A GEOMETRIC STUDY OF BERTRAND CURVES IN EUCLIDEAN 3-SPACE

Ranjhan Ali; Israr Ahmed; Fayaz Hussain Bhatti; Irshad Ali; Muhammad Arif; Soyam Kapoor; Manav Dev; Bisma Memon

doi:10.71146/kjmr846

Authors

Ranjhan Ali Department of Mathematics, Shah Abdul Latif University Khairpur, Sindh, Pakistan. Author
Israr Ahmed Department of Mathematics, Shah Abdul Latif University Khairpur, Sindh, Pakistan. Author
Fayaz Hussain Bhatti Lecturer Mathematics, Cadet College Pano Aqil, Sindh, Pakistan. Author
Irshad Ali MS Scholar, Department of Mathematics, Shah Abdul Latif University Khairpur, Sindh, Pakistan. Author
Muhammad Arif Crescent College of Accountancy Lahore, Punjab, Pakistan Author
Soyam Kapoor Department of Computer System Engineering, IBA Sukkur University, Sindh, Pakistan. Author
Manav Dev Department of Computer System Engineering, IBA Sukkur University, Sindh, Pakistan. Author
Bisma Memon Department of Computer System Engineering, IBA Sukkur University, Sindh, Pakistan. Author

DOI:

https://doi.org/10.71146/kjmr846

Keywords:

Bertrand Curves, Differential Geometry, Curvature, Torsion, Euclidean 3-Space, Freenet Frame, Curve Theory

Abstract

It is a differential geometrical work on Bertrand curves in the Euclidean space, in three dimensions. The so-called Bertrand curves are regarded as special curves of space; the two curves have a mate curve of which the two curves are believed to coincide at the same principal normal vectors. It is devoted to the investigation of the most significant correspondence between curvature and torsion which defines the appearance and the labours of Bertrand curves.

The article applies the Freenet-Serret model to describe the geometrical properties of curves, and to study what conditions to apply to obtain Bertrand curves. The results show that the dependence between curvature and torsion must be linear to have Bertrand property. Such curves are analysed under various analysis situations with the objective of establishing their stability, predictability and consistency of structure.

The results reveal that the shapes of the Bertrand curves are much more regular and stable geometries than the underlying space curves, and may be used in geometric modelling, in computer graphics and in mechanical design. Moreover, the study concludes that parameters are to be controlled because despite even the smallest variations, the presence of Bertrand mate curves can be affected.

Such is a piece, which has led to the study of the subject of differential geometry in giving a clear and detailed insight to the Bertrand curves in the classical view and in the modern-day analysis view of the object. New research in non-Euclidean geometries as well as applied mathematics is also a result of the research.

Downloads

Download data is not yet available.

References

[1]. Eren, K., Ersoy, S., & Stanković, M. S. (2025). Bertrand-like curves in Euclidean 3-space. Filomat, 39(22), 7697-7705.

[2]. Nakatsuyama, N., & Takahashi, M. (2024). Bertrand types of regular curves and Bertrand framed curves in the Euclidean 3-space. arXiv preprint arXiv:2403.19138.

[3]. Mofarreh, F., & Abdel-Baky, R. A. (2023). Surface pencil pair interpolating Bertrand pair as common asymptotic curves in Euclidean 3-space. Mathematics, 11(16), 3495.

[4]. Erdem, H. A., & Ilarslan, K. (2023). Spacelike Bertrand curves in Minkowski 3-space revisited. Analele ştiinţifice ale Universităţii" Ovidius" Constanţa. Seria Matematică, 31(3), 87-109

[5]. Aydın, M. E., Bektaş, M., Öğremiş, A., & Yokuş, A. (2021). Differential geometry of curves in Euclidean 3-space with fractional order. International Electronic Journal of Geometry, 14(1), 132-144.

[6]. ELSHARKAWY, A., & ELSHARKAWY, N. (2025). BERTRAND AND MANNHEIM CURVES OF NORMAL-INTEGRAL AND BINORMAL-INTEGRAL CURVES IN EUCLIDEAN 3-SPACE. Bulletin of Mathematical Analysis & Applications, 17(4).

[7]. Erdem, H. A., Uçum, A., Ilarslan, K., & CAMCI, Ç. (2023). New approach to timelike Bertrand curves in 3-dimensional Minkowski space. Carpathian Mathematical Publications, 15(2).

[8]. Pal, B., & Kumar, S. (2023). Ruled like surfaces in three-dimensional Euclidean space. In Annales Mathematicae et Informaticae (Vol. 59, pp. 83-101). Eszterházy Károly Egyetem Líceum Kiadó.

[9]. Nazra, S. H., & Abdel-Baky, R. A. (2023). A Surface Pencil with Bertrand Curves as Joint Curvature Lines in Euclidean Three-Space. Symmetry, 15(11), 1986.

[10]. Aldossary, M. T., & Abdel-Baky, R. A. (2023). Surfaces Family with Bertrand Curves as Common Asymptotic Curves in Euclidean 3–Space E3. Symmetry, 15(7), 1440.

[11]. Almoneef, A. A., & Abdel-Baky, R. A. (2024). Bertrand Offsets of Slant Ruled Surfaces in Euclidean 3-Space. Symmetry, 16(2), 235.

[12]. Almoneef, A. A., & Abdel-Baky, R. A. (2025). Geometric analysis of slant timelike-ruled surfaces and Bertrand offsets in Minkowski 3-space. AIP Advances, 15(8).

[13]. Elsayied, H. K., Altaha, A. A., & Elsharkawy, A. (2022). Bertrand curves with the modified orthogonal frame in Minkowski 3-space E3 1. Rev. Edu, 392(6), 43-55.

[14]. ALMAZ, F. (2024). Research on space-like Bertrand curve pair in 3D light like cone. Punjab University Journal of Mathematics, 56(7), 357-368.

[15]. HA, E. (2023). New approach to timelike Bertrand curves in 3-dimensional Minkowski space. Carpathian Mathematical Publications/Karpats' kì Matematičnì Publìkacìï, 15(2).

[16]. Sun, J., & Zhao, Y. (2021). The Geometrical Characterizations of the Bertrand Curves of the Null Curves in Semi-Euclidean 4-Space. Mathematics, 9(24), 3294.

[17]. Aldossary, M. T. (2024). Surface family mate with Bertrand mate as mutual curvature lines in Galilean 3-space G3. Appl. Math, 18(4), 885-893.

[18]. Kaya, O., & Önder, M. (2021). Generalized normal ruled surface of a curve in the Euclidean 3-space. Acta Universitatis Sapientiae, Mathematica, 13(1).

[19]. TURHAN, T., & TOPDAL, H. T. (2023). On Geometry Of Some Curve Pairs According To Type of Bishop Frame In Lorentz 3-Space. Journal of Pharmaceutical Negative Results, 14(4).

[20]. Nazra, S. H., & Abdel-Baky, R. A. (2023). Bertrand offsets of ruled surfaces with Blaschke frame n Euclidean 3-space. Axioms, 12(7), 649.

[21]. Bilgin, B., & Camcı, Ç. (2022). Timelike V-Bertrand Curves in Minkowski 3-Space $ E_ {1}^{3} $. Journal of New Theory, (38), 14-24.

[22]. GÜR, S., & BEKTAŞ, M. (2023). INVOLUTE CURVES OF ANY NON-UNIT SPEED CURVE IN EUCLIDEAN 3-SPACE. INTERNATIONAL STUDIES IN SCIENCE AND MATHEMATICS, 177.