Research Article
Surