Construction and analysis of the quasi-ruled surfaces based on the quasi-focal curves in $ \mathbb{R}^{3} $

Samah Gaber; Asmahan Essa Alajyan; Adel H. Sorour; Samah Gaber; Asmahan Essa Alajyan; Adel H. Sorour

doi:10.3934/math.2025258

AIMS Mathematics

2025, Volume 10, Issue 3: 5583-5611. doi: 10.3934/math.2025258

Previous Article Next Article

Research article

Construction and analysis of the quasi-ruled surfaces based on the quasi-focal curves in $ \mathbb{R}^{3} $

1.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa $ 31982 $, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Assiut University, Assiut $ 71516 $, Egypt
3.
Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt

Received: 31 December 2024 Revised: 20 February 2025 Accepted: 24 February 2025 Published: 13 March 2025
MSC : 53A04, 53A05, 53A10

This paper presents the concept of a quasi-ruled surface, which is a ruled surface generated by a base curve and a ruling, both of which are defined by the quasi-frame (q-frame). This study begins with the original curve defined by the q-frame, and then we focus on the focal curve of the original curve, which serves as the base curve of the ruled surface. We define the focal curve by the q-frame, so the terminology quasi-focal curve is used in this paper. This paper investigates the formation and properties of the quasi-ruled surface (QRS) using a quasi-focal curve (QFC) as the base curve (directrix). The ruling of the surface is expressed in terms of the q-frame associated with the QFC. A variety of QRS types are discussed in this study, including the osculating, normal, and rectifying types. In addition, the types of a quasi-tangent developable surface, a quasi-principal normal surface, and a quasi-binormal ruled surface will also be discussed. The geometric properties of these surfaces, such as the first and second fundamental quantities, Gaussian curvature, mean curvature, second Gaussian curvature, and second mean curvature, are described. The conditions for their developability and minimality are derived. Moreover, we provide an example that includes the study of geometric properties and clear visualizations of these novel types of QRS.
Citation: Samah Gaber, Asmahan Essa Alajyan, Adel H. Sorour. Construction and analysis of the quasi-ruled surfaces based on the quasi-focal curves in $ \mathbb{R}^{3} $[J]. AIMS Mathematics, 2025, 10(3): 5583-5611. doi: 10.3934/math.2025258

Related Papers:

Abstract

This paper presents the concept of a quasi-ruled surface, which is a ruled surface generated by a base curve and a ruling, both of which are defined by the quasi-frame (q-frame). This study begins with the original curve defined by the q-frame, and then we focus on the focal curve of the original curve, which serves as the base curve of the ruled surface. We define the focal curve by the q-frame, so the terminology quasi-focal curve is used in this paper. This paper investigates the formation and properties of the quasi-ruled surface (QRS) using a quasi-focal curve (QFC) as the base curve (directrix). The ruling of the surface is expressed in terms of the q-frame associated with the QFC. A variety of QRS types are discussed in this study, including the osculating, normal, and rectifying types. In addition, the types of a quasi-tangent developable surface, a quasi-principal normal surface, and a quasi-binormal ruled surface will also be discussed. The geometric properties of these surfaces, such as the first and second fundamental quantities, Gaussian curvature, mean curvature, second Gaussian curvature, and second mean curvature, are described. The conditions for their developability and minimality are derived. Moreover, we provide an example that includes the study of geometric properties and clear visualizations of these novel types of QRS.

