space. Int. Electron. J. Geom. 2014, 7 (1), 44–107. doi:10.48550/arXiv.0810.3351
  15. Lucas P., Ortega-Yagües J.A. Bertrand curves in the three-dimensional sphere. J. Geom. Phys. 2012, 62 (9),–1914. doi:10.1016/j.geomphys.2012.04.007
  16. Lucas P., Ortega-Yagües J.A. Bertrand Curves in Non-Flat 3-Dimensional (Riemannian and Lorentzian) Space forms. Bull. Korean Math. Soc. 2013, 50 (4), 1109–1126. doi:10.4134/BKMS.2013.50.4.1109
  17. Lucas P., Ortega-Yagües J.A. A variational characterization and geometric integration for Bertrand curves. J. Math. Phys. 2013, 54 (4), 043508. doi:10.1063/1.4800767
  18. Matsuda H., Yorozu S. Notes on Bertrand curves. Yokohama Math. J. 2003, 50 (1-2), 41–58.
  19. Papaioannou S.G., Kiritsis D. An application of bertrand curves and surfaces to CAD/CAM. Comput. Aided Geom. Design 1985, 17 (8), 348–352. doi:10.1016/0010-4485(85)90025-9
  20. Pears L.R. Bertrand curves in Riemannian space. J. Lond. Math. Soc. (2) 1935, 1-10 (3), 180–183. doi:10.1112/jlms/s1-10.2.180
  21. Ravani B., Ku T.S. Bertrand offsets of ruled and developable surfaces. Comput.-Aided Des. 1991, 23 (2), 145–152. doi:10.1016/0010-4485(91)90005-H
  22. Razzaboni A. Delle superficie nelle quali un sistema di geodetiche sono curve del Bertrand. Bologna Mem. 1903, 10, 539–548.
  23. Schief W.K. On the integrability of Bertrand curves and Razzaboni surfaces. J. Geom. Phys. 2003, 45 (1-2), 130–150. doi:10.1016/S0393-0440(02)00130-4
  24. Sun J., Pei D. Null Cartan Bertrand curves of AW(k)-type in Minkowski 4-space. Phys. Lett. A 2012, 376 (33), 2230–2233. doi:10.1016/j.physleta.2012.05.044
  25. Tanriöver N. Some properties of Bertrand curves in Lorentzian \(n\)-space \(L^{n}\). Int. J. Geom. Methods Mod. Phys. 2016, 13 (5), 1650064. doi:10.1142/S021988781650064X
  26. Tuna Aksoy A. Null quaternionic Bertrand curves in Minkowski Space \(\mathbb{R}_{1}^{4}\). Acta Phys. Polon. A 2016, 130 (1), 256–258. doi:10.12693/APhysPolA.130.256
  27. Uçum A., Keçilioğlu O., İlarslan K. Generalized Pseudo Null Bertrand curves in Semi-Euclidean 4-Space with index 2. Rend. Circ. Mat. Palermo (2) 2016, 65 (3), 459–472. doi:10.1007/s12215-016-0246-x
  28. Uçum A., Keçilioğlu O., İlarslan K. Generalized Bertrand curves with timelike \((1,3)\)-normal plane in Minkowski space-time. Kuwait J. Sci. 2015, 42 (3), 10–27.
  29. Uçum A., İlarslan K. On timelike Bertrand curves in Minkowski 3-space. Honam Math. J. 2016, 38 (3), 467–477. doi:10.5831/HMJ.2016.38.3.467
  30. Uçum A., İlarslan K., Sakaki M. On \((1,3)\)-Cartan null Bertrand curves in semi-Euclidean 4-space with index 2. J. Geom. 2016,107, 579–591. doi:10.1007/s00022-015-0290-2
  31. Walrave J. Curves and Surfaces in Minkowski Space. K.U. LEUVEN, Faculteit Der Wetenschappen, Ph. D. Thesis, 1995.
  32. Saint Venant B. Mémoire sur les lignes courbes non planes. J. Éc. Polytech. Math. 1845, 18, 1–76.
  33. Xu C., Cao X., Zhu P. Bertrand curves and Razzaboni surfaces in Minkowski 3-space. Bull. Korean Math. Soc. 2015, 52 (2), 377–394.
  34. Yilmaz M., Bektas M. General properties of Bertrand curves in Riemann-Otsuki space. Nonlinear Anal. 2008, 69 (10), 3225–3231. doi:10.1016/j.na.2007.10.003