Geometric analysis of translation surfaces based on special curves in the modified orthogonal frame

Burçin Saltık Baek; Nural Yüksel; Burçin Saltık Baek; Nural Yüksel

doi:10.3934/math.2025748

AIMS Mathematics

2025, Volume 10, Issue 7: 16676-16691. doi: 10.3934/math.2025748

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Geometric analysis of translation surfaces based on special curves in the modified orthogonal frame

Burçin Saltık Baek ^,,
Nural Yüksel

Department of Mathematics, Erciyes University, Kayseri 38030, Turkey

Received: 25 May 2025 Revised: 27 June 2025 Accepted: 08 July 2025 Published: 24 July 2025
MSC : 53A04, 53A05

The geometry of translation surfaces generated by curves related by the modified orthogonal frame in Euclidean 3-space was investigated in this paper. We described the geometric features leading to minimality and developability by taking into account particular pairs as Bertrand, Mannheim, and involute-evolute curves. Furthermore, we revealed new connections between surface behavior and curvature requirements by incorporating adjoint curves into the building process. Understanding translation surfaces in the wider context of differential geometry has advanced considerably due to the examples and theorems that have been presented.
- translation surfaces,
- adjoint curve,
- modified orthogonal frame,
- Bertrand partner curves,
- Mannheim partner curves,
- involute-evolute curves
Citation: Burçin Saltık Baek, Nural Yüksel. Geometric analysis of translation surfaces based on special curves in the modified orthogonal frame[J]. AIMS Mathematics, 2025, 10(7): 16676-16691. doi: 10.3934/math.2025748

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Abstract

The geometry of translation surfaces generated by curves related by the modified orthogonal frame in Euclidean 3-space was investigated in this paper. We described the geometric features leading to minimality and developability by taking into account particular pairs as Bertrand, Mannheim, and involute-evolute curves. Furthermore, we revealed new connections between surface behavior and curvature requirements by incorporating adjoint curves into the building process. Understanding translation surfaces in the wider context of differential geometry has advanced considerably due to the examples and theorems that have been presented.

