Year 2024,
Volume: 7 Issue: 1, 61 - 77, 30.06.2024
Orhan Karacan
,
Bengü Bayram
References
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- A. T. Ali, “Special Smarandache curve in the Euclidean space”, International J. Math. Combin., vol. 2, pp. 30-36, 2010.
- S. Şenyurt, S. Sivas, “An application of Smarandache curve”, Ordu Univ. J. Sci. Tech., vol. 3, no. 1, pp. 46-60, 2013.
- V. Bulut, A. Caliskan, “Spherical images of special Smarandache curves in E^3”, International J. Math. Combin., vol. 3, pp. 43-54, 2015.
H. S. Abdel-Aziz, M. Khaalifa Saad, “Computation of Smarandache curves according to Darboux frame in Minkowski 3-space”, Journal of the Egyptian Mathematical Society, vol. 25, pp. 382-390, 2017.
- M. Elzawy, “Smarandache curves in Euclidean 4-space E^4 ”, Journal of the Egyptian Mathematical Society, vol. 25, pp. 268-271, 2017.
- S. Şenyurt, B. Öztürk, “Smarandache curves of Anti-Salkowski curve according to Frenet frame”, Proceedings of The International Conference on Mathematical Studies and Application, Karamanoğlu Mehmetbey University, Karaman, Turkey, 4-6 October 2018.
- S. Lal, “Smarandache curves in Euclidean space of parallel transport frame”, Journal of Energing Technologies and Innovative Research, vol. 6, no. 7, 2019.
- M. Altın, A. Kazan and H. B. Karadağ, “Hypersurface families with Smarandache curves in Galilean 4-space”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 70, no. 2, pp. 744-761, 2021.
- Ş. Alıç, B. Yılmaz, “Smarandache curves according to alternative frame in E^3”, Journal of Universal Mathematics, vol. 4, no. 2, pp. 140-156, 2021.Guo, Q., Chen, Z., Liu, P., Li, Y., Hu, J., Gao, Y., Li, X. (2021). Influence of basalt fiber on mode I and II fracture properties of asphalt mixture at medium and low temperatures. Theor. Appl. Fract. Mech., V.112, 102884.
- S. Kaya Nurkan, İ. A. Güven, “A New approach for Smarandache curve”, Turk. J. Math. Comput. Sci., vol. 14, no. 1, pp. 155-165, 2022.
- S. Şenyurt, D. Canlı, E. Çan and S. G. Mazlum, “Some special Smarandache Ruled surfaces by Frenet frame in E^3-2”, Honam Mathematical Journal, vol. 44, no. 4, pp. 594-617, 2022.
- S. Şenyurt, K. H. Ayvacı and D. Canlı, “Smarandache curves according to Flc-frame in Euclidean 3-space”, Fundamentals of Contemporary Mathematical Sciences, vol. 4, no. 1, pp. 16-30, 2023.
- G. Tzitzeica, “Sur certaines courbes gauches”, Annales scientifiques de l'École Normale Supérieure, vol. 28, no. 3, pp. 9-32, 1911.
- A. Bobe, W. G. Boskoff and M. G. Ciuca, “Tzitzeica type centro-affine invariants in Minkowski space”, An. St. Univ. Ovidius Constanta, vol. 20, no. 2, pp. 27-34, 2012.
- M. Crasmareanu, “Cylindrical Tzitzeica curves implies forced harmonic oscillators”, Balkan Journal of Geometry and Its Applications, vol. 7, no. 1, pp. 37-42, 2002.
- M. K. Karacan, B. Bükcü, “On the hyperbolic cylindrical Tzitzeica curves in Mikowski 3-space”, BAÜ FBE Dergisi, vol. 10, no. 1, pp. 46-51, 2009.
- M. K. Karacan, B. Bükcü, “On the elliptic cylindrical Tzitzeica curves in Mikowski 3-space”, Scientia Magna, vol. 5, no. 3, pp. 44-48, 2009.
- N. Bila, “Symmetry reductions for the Tzitzeica curve equation”, Math and Computer Science Working Papers, vol. 16, 2012.
- B. Bayram, E. Tunç, K. Arslan, G. Öztürk, “On Tzitzeica curves in Euclidean 3-space E^3”, Facta Universitatis, Series Mathematics and Informatics, vol. 33, no. 3, pp. 409-416, 2018.
- B. Bayram, E. Tunç, “On tzitzeica surfaces in euclidean 3-space E^3”, Journal of Balikesir University Institute of Science and Technology, vol. 23, no. 1, pp. 277-290, 2021.
