Tzitzeica Curves According to Modified Orthogonal Frames

Esra Damar

doi:10.17776/csj.1751734

Tzitzeica Curves According to Modified Orthogonal Frames

Abstract

This paper explores Tzitzeica curves in three-dimensional Euclidean space through the use of modified orthogonal frames constructed via curvature and torsion functions. First, the fundamental differences between the classical Frenet frame and the modified orthogonal frame are clarified. Following this, the necessary and sufficient conditions under which a curve can be classified as a Tzitzeica curve relative to the modified frame are established. The study also presents characterization theorems for spherical Tzitzeica curves and Tz-general helices, supported by differential geometric expressions involving curvature and torsion functions. Furthermore, special cases such as anti-Salkowski curves and rectifying curves are examined.

Keywords

References

[1] Sasai T., The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations, Tohoku Math. J., 36(1) (1984) 17-24.
[2] Bükcü B., Karacan M. K., On the modified orthogonal frame with curvature and torsion in 3-space, Math. Sci. Appl. E-Notes, 4(1) (2016) 184-188.
[3] Bükçü B., Karacan M.K., Spherical curves with modified orthogonal frame, J. New Results Sci., 5(10) (2016) 60-68.
[4] Lone M.S., Es H., Karacan M.K., Bükçü B., On some curves with modified orthogonal frame in Euclidean 3-space, Iran. J. Sci. Technol. Trans. A Sci., 43 (2019) 1905-1916.
[5] Gür Mazlum S., Bektaş M., On the modified orthogonal frames of the non-unit speed curves in Euclidean space , Turkish Journal of Science, 7(2) (2022) 58-74.
[6] Gür Mazlum S., Şenyurt S., Bektaş M., Salkowski curves and their modified orthogonal frames in , Journal of New Theory, 40 (2022) 12-26.
[7] Yüksel N., Karacan M.K., Demirkıran T., Spherical curves with modified orthogonal frame with torsion, Turkish Journal of Science, 7(3) (2022) 177- 184.
[8] Damar E., Baek B. S., Oğraş N., Yüksel N., AW (k)-type curves in modified orthogonal frame, Turkish Journal of Nature and Science, 13(4) (2024) 33-40.

[9] Tzitzeica G., On a Euler’s Theorem (romanian), Gazeta Matematic˘a, 13, (1907) 293– 294.
[10] Tzitzeica G., Sur une nouvelle classe de surfaces, Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, 150 (1911) 955–956.
[11] Agnew A., Bobe A., Boskoff W., Suceava B., Tzitzeica curves and surfaces, The Mathematica Journal, 12 (2010) 1-18.
[12] Bayram B., Tunç E., Arslan K., Öztürk G., On Tzitzeica curves in Euclidean 3-space , Facta Univ. Ser. Math. Inform., 33(3) (2018) 409-416.
[13] Karacan O., Bayram B., Tzitzeica Smarandache Curves in Euclidean 3-Space, Natural and Applied Sciences Journal, 7(1) (2024) 61-77.
[14] Crâsmareanu M., Cylindrical Tzitzeica curves implies forced harmonic oscillators, Balkan J. Geom. Its Appl.,7 (2002) 37-42.
[15] Eren K., Ersoy S., On curve pairs of Tzitzeica type, Adv. Appl. Math. Sci., 22(9) (2023) 2009-2021.
[16] Tunç E., Bayram B., A New characterization of Tzitzeica curves in Euclidean 4-space, Fundam. Contemp. Math. Sci., 4(2) (2023) 77-86.
[17] Tunç E., Bayram B., A Note On Tzıtzeıca Curves In Euclıdean 4-Space 𝔼4. And Revıews In Scıence And Mathematıcs, (2023) 141-166.
[18] Karacan M.K., Bükcü B., On the elliptic cylindrical Tzitzeica curves in Minkowski 3-space, Scientia Magna, 5 (2009) 44-48.
[19] Aydın M.E., Ergüt M., Non-null curves of Tzitzeica type in Minkowski 3-space, Rom. J. Math. Comput. Sci., 4(1) (2014) 81-90.
[20] Kaymanlı G. U., Şen G. N., Ekici C., Tzitzeica curves with q-frame in three-dimensional Minkowski space, Commun. Fac. Sci. Univ. Ankara Ser. A1 Math. Stat., 73(4) (2024) 957-968.
[21] Özen K.E., İşbilir Z., Tosun M., Characterization of Tzitzéica curves using positional adapted frame, Konuralp J. Math., 10(2) 2(2022) 60-268.
[22] Özen K.E., Tosun M., Avcı K., Type 2-positional adapted frame and its application to Tzitzeica and Smarandache curves, Karatekin Uni. J. Sci., 1(1) (2022) 42-53.
[23] Eren K., Ersoy S., Characterizations of Tzitzeica curves using Bishop frames. Mathematical Methods in the Applied Sciences, 45(18) (2022) 12046-12059.
[24] Şenyurt S., Eren K., Ayvacı K.H., Characterizations of Tzitzeica curves using FLC frame, Sigma J. Eng. Nat. Sci., 42(1) (2024) 37-41.
[25] Yazıcı D.B., Karakuş S.Ö., Tosun M., On framed Tzitzeica curves in Euclidean space, Facta Univ. Ser. Math. Inform., 37(3) (2022) 307-319.
[26] Barros M., General helices and a theorem of Lancret. Proceedings of the American Mathematical Society, 125(5) (1997) 1503-1509.
[27] Salkowski E., Zur Transformation von Raumkurven, Math. Ann., 66(1909) 517-557.

Details

Primary Language

English

Subjects

Algebraic and Differential Geometry

Journal Section

Research Article

Authors

Esra Damar ^*
0000-0002-0743-8545
Türkiye

Publication Date

September 30, 2025

Submission Date

July 26, 2025

Acceptance Date

September 22, 2025

Published in Issue

Year 2025 Volume: 46 Number: 3

DOI

https://doi.org/10.17776/csj.1751734

IZ

https://izlik.org/JA35BM67BU

Cite

RIS / Bibtex

APA

Damar, E. (2025). Tzitzeica Curves According to Modified Orthogonal Frames. Cumhuriyet Science Journal, 46(3), 595-600. https://doi.org/10.17776/csj.1751734

Cited By

Characterization of Tzitzeica Curves by Using q-frame in Euclidean 3-space

Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi

https://doi.org/10.19113/sdufenbed.1805264