References

[1]	J. M. Bertrand, Mémoire sur la théorie des courbes à double courbure, Comptes Rendus, 15 (1850), 332–350.
[2]	A. Mannheim, De l'emploi de la courbe représentative de la surface des normales principales d'une courbe gauche pour la démonstration de propriétés relatives à cette courbure, C. R. Comptes Rendus Séances l'Acad. Sci., 86 (1878), 1254–1256.
[3]	M. P. do Carmo, Differential geometry of curves and surfaces, New Jersey: Prentice-Hall, Inc., 1976.
[4]	D. J. Struik, Lectures on classical differential geometry, 2 Eds., Dover Publications, 1988.
[5]	B. O'Neill, Elementary differential geometry, 2 Eds., Academic Press, 2006.
[6]	H. L. Liu, Translation surfaces with constant mean curvature in 3-dimensional spaces, J. Geom., 64 (1999), 141–149. https://doi.org/10.1007/BF01229219 doi: 10.1007/BF01229219
[7]	D. W. Yoon, On the Gauss map of translation surfaces in Minkowski 3-space, Taiwanese J. Math., 6 (2002), 389–398.
[8]	Y. Yuan, H. L. Liu, Some new translation surfaces in 3-Minkowski space, J. Math. Res. Exp., 31 (2011), 1123–1128.
[9]	H. Liu, Y. Yu, Affine translation surfaces in Euclidean 3-space, Proc. Japan Acad., 89 (2013), 111–113.
[10]	M. Çetin, Y. Tunçer, N. Ekmekçi, Translation surfaces in Euclidean 3-space, Int. J. Phys. Math. Sci., 5 (2011), 49–56.
[11]	M. Çetin, H. Kocayığit, M. Önder, Translation surfaces according to Frenet frame in Minkowski 3-space, Int. J. Phys. Sci., 7 (2012), 6135–6143.
[12]	G. S. Atalay, F. Güler, E. Kasap, Translation surfaces according to Bishop frame in Euclidean 3-space, J. Math. Comput. Sci., 4 (2014), 50–57.
[13]	H. N. Abd-Ellah, Evolution of translation surfaces in Euclidean 3-space, Appl. Math. Inf. Sci., 9 (2015), 661–668. http://dx.doi.org/10.12785/amis/090214 doi: 10.12785/amis/090214
[14]	A. T. Ali, H. S. Abdel Aziz, A. H. Sorour, On curvatures and points of the translation surfaces in Euclidean 3-space, J. Egypt. Math. Soc., 23 (2015), 167–172. http://dx.doi.org/10.1016/j.joems.2014.02.007 doi: 10.1016/j.joems.2014.02.007
[15]	R. López, Ó. Perdomo, Minimal translation surfaces in Euclidean space, J. Geom. Anal., 27 (2017), 2926–2937. https://doi.org/10.1007/s12220-017-9788-1 doi: 10.1007/s12220-017-9788-1
[16]	M. E. Aydın, M. A. Külahçı, A. O. Öğrenmiş, Constant curvature translation surfaces in Galilean 3-space, Int. Electron. J. Geome., 12 (2019), 9–19. https://doi.org/10.36890/iejg.545741 doi: 10.36890/iejg.545741
[17]	M. S. Lone, H. Es, M. K. Karacan, B. Bükcü, On some curves with modified orthogonal frame in Euclidean 3-space, Iran J. Sci. Technol. Trans. Sci., 43 (2019), 1905–1916. https://doi.org/10.1007/s40995-018-0661-2 doi: 10.1007/s40995-018-0661-2
[18]	M. S. Lone, H. Es, M. K. Karacan, B. Bükcü, Mannheim curves with modified orthogonal frame in Euclidean 3-space, Turkish J. Math., 43 (2019), 648–663. http://dx.doi.org/10.3906/mat-1807-177 doi: 10.3906/mat-1807-177
[19]	S. K. Nurkan, İ. A. Güven, M. K. Karacan, Characterizations of adjoint curves in Euclidean 3-space, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci., 89 (2019), 155–161. https://doi.org/10.1007/s40010-017-0425-y doi: 10.1007/s40010-017-0425-y
[20]	Ş. Özel, M. Bektaş, On the characterisations of curves with modified orthogonal frame in $E^3$, J. New Theory, 40 (2022), 54–59. https://doi.org/10.53570/jnt.1148933 doi: 10.53570/jnt.1148933
[21]	A. Has, B. Yılmaz, Translation surfaces generating with some partner curve, Commun. Fac. Sci. Univ., 73 (2024), 165–177. https://doi.org/10.31801/cfsuasmas.1299533 doi: 10.31801/cfsuasmas.1299533
[22]	A. T. Ali, Minimal timelike translation surfaces in Minkowski 3-spaces, Honam Math. J., 47 (2025), 70–85. https://doi.org/10.5831/HMJ.2025.47.1.70 doi: 10.5831/HMJ.2025.47.1.70
[23]	S. Ismoilov, B. Sultanov, B. Mamadaliyev, A. Kurudirek, Translation surfaces with non-zero constant total curvature in multidimensional isotropic space, J. Appl. Math. Inform., 43 (2025), 191–203. https://doi.org/10.14317/jami.2025.191 doi: 10.14317/jami.2025.191
[24]	T. Sasai, The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations, Tohoku Math. J., 36 (1984), 17–24. https://doi.org/10.2748/tmj/1178228899 doi: 10.2748/tmj/1178228899
[25]	B. Bükcü, M. K. Karacan, On the modified orthogonal frame with curvature and torsion in 3-space, Math. Sci. Appl. E-Notes, 4 (2016), 184–188. https://doi.org/10.36753/mathenot.421429 doi: 10.36753/mathenot.421429
[26]	K. Eren, H. H. Kosal, Evolution of space curves and the special ruled surfaces with modified orthogonal frame, AIMS Mathematics, 5 (2020), 2027–2039. https://doi.org/10.3934/math.2020134 doi: 10.3934/math.2020134
[27]	B. Bükcü, M. K. Karacan, Spherical curves with modified orthogonal frame, J. New Res. Sci., 5 (2016), 60–68.
[28]	E. Damar, B. S. Baek, N. Oğraş, N. Yüksel, AW(k)-type curves in modified orthogonal frame, Turk. J. Nat. Sci., 13 (2024), 33–40. https://doi.org/10.46810/tdfd.1468641 doi: 10.46810/tdfd.1468641
[29]	K. Eren, S. Ersoy, On characterization of Smarandache curves constructed by modified orthogonal frame, Math. Sci. Appl. E-Notes, 12 (2024), 101–112. https://doi.org/10.36753/mathenot.1409228 doi: 10.36753/mathenot.1409228
[30]	G. Ş. Atalay, A new approach to special curved surface families according to modified orthogonal frame, AIMS Mathematics, 9 (2024), 20662–20676. http://doi.org/10.3934/math.20241004 doi: 10.3934/math.20241004
[31]	B. S. Baek, E. Damar, N. Oğraş, N. Yüksel, Ruled surfaces of adjoint curve with the modified orthogonal frame, J. New Theory, 49 (2024), 69–82. https://doi.org/10.53570/jnt.1583283 doi: 10.53570/jnt.1583283
[32]	E. Damar, B. S. Baek, N. Yüksel, N. Oğraş, Tubular surfaces of adjoint curves according to the modified orthogonal frame, Black Sea J. Eng. Sci., 8 (2025), 206–213. https://doi.org/10.34248/bsengineering.1582756 doi: 10.34248/bsengineering.1582756
[33]	M. Akyigit, K. Eren, H. H. Kosal, Tubular surfaces with modified orthogonal frame in Euclidean 3-space, Honam Math. J., 43 (2021), 453–463. https://doi.org/10.5831/HMJ.2021.43.3.453 doi: 10.5831/HMJ.2021.43.3.453
[34]	A. Elsharkawy, M. Turan, H. G. Bozok, Involute-evolute curves with a modified orthogonal frame in the Galilean space $ G_3 $, Ukr. Math. J., 76 (2025), 1625–1636. http://dx.doi.org/0.1007/s11253-025-02411-5
[35]	M. Arıkan, S. K. Nurkan, Adjoint curve according to modified orthogonal frame with torsion in 3-space, Uşak Univ. J. Sci. Nat. Sci., 4 (2020), 54–64. https://doi.org/10.47137/usufedbid.798189 doi: 10.47137/usufedbid.798189

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