Tubular surface generated by a curve lying on a regular surface and its characterizations

A. A. Abdel-Salam; M. I. Elashiry; M. Khalifa Saad; A. A. Abdel-Salam; M. I. Elashiry; M. Khalifa Saad

doi:10.3934/math.2024594

AIMS Mathematics

2024, Volume 9, Issue 5: 12170-12187. doi: 10.3934/math.2024594

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Tubular surface generated by a curve lying on a regular surface and its characterizations

1.
Department of Mathematics, Applied College, Northern Border University, Rafha, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Sohag University, 82524 Sohag, Egypt
3.
Department of Mathematics, Faculty of Science and Arts, Northern Border University, Rafha, KSA
4.
Department of Mathematics, Faculty of Science, Fayoum University, El-Fayoum, Egypt
5.
Department of Mathematics, Faculty of Science, Islamic University of Madinah, KSA

Received: 22 December 2023 Revised: 11 March 2024 Accepted: 21 March 2024 Published: 28 March 2024
MSC : 53A04, 53A05

In this research, we have constructed and studied special tubular surfaces in Euclidean 3-space $ \mathbb{R}^{3} $. We examined the singularities and geometrical properties of these surfaces. We achieved some significant results for these surfaces via Darboux frame. Also, we have proposed a few geometric invariants that illustrate the geometric characteristics of these surfaces, such as tubular Weingarten surfaces, using the traditional methods of differential geometry. Additionally, taking advantage of the singularity theory, we have given the classification of generic singularities of these surfaces. At last, we have presented some computational examples as an instance of use to validate our theoretical findings.
- tubular surfaces,
- Weingarten surfaces,
- singularities,
- Darboux frame
Citation: A. A. Abdel-Salam, M. I. Elashiry, M. Khalifa Saad. Tubular surface generated by a curve lying on a regular surface and its characterizations[J]. AIMS Mathematics, 2024, 9(5): 12170-12187. doi: 10.3934/math.2024594

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Abstract

In this research, we have constructed and studied special tubular surfaces in Euclidean 3-space $ \mathbb{R}^{3} $. We examined the singularities and geometrical properties of these surfaces. We achieved some significant results for these surfaces via Darboux frame. Also, we have proposed a few geometric invariants that illustrate the geometric characteristics of these surfaces, such as tubular Weingarten surfaces, using the traditional methods of differential geometry. Additionally, taking advantage of the singularity theory, we have given the classification of generic singularities of these surfaces. At last, we have presented some computational examples as an instance of use to validate our theoretical findings.

References

[1]	M. P. Do Carmo, Differential geometry of curves and surfaces, Englewood Cliffs: Prentice Hall, 1976.
[2]	G. Farin, Curves and surfaces for computer aided geometric design, Academic Press, 1990.
[3]	F. Dogan, Y. Yaylı, On the curvatures of tubular surface with Bishop frame, Commun. Fac. Sci. Univ., 60 (2011), 59–69. https://doi.org/10.1501/Commua1_0000000669 doi: 10.1501/Commua1_0000000669
[4]	M. Dede, Tubular surfaces in Galilean space, Math. Commun., 18 (2013), 209–217.
[5]	M. K. Karacan, D. W. Yoon, Y. Tuncer, Tubular surfaces of Weingarten types in Minkowski 3-space, Gen. Math. Notes, 22 (2014), 44–56.
[6]	M. Dede, C. Ekici, H. Tozak, Directional tubular surfaces, Int. J. Algebra, 9 (2015), 527–535. https://doi.org/10.12988/ija.2015.51274
[7]	A. H. Sorour, Weingarten tube-like surfaces in Euclidean 3-space, Stud. U. Babeş-Bol. Mat., 61 (2016), 239–250.
[8]	A. Cakmak, O. Tarakci, On the tubular surfaces in $E^{3}$, New Trends Math. Sci., 1 (2017), 40–50. https://doi.org/10.20852/ntmsci.2017.124 doi: 10.20852/ntmsci.2017.124
[9]	T. Maekawa, N. M. Patrikalakis, T. Sakkalis, G. Yu, Analysis and applications of pipe surfaces, Comput. Aided Geom. D., 15 (1998), 437–458. https://doi.org/10.1016/S0167-8396(97)00042-3 doi: 10.1016/S0167-8396(97)00042-3
[10]	Z. Xu, R. Feng, G. J. Sun, Analytic and algebraic properties of canal surfaces, J. Comput. Appl. Math., 195 (2006), 220–228. https://doi.org/10.1016/j.cam.2005.08.002 doi: 10.1016/j.cam.2005.08.002
[11]	F. Doğan, Y. Yayli, Tubes with Darboux frame, Int. J. Contemp. Math. Sciences, 16 (2012), 751–758.
[12]	F. Ateş, E. Kocakuşakli, Î. Gök, Y. Yayli, A study of the tubular surfaces constructed by the spherical indicatrices in Euclidean 3-space, Turk. J. Math., 42 (2018), 1711–1725. https://doi.org/10.3906/mat-1610-101 doi: 10.3906/mat-1610-101
[13]	Y. Gülsüm, Y. Salim, Characteristic properties of the ruled surface with Darboux frame in $E^{3}$, Kuwait J. Sci., 42 (2015), 14–30.
[14]	C. Baikoussis, T. Koufogiorgos, On the inner curvature of the second fundamental form of helicoidal surfaces, Arch. Math., 68 (1997), 169–176. https://doi.org/10.1007/s000130050046 doi: 10.1007/s000130050046
[15]	M. I. Munteanu, A. I. Nistor, Polynomial translation Weingarten surfaces in 3-dimensional Euclidean space, In: Differential Geometry, Proceedings of the VIII International Colloquium Hackensack, World Scientific, 2009,316–320. https://doi.org/10.1142/9789814261173_0034
[16]	J. W. Bruce, P. J. Giblin, Curves and singularities, Cambridge University Press, 1992. http://dx.doi.org/10.1017/CBO9781139172615
[17]	I. R. Porteous, Geometric differentiation for the intelligence of curves and surfaces, Cambridge University Press, 2001.

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