| Title:
|
On the characterizations of special adjoint curves and surface family for the modified orthogonal frame with torsion (English) |
| Author:
|
Şaffak Atalay, Gülnur |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0011-4642 |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
151 |
| Issue:
|
4 |
| Year:
|
2026 |
| Pages:
|
667-682 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this context, the main interest of this paper is to construct a family of surfaces in $E^3$ space which is the adjoint pair of a given curve. Then, using the modified orthogonal frame with torsion, we express the necessary conditions for the surface family that recognize the adjoint curve as a geodesic curve, an asymptotic curve and a line of curvature. Also, the singular points of the surface given by the parametric equation are expressed. It is concluded that the adjoint curve of the alpha curve whose torsion is not constant can not be geodesic on the surface under any circumstances. In addition, the case where the adjoint curve of the unit velocity alpha curve with nonconstant torsion with respect to the modified orthogonal frame with torsion is the Salkowski curve, is also exemplified. By the way, this work is illustrated by several examples. (English) |
| Keyword:
|
the modified orthogonal frame with torsion |
| Keyword:
|
adjoint curve |
| Keyword:
|
Salkowski curve |
| Keyword:
|
geodesic curve |
| Keyword:
|
asymptotic curve |
| Keyword:
|
line of curvature |
| Keyword:
|
surface family |
| MSC:
|
53A04 |
| MSC:
|
53A05 |
| MSC:
|
53Z50 |
| DOI:
|
10.21136/MB.2025.0068-25 |
| . |
| Date available:
|
2026-09-23T08:26:31Z |
| Last updated:
|
2026-09-23 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153773 |
| . |
| Reference:
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[1] Akyigit, M., Eren, K., Kosal, H. H.: Tubular surfaces with modified orthogonal frame in Euclidean 3-space.Honam Math. J. 43 (2021), 453-463. Zbl 1549.53020, MR 4331705, 10.5831/HMJ.2021.43.3.453 |
| Reference:
|
[2] kan, M. Arı, Nurkan, S. K.: Adjoint curve according to the modified orthogonal frame with torsion in 3-space.Uşak Univ. J. Sci. Natural Sci. 4 (2020), 54-64. 10.47137/usufedbid.798189 |
| Reference:
|
[3] Atalay, G. Ş.: A new approach to special curved surface families according to the modified orthogonal frame.AIMS Math. 9 (2024), 20662-20676. MR 4765506, 10.3934/math.20241004 |
| Reference:
|
[4] Atalay, G. S.: A study of tubular surfaces according to the modified orthogonal frame with torsion.Honam Math. J. 46 (2024), 279-290. Zbl 07939324, MR 4754007, 10.5831/HMJ.2024.46.2.279 |
| Reference:
|
[5] Atalay, G. S., Murat, H.: Surface families with common Mannheim-B curvature line.J. Sci. Arts 24 (2024), 357-368. 10.46939/J.Sci.Arts-24.2-a10 |
| Reference:
|
[6] Atalay, G. Ş., Şimşek, K. H. A.: Family of surfaces with a common special polynomial curves.Bol. Soc. Parana. Mat. (3) 43 (2025), 14 pages. MR 4860997, 10.5269/bspm.70672 |
| Reference:
|
[7] Ayvacı, K. H., Atalay, G. Ş.: Surface family with a common Mannheim B-pair asymptotic curve.Int. J. Geom. Methods Mod. Phys. 20 (2023), Article ID 2350229, 13 pages. Zbl 1534.53005, MR 4663738, 10.1142/S0219887823502298 |
| Reference:
|
[8] Azak, A. Z.: Involute-evolute curves according to the modified orthogonal frame.J. Sci. Arts 55 (2021), 385-394. 10.46939/J.Sci.Arts-21.2-a06 |
| Reference:
|
[9] Bayram, E., Bilici, M.: Surface family with a common involute asymptotic curve.Int. J. Geom. Methods Mod. Phys. 13 (2016), Article ID 1650062, 9 pages. Zbl 1338.53007, MR 3489226, 10.1142/S0219887816500626 |
| Reference:
|
[10] Bukcu, B., Karacan, M. K.: On the modified orthogonal frame with curvature and torsion in 3-space.Math Sci. Appl. E-Notes 4 (2016), 184-188. Zbl 1421.53005, MR 3516546, 10.36753/mathenot.421429 |
| Reference:
|
