The study of a family of equiform Bishop spherical image ruled surfaces created by some specific curves such as spherical image in Minkowski 3-space using equiform Bishop frame of that curve is presented in this paper. We also offer the necessary criteria for these surfaces to be equiform Bishop developable and equiform Bishop minimum in relation to equiform Bishop curvatures, as well as when the curve is enclosed in a plane. Finally, we provide an example, such as these surfaces.
Citation: Emad Solouma, Mohamed Abdelkawy. Family of ruled surfaces generated by equiform Bishop spherical image in Minkowski 3-space[J]. AIMS Mathematics, 2023, 8(2): 4372-4389. doi: 10.3934/math.2023218
The study of a family of equiform Bishop spherical image ruled surfaces created by some specific curves such as spherical image in Minkowski 3-space using equiform Bishop frame of that curve is presented in this paper. We also offer the necessary criteria for these surfaces to be equiform Bishop developable and equiform Bishop minimum in relation to equiform Bishop curvatures, as well as when the curve is enclosed in a plane. Finally, we provide an example, such as these surfaces.
| [1] | M. Aydin, M. Ergut, The equiform differential geometry of curves in 4-dimensional galilean space $G_4$, Stud. Univ. Babes-Bolyai Math., 58 (2013), 399–406. |
| [2] |
I. Al-Dayel, E. Solouma, Characteristic properties of type-2 Smarandache ruled surfaces according to the type-2 Bishop frame in $E^3$, Adv. Math. Phys., 2021 (2021), 8575443. https://doi.org/10.1155/2021/8575443 doi: 10.1155/2021/8575443
|
| [3] |
I. Al-Dayel, E. Solouma, M. Khan1, On geometry of focal surfaces due to B-Darboux and type-2 Bishop frames in Euclidean 3-space, AIMS Mathematics, 7 (2022), 13454–13468. https://doi:10.3934/math.2022744 doi: 10.3934/math.2022744
|
| [4] |
R. Bishop, There is more than one way to frame a curve, The American Mathematical Monthly, 82 (1975), 246–251. https://doi.org/10.2307/2319846 doi: 10.2307/2319846
|
| [5] |
B. Bukcu, M. Karacan, Bishop frame of the spacelike curve with a spacelike principal normal in Minkowski 3-space, Commun. Fac. Sci. Univ., 57 (2008), 13–22. https://doi.org/10.1501/Commua1_0000000185 doi: 10.1501/Commua1_0000000185
|
| [6] | J. Barbosa, A. Gervasio Colares, Minimal surfaces in $R^3$, Berlin: Springer Verlag, 1986. https://doi.org/10.1007/BFb0077105 |
| [7] | M. Do Carmo, Differential geometry of curves and surfaces, 2Eds, Dover: Courier Dover Publications, 2016. |
| [8] |
F. Dillen, W. Sodsiri, Ruled surfaces of Weingarten type in Minkowski 3-space, J. Geom., 83 (2005), 10–21. https://doi.org/10.1007/s00022-005-0002-4 doi: 10.1007/s00022-005-0002-4
|
| [9] |
O. Gursoy, On the integral invariants of a closed ruled surface, J. Geome., 39 (1990), 80–91. https://doi.org/10.1007/BF01222141 doi: 10.1007/BF01222141
|
| [10] |
G. Hu, H. Cao, J. Wu, G. Wei, Construction of developable surfaces using generalized $C$-Bézier bases with shape parameters, Comp. Appl. Math., 39 (2020), 157. https://doi.org/10.1007/s40314-020-01185-9 doi: 10.1007/s40314-020-01185-9
|
| [11] |
H. Kocayigit, M. Cetin, Spacelike curves of constant breadth according to Bishop frame in Minkowski 3-space, Mathematical Sciences and Applications E-Notes, 3 (2015), 86–93. https://doi.org/10.36753/mathenot.421222 doi: 10.36753/mathenot.421222
|
| [12] |
O. Kose, Contribution to the theory of integral invariants of a closed ruled surface, Mech. Mach. Theory, 32 (1997), 261–277. https://doi.org/10.1016/S0094-114X(96)00034-1 doi: 10.1016/S0094-114X(96)00034-1
|
| [13] |
Y. Kim, D. Yoon, Classification of ruled surfaces in Minkowski 3-space, J. Geom. Phys., 49 (2004), 89–100. https://doi.org/10.1016/S0393-0440(03)00084-6 doi: 10.1016/S0393-0440(03)00084-6
|
| [14] |
A. Kucuk, On the developable time-like trajectory ruled surfaces in Lorentz 3-space $E^3_1$, Appl. Math. Comput., 157 (2004), 483–489. https://doi.org/10.1016/j.amc.2003.09.001 doi: 10.1016/j.amc.2003.09.001
|
| [15] |
R. López, Differential geometry of curves and surfaces in Lorentz-Minkowski space, Int. Electron. J. Geom., 7 (2014), 44–107. https://doi.org/10.36890/iejg.594497 doi: 10.36890/iejg.594497
|
| [16] |
W. Lam, Minimal surfaces from infinitesimal deformations of circle packings, Adv. Math., 362 (2020), 106939. https://doi.org/10.1016/j.aim.2019.106939 doi: 10.1016/j.aim.2019.106939
|
| [17] | B. O'Neill, Semi-Riemannian geometry with applications to relativity, New York: Academic press, 1983. |
| [18] |
S. Ouarab, Smarandache ruled surfaces according to Frenet-Serret frame of a regular curve in $E^3$, Abstr. Appl. Anal., 2021 (2021), 1–8. https://doi.org/10.1155/2021/5526536 doi: 10.1155/2021/5526536
|
| [19] | E. Solouma, Generalized Smarandache curves of spacelike and equiform spacelike curves via timelike second binormal in $R_1^4$, Appl. Appl. Math., 15 (2020), 1369–1380. |
| [20] |
E. Solouma, W. Mahmoud, On spacelike equiform Bishop Smarandache curves on $S_1^2$, J. Egypt. Math. Soc., 27 (2019), 7. https://doi.org/10.1186/s42787-019-0009-x doi: 10.1186/s42787-019-0009-x
|
| [21] |
E. Solouma, Equiform spacelike Smarandache curves of anti-Eqiform Salkowski curve according to Equiform frame, International Journal of Mathematical Analysis, 15 (2021), 43–59. https://doi.org/10.12988/ijma.2021.912141 doi: 10.12988/ijma.2021.912141
|
| [22] |
E. 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
|
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Emad Solouma, Mohamed Abdelkawy. Family of ruled surfaces generated by equiform Bishop spherical image in Minkowski 3-space[J]. AIMS Mathematics, 2023, 8(2): 4372-4389. doi: 10.3934/math.2023218
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