- B. Bayram, E. Tunç, “A New characterization of Tzitzeica curves in Euclidean 4-space”, Fundamentals of Contemporary Mathematical Sciences, vol. 4, no. 2, pp. 77-86, 2023
- B. Bayram, E. Tunç, “A Note on Tzitzeica curve in Euclidean 4-spaces E^4”, International Theory, Research and Reviews in Science and Mathematics, Chapter. 10, Serüven Yayınevi, pp. 141-166, 2023
- G. Öztürk, S. Büyükkütük, İ. Kişi, “A Characterization of Curves in Galilean 4-Space”, Bulletin of Irannian Mathematical Society, vol. 43, no. 3, pp. 771-780, 2017.
- H. Gluck, “Higher curvatures of curves in Euclidean space”, The American Mathematical Monthly, vol. 73, no. 7, pp. 243-245, 1966.
- A. Gray, “Modern differential geometry of curves and surfaces”, CRC Press, 1993.
- F. Klein, S. Lie, “Uber diejenigen ebenenen kurven welche durch ein geschlossenes system von einfach unendlich vielen vartauschbaren linearen Transformationen in sich übergehen”, The American Mathematical Monthly, vol. 4, pp. 50-84, 1871.
- G. Öztürk, K. Arslan, H. Hacısalihoğlu, “A characterization of ccr-curves in R^n”, Proceedings of the Estonian Academy of Sciences, vol. 57, pp. 217-224, 2008.
- B. Y. Chen, “When does the position vector of a space curve always lies in its rectifying plane?”, The American Mathematical Monthly, vol. 110, pp. 147-152, 2003.
- K. İlarslan, E. Nesovic, “Some characterizations of osculating curves in the Euclidean spaces”, Demonstratio Mathematica, vol. 16, no. 4, pp. 931-939, 2008.
- O. Karacan, Tzitzeica Smarandache Curves in Three Dimensional Euclidean Space, M.S. thesis, Instutıte of Science, Balıkesir University, Balıkesir, Türkiye, 2024
Tzitzeica Smarandache Curves in Euclidean 3- Space
Year 2024,
Volume: 7 Issue: 1, 61 - 77, 30.06.2024
Orhan Karacan
,
Bengü Bayram
Abstract
The aim of this study is to examine the relations between Tzitzeica curves and Smarandache curves in Euclidean space. In addition, the necessary and sufficient conditions for Smarandache curves to be Tzitzeica curves in 3-dimensional Euclidean space are investigated and examples are given.
Thanks
Destekleri için Prof. Dr. Bengü BAYRAM hocama teşekkürlerimi sunuyorum.
References
- M. Turgut, S. Yilmaz, “Smarandache curves in Minkowski space-time”, International J. Math. Combin., vol. 3, pp. 51-55, 2008.
- A. T. Ali, “Special Smarandache curve in the Euclidean space”, International J. Math. Combin., vol. 2, pp. 30-36, 2010.
- S. Şenyurt, S. Sivas, “An application of Smarandache curve”, Ordu Univ. J. Sci. Tech., vol. 3, no. 1, pp. 46-60, 2013.
- V. Bulut, A. Caliskan, “Spherical images of special Smarandache curves in E^3”, International J. Math. Combin., vol. 3, pp. 43-54, 2015.
H. S. Abdel-Aziz, M. Khaalifa Saad, “Computation of Smarandache curves according to Darboux frame in Minkowski 3-space”, Journal of the Egyptian Mathematical Society, vol. 25, pp. 382-390, 2017.
- M. Elzawy, “Smarandache curves in Euclidean 4-space E^4 ”, Journal of the Egyptian Mathematical Society, vol. 25, pp. 268-271, 2017.
- S. Şenyurt, B. Öztürk, “Smarandache curves of Anti-Salkowski curve according to Frenet frame”, Proceedings of The International Conference on Mathematical Studies and Application, Karamanoğlu Mehmetbey University, Karaman, Turkey, 4-6 October 2018.
- S. Lal, “Smarandache curves in Euclidean space of parallel transport frame”, Journal of Energing Technologies and Innovative Research, vol. 6, no. 7, 2019.
- M. Altın, A. Kazan and H. B. Karadağ, “Hypersurface families with Smarandache curves in Galilean 4-space”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 70, no. 2, pp. 744-761, 2021.
- Ş. Alıç, B. Yılmaz, “Smarandache curves according to alternative frame in E^3”, Journal of Universal Mathematics, vol. 4, no. 2, pp. 140-156, 2021.Guo, Q., Chen, Z., Liu, P., Li, Y., Hu, J., Gao, Y., Li, X. (2021). Influence of basalt fiber on mode I and II fracture properties of asphalt mixture at medium and low temperatures. Theor. Appl. Fract. Mech., V.112, 102884.