[11] Choi, J. H., Kim, Y. H.: Associated curves of a Frenet curve and their applications.Appl. Math. Comput. 218 (2012), 9116-9124. Zbl 1245.53010, MR 2923010, 10.1016/j.amc.2012.02.064 |
| Reference:
|
[12] Desmukh, S., Chen, B.-Y., Alghanemi, A.: Natural mates of Frenet curves in Euclidean 3-space.Turk. J. Math. 42 (2018), 2826-2840. Zbl 1424.53020, MR 3866194, 10.3906/mat-1712-34 |
| Reference:
|
[13] Carmo, M. P. do: Differential Geometry of Curves and Surfaces.Prentice-Hall, Englewood Cliffs (1976). Zbl 0326.53001, MR 0394451 |
| Reference:
|
[14] Eren, K.: A study of the evolution of space curves with modified orthogonal frame in Euclidean 3-space.Appl. Math. E-Notes 22 (2022), 281-286. Zbl 1521.53002, MR 4483044 |
| Reference:
|
[15] Eren, K.: New representation of Hasimoto surfaces with the modified orthogonal frame.Konuralp J. Math. 10 (2022), 69-72. MR 4416309 |
| Reference:
|
[16] Eren, K., Ersoy, S.: On characterization of Smarandache curves constructed by modified orthogonal frame.Math. Sci. Appl. E-Notes 12 (2024), 101-112. 10.36753/mathenot.1409228 |
| Reference:
|
[17] Eren, K., Kosal, H. H.: Evolution of space curves and the special ruled surfaces with the modified orthogonal frame.AIMS Math. 5 (2020), 2027-2039. Zbl 1484.53021, MR 4143188, 10.3934/math.2020134 |
| Reference:
|
[18] Jiang, X., Jiang, P., Meng, J., Wang, K.: Surface pencil pair interpolating Bertrand pair as common asymptotic curves in Galilean space $G_3$.Int. J. Geom. Methods Mod. Phys. 18 (2021), Article ID 2150114, 11 pages. Zbl 7830899, MR 4273916, 10.1142/S0219887821501140 |
| Reference:
|
[19] Li, C.-Y., Wang, R.-H., Zhu, C.-G.: Parametric representation of a surface pencil with a common line of curvature.Comput. Aided Des. 43 (2011), 1110-1117. 10.1016/j.cad.2011.05.001 |
| Reference:
|
[20] Li, Y., Eren, K., Ersoy, S., Savić, A.: Modified sweeping surfaces in Euclidean 3-space.Axioms 13 (2024), 800-815. 10.3390/axioms13110800 |
| Reference:
|
[21] Lipschutz, M. M.: Theory and Problems of Differential Geometry.Schaum's Outline Series. McGraw-Hill, New York (1969). Zbl 0202.52301 |
| Reference:
|
[22] Nurkan, S. K., Güven, I.: A new approach for Smarandache curves.Turk. J. Math. Comput. Sci. 14 (2022), 155-165. 10.47000/tjmcs.1004423 |
| Reference:
|
[23] Nurkan, S. K., Güven, I. A., Karacan, M. K.: Characterizations of adjoint curves in Euclidean 3-space.Proc. Natl. Acad. Sci. India, Sect. A, Phys. Sci. 89 (2019), 155-161. MR 3924197, 10.1007/s40010-017-0425-y |
| Reference:
|
[24] O'Neill, B.: Elementary Differential Geometry.Elsevier/Academic Press, Amsterdam (2006). Zbl 1208.53003, MR 2351345 |
| Reference:
|
[25] Sasai, T.: The fundamental theorem of analytic space curves and apparent singularities of Fuchsian differential equations.Tohoku Math. J., II. Ser. 36 (1984), 17-24. Zbl 0543.53001, MR 0733615, 10.2748/tmj/1178228899 |
| Reference:
|
[26] Şenatalar, M.: Diferansiyel Geometri Eğriler ve Yüzeyler Teorisi.İstanbul Devlet Mühendislik Ve Mimarlık Akademisi Yayınları(1977), Turkish. |
| Reference:
|
[27] Struik, D. J.: Lectures on Classical Differential Geometry.Dover Books on Advanced Mathematics. Dover, New York (1988). Zbl 0697.53002, MR 0939369 |
| Reference:
|
[28] Wang, G.-J., Tang, K., Tai, C.-L.: Parametric representation of a surface pencil with a common spatial geodesic.Comput. Aided Des. 36 (2004), 447-459. Zbl 1206.65065, 10.1016/S0010-4485(03)00117-9 |
| Reference:
|
[29] Weatherburn, G. E.: Differential Geometry of Three Dimensions.Cambridge University Press, Cambridge (1927),\99999JFM99999 53.0658.08. |
| Reference:
|
[30] Willmore, T. J.: An Introduction to Differential Geometry.Clarendon Press, Oxford (1959). Zbl 0086.14401, MR 0159265 |
| Reference:
|
[31] Yakut, A. T., lay, A. Kızı: On the curve evolution with a new the modified orthogonal Saban frame.AIMS Math. 9 (2024), 29573-29586. MR 4811012, 10.3934/math.20241432 |
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