In Euclidean 3-space $ {\mathrm{E}}^3 $, a canonical subject is the focal surface of such a cliched space curve, which would be a two-dimensional corrosive with Lagrangian discontinuities. The tubular surfaces with respect to the B-Darboux frame and type-2 Bishop frame in $ {\mathrm{E}}^3 $ are given. These tubular surfaces' focal surfaces are then defined. For such types of surfaces, we acquire some results becoming Weingarten, flat, linear Weingarten conditions and we demonstrate that in $ {\mathrm{E}}^3 $, a tubular surface has no minimal focal surface. We also provide some examples of these types of surfaces.
Citation: Ibrahim AL-Dayel, Emad Solouma, Meraj Khan. On geometry of focal surfaces due to B-Darboux and type-2 Bishop frames in Euclidean 3-space[J]. AIMS Mathematics, 2022, 7(7): 13454-13468. doi: 10.3934/math.2022744
In Euclidean 3-space $ {\mathrm{E}}^3 $, a canonical subject is the focal surface of such a cliched space curve, which would be a two-dimensional corrosive with Lagrangian discontinuities. The tubular surfaces with respect to the B-Darboux frame and type-2 Bishop frame in $ {\mathrm{E}}^3 $ are given. These tubular surfaces' focal surfaces are then defined. For such types of surfaces, we acquire some results becoming Weingarten, flat, linear Weingarten conditions and we demonstrate that in $ {\mathrm{E}}^3 $, a tubular surface has no minimal focal surface. We also provide some examples of these types of surfaces.
[1] | L. R. Bishop, There is more than one way to frame a curve, Amer. Math. Mon., 82 (1975), 246–251. https://doi.org/10.1080/00029890.1975.11993807 doi: 10.1080/00029890.1975.11993807 |
[2] | J. L. M. Barbosa, A. G. Colares, Minimal surfaces in $R^3$, Springer, 1986. |
[3] | E. M. Solouma, Investigation of non-lightlike tubular surfaces with Darboux frame in Minkowski 3-space, Commun. Math. Model. Appl., 4 (2016), 58–65. |
[4] | G. Darboux, Leçons sur la théorie générale des surfaces, Paris: Gauthier-Villars, 1896. |
[5] | M. P. Do Carmo, Differential geometry of curves and surfaces: Revised and updated second edition, Courier Dover Publications, 2016. |
[6] | M. Dede, M. Ç. Aslan, C. Ekici, On a variational problem due to the $B$-Darboux frame in Euclidean 3-space, Math. Methods Appl. Sci., 44 (2021), 12630–2639. https://doi.org/10.1002/mma.7567 doi: 10.1002/mma.7567 |
[7] | G. H. Georgiev, M. D. Pavlov, Focal and generalized focal surfaces of parabolic cylinders, ARPN J. Eng. Appl. Sci., 13 (2018), 4458–4465. |
[8] | S. Hu, Z. Wang, X. Tang, Tubular surfaces of center curves on spacelike surfaces in Lorentz-Minkowski 3-space, Math. Methods Appl. Sci., 42 (2019), 3136–3166. https://doi.org/10.1002/mma.5574 doi: 10.1002/mma.5574 |
[9] | G. Hu, H. Cao, J. Wu, G. Wei, Construction of developable surfaces using generalized $C$-Bézier bases with shape parameters, Comput. Appl. Math., 39 (2020), 157. https://doi.org/10.1007/s40314-020-01185-9 doi: 10.1007/s40314-020-01185-9 |
[10] | S. Kiziltuǧ, S. Kya, Ö. Tarakci, Tube surfaces with type-2 Bishop frame of Weingarten types in $E^3$, Int. J. Math. Anal., 7 (2013), 9–18. http://dx.doi.org/10.12988/ijma.2013.13002 doi: 10.12988/ijma.2013.13002 |
[11] | S. Liu, Z. Wang, Generalized focal surfaces of spacelike curves lying in lightlike surfaces, Math. Methods Appl. Sci., 44 (2020), 7501–7525. https://doi.org/10.1002/mma.6296 doi: 10.1002/mma.6296 |
[12] | H. Liu, J. Miao, Geometric invariants and focal surfaces of spacelike curves in de Sitter space from a caustic viewpoint, AIMS Math., 6 (2021), 3177–3204. https://doi.org/10.3934/math.2021192 doi: 10.3934/math.2021192 |
[13] | B. O'Neill, Elementary differential geometry, New York: Academic Press Inc., 1966. |
[14] | G. Öztürk, K. Arslan, On focal curves in Euclidean $n$-space $\mathbb{R}^n$, Novi Sad J. Math., 46 (2016), 35–44. |
[15] | J. Qian, J. Liu, X. Fu, S. D. Jung, Geometric characterizations of canal surfaces with Frenet center curves, AIMS Math., 6 (2021), 9476–9490. https://doi.org/10.3934/math.2021551 doi: 10.3934/math.2021551 |
[16] | E. M. Solouma, I. Al-Dayel, Harmonic evolute surface of tubular surfaces via B-Darboux frame in Euclidean 3-space, Adv. Math. Phys., 2021 (2021), 5269655. https://doi.org/10.1155/2021/5269655 doi: 10.1155/2021/5269655 |
[17] | Z. Wang, D. Pei, L. Chen, L. Kong, Q. Han, Singularities of focal surfaces of null Cartan curves in Minkowski 3-space, Abstr. Appl. Anal., 2021 (2021), 823809. https://doi.org/10.1155/2012/823809 doi: 10.1155/2012/823809 |
[18] | S. Yılmaz, M. Turgut, A new version of Bishop frame and an application to spherical images, J. Math. Anal. Appl., 371 (2010), 764–776. https://doi.org/10.1016/j.jmaa.2010.06.012 doi: 10.1016/j.jmaa.2010.06.012 |