References

[1]	Y. Tuncer, N. Ekmekci, A study on ruled surfaces in Euclidean 3-space, J. Dyn. Syst. Geom. The., 8 (2010), 49–57. https://doi.org/10.1080/1726037X.2010.10698577 doi: 10.1080/1726037X.2010.10698577
[2]	I. A. Guven, S. Kaya, H. H. Hacisalihoglu, On closed ruled surfaces concerned with dual Frenet and Bishop frames, J. Dyn. Syst. Geom. The., 9 (2011), 67–74. https://doi.org/10.1080/1726037X.2011.10698593 doi: 10.1080/1726037X.2011.10698593
[3]	Y. Yu, H. Liu, S. D. Jung, Structure and characterization of ruled surfaces in Euclidean 3-space, Appl. Math. Comput., 233 (2014), 252–259. https://doi.org/10.1016/j.amc.2014.02.006 doi: 10.1016/j.amc.2014.02.006
[4]	Y. Tuncer, Ruled surfaces with the Bishop frame in Euclidean 3-space, Gen. Math. Notes, 26 (2015), 74–83. https://doi.org/10.1007/s40010-018-0546-y doi: 10.1007/s40010-018-0546-y
[5]	M. Masal, A. Z. Azak, The ruled surfaces according to type-2 Bishop frame in the Euclidean 3-space $\mathbb{E}^3$, Math. Sci. Appl. E-Notes, 3 (2018), 74–83. https://doi.org/10.36753/mathenot.421334 doi: 10.36753/mathenot.421334
[6]	G. Y. Senturk, S. Yuce, Characteristic properties of the ruled surface with Darboux frame in $\mathbb{E}^3$, Kuwait J. Sci. Eng., 42 (2015), 14–33. Available from: https://www.researchgate.net/publication/278686054_Characteristic_properties_of_the_ruled_surface_with_Darboux_frame_in_E3.
[7]	Y. Unluturk, M. Cimdiker, C. Ekici, Characteristic properties of the parallel ruled surfaces with Darboux frame in Euclidean 3-space, CMMA, 1 (2016), 26–43. Available from: https://dergipark.org.tr/en/pub/cmma/issue/27697/292077.
[8]	H. Liu, Y. Liu, S. D. Jung, Ruled invariants and associated ruled surfaces of a space curve, Appl. Math. Comput., 348 (2019), 479–486. https://doi.org/10.1016/j.amc.2018.12.011 doi: 10.1016/j.amc.2018.12.011
[9]	U. G. Kaymanli, C. Ekici, M. Dede, Directional evolution of the ruled surfaces via the evolution of their directrix using q-frame along a timelike space curve, Avrupa Bilim ve Teknoloji Derg., 20 (2019), 392–396. https://doi.org/10.31590/ejosat.681674 doi: 10.31590/ejosat.681674
[10]	S. Ouarab, A. O. Chahdi, Some characteristic properties of ruled surfaces with Frenet frame of an arbitrary non-cylindrical ruled surface in Euclidean 3-space, Int. J. Appl. Phys. Math., 10 (2020), 16–24. https://doi.org/10.17706/ijapm.2020.10.1.16-24 doi: 10.17706/ijapm.2020.10.1.16-24
[11]	C. Ekici, G. U. Kaymanli, S. Okur, A new characterization of ruled surfaces according to q-frame vectors in Euclidean 3-space, Int. J. Math. Combin., 3 (2017), 20–31. Available from: https://fs.unm.edu/IJMC/A_New_Characterization_of_Ruled_Surfaces_According_to_q-Frame_Vectors_in_Euclidean_3-Space.pdf.
[12]	S. Senyurt, D. Canli, E. Can, Smarandache-based ruled surfaces with the Darboux vector according to Frenet frame in $\mathbb{E}^3$, J. New Theory, 39 (2022), 8–18. Available from: https://www.researchgate.net/publication/362039701_Smarandache-Based_Ruled_Surfaces_with_the_Darboux_Vector_According_to_Frenet_Frame_in_E3#fullTextFileContent.
[13]	H. K. Samanci, The quasi-frame of the rational and polynomial Bezier curve by algorithm method in Euclidean space, Eng. Computations, 40 (2023), 1698–1722. https://doi.org/10.1108/EC-04-2022-0193 doi: 10.1108/EC-04-2022-0193
[14]	G. U. Kaymanli, C. Ekici, Evolutions of the ruled surfaces along a spacelike space curve, Punjab Univ. J. Math., 54 (2022), 221–232. https://doi.org/10.52280/pujm.2022.540401 doi: 10.52280/pujm.2022.540401
[15]	B. Pal, S. Kumar, Ruled like surfaces in three-dimensional Euclidean space, Ann. Math. Inform., 59 (2022), 83–101. https://doi.org/10.33039/ami.2022.12.011 doi: 10.33039/ami.2022.12.011
[16]	S. Gaber, A. H. Sorour, A. A. A. Salam, Construction of ruled surfaces from the W-curves and their characterizations in $\mathbb{R}^3$, Symmetry, 16 (2023), 509. https://doi.org/10.3390/sym16050509 doi: 10.3390/sym16050509
[17]	T. Korpinar, On quasi focal curves with quasi frame in space, Bol. Soc. Parana. Mat., 41 (2023), 1–3. https://doi.org/10.5269/bspm.50873 doi: 10.5269/bspm.50873
[18]	O. N. Barrett, Elementary differential geometry, 2 Eds., Academic Press, 2006. Available from: http://staff.ustc.edu.cn/spliu/2018DG/ONeill.pdf.
[19]	M. Dede, C. Ekici, A. Gorgulu, Directional q-frame along a space curve, IJARCSSE, 5 (2015), 775–780. https://doi.org/10.31590/ejosat.681674 doi: 10.31590/ejosat.681674
[20]	M. Dede, C. Ekici, H. Tozak, Directional tubular surfaces, Int. J. Algebr., 9 (2015), 527–535. https://doi.org/10.12988/ija.2015.51274 doi: 10.12988/ija.2015.51274
[21]	M. D. Carmo, Differential geometry of curves and surfaces, NJ, Englewood Cliffs: Prentice Hall, 1976. Available from: http://www2.ing.unipi.it/griff/files/dC.pdf.
[22]	O. B. Kalkan, S. Senyurt, Osculating type ruled surfaces with type-2 Bishop frame in $\mathbb{E}^3$, Symmetry, 16 (2024), 498. https://doi.org/10.3390/sym16040498 doi: 10.3390/sym16040498
[23]	O. Kaya, M. Onder, Generalized normal ruled surface of a curve in the Euclidean 3-space, Acta U. Sapientiae-Math., 13 (2021), 217–238. https://doi.org/10.2478/ausm-2021-0013 doi: 10.2478/ausm-2021-0013
[24]	Z. Isbilir, B. D. Yazici, M. Tosun, Generalized rectifying ruled surfaces of special singular curves, An. Sti. U. Ovid. Co-Mat., 31 (2023), 177–206. https://doi.org/10.2478/auom-2023-0038 doi: 10.2478/auom-2023-0038
[25]	S. Verpoort, The geometry of the second fundamental form: Curvature properties and variational aspects, Ph. D. Thesis, Belgium: Katholieke Universiteit Leuven, 2008. Available from: https://lirias.kuleuven.be/bitstream/1979/1779/2/hierrrissiedan!.pdf.
[26]	D. W. Yoon, On the second Gaussian curvature of ruled surfaces in Euclidean 3-space, Tamkang J. Math., 37 (2006), 221–226. https://doi.org/10.5556/j.tkjm.37.2006.167 doi: 10.5556/j.tkjm.37.2006.167
[27]	D. E. Blair, T. Koufogiorgos, Ruled surfaces with vanishing second Gaussian curvature, Monatsh. Math., 113 (1992), 177–181. https://doi.org/10.1007/BF01641765 doi: 10.1007/BF01641765

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)