- S. Kaya Nurkan, İ. A. Güven, “A New approach for Smarandache curve”, Turk. J. Math. Comput. Sci., vol. 14, no. 1, pp. 155-165, 2022.
- S. Şenyurt, D. Canlı, E. Çan and S. G. Mazlum, “Some special Smarandache Ruled surfaces by Frenet frame in E^3-2”, Honam Mathematical Journal, vol. 44, no. 4, pp. 594-617, 2022.
- S. Şenyurt, K. H. Ayvacı and D. Canlı, “Smarandache curves according to Flc-frame in Euclidean 3-space”, Fundamentals of Contemporary Mathematical Sciences, vol. 4, no. 1, pp. 16-30, 2023.
- G. Tzitzeica, “Sur certaines courbes gauches”, Annales scientifiques de l'École Normale Supérieure, vol. 28, no. 3, pp. 9-32, 1911.
- A. Bobe, W. G. Boskoff and M. G. Ciuca, “Tzitzeica type centro-affine invariants in Minkowski space”, An. St. Univ. Ovidius Constanta, vol. 20, no. 2, pp. 27-34, 2012.
- M. Crasmareanu, “Cylindrical Tzitzeica curves implies forced harmonic oscillators”, Balkan Journal of Geometry and Its Applications, vol. 7, no. 1, pp. 37-42, 2002.
- M. K. Karacan, B. Bükcü, “On the hyperbolic cylindrical Tzitzeica curves in Mikowski 3-space”, BAÜ FBE Dergisi, vol. 10, no. 1, pp. 46-51, 2009.
- M. K. Karacan, B. Bükcü, “On the elliptic cylindrical Tzitzeica curves in Mikowski 3-space”, Scientia Magna, vol. 5, no. 3, pp. 44-48, 2009.
- N. Bila, “Symmetry reductions for the Tzitzeica curve equation”, Math and Computer Science Working Papers, vol. 16, 2012.
- B. Bayram, E. Tunç, K. Arslan, G. Öztürk, “On Tzitzeica curves in Euclidean 3-space E^3”, Facta Universitatis, Series Mathematics and Informatics, vol. 33, no. 3, pp. 409-416, 2018.
- B. Bayram, E. Tunç, “On tzitzeica surfaces in euclidean 3-space E^3”, Journal of Balikesir University Institute of Science and Technology, vol. 23, no. 1, pp. 277-290, 2021.
- B. Bayram, E. Tunç, “A New characterization of Tzitzeica curves in Euclidean 4-space”, Fundamentals of Contemporary Mathematical Sciences, vol. 4, no. 2, pp. 77-86, 2023
- B. Bayram, E. Tunç, “A Note on Tzitzeica curve in Euclidean 4-spaces E^4”, International Theory, Research and Reviews in Science and Mathematics, Chapter. 10, Serüven Yayınevi, pp. 141-166, 2023
- G. Öztürk, S. Büyükkütük, İ. Kişi, “A Characterization of Curves in Galilean 4-Space”, Bulletin of Irannian Mathematical Society, vol. 43, no. 3, pp. 771-780, 2017.
- H. Gluck, “Higher curvatures of curves in Euclidean space”, The American Mathematical Monthly, vol. 73, no. 7, pp. 243-245, 1966.
- A. Gray, “Modern differential geometry of curves and surfaces”, CRC Press, 1993.
- F. Klein, S. Lie, “Uber diejenigen ebenenen kurven welche durch ein geschlossenes system von einfach unendlich vielen vartauschbaren linearen Transformationen in sich übergehen”, The American Mathematical Monthly, vol. 4, pp. 50-84, 1871.
- G. Öztürk, K. Arslan, H. Hacısalihoğlu, “A characterization of ccr-curves in R^n”, Proceedings of the Estonian Academy of Sciences, vol. 57, pp. 217-224, 2008.
- B. Y. Chen, “When does the position vector of a space curve always lies in its rectifying plane?”, The American Mathematical Monthly, vol. 110, pp. 147-152, 2003.
- K. İlarslan, E. Nesovic, “Some characterizations of osculating curves in the Euclidean spaces”, Demonstratio Mathematica, vol. 16, no. 4, pp. 931-939, 2008.
- O. Karacan, Tzitzeica Smarandache Curves in Three Dimensional Euclidean Space, M.S. thesis, Instutıte of Science, Balıkesir University, Balıkesir, Türkiye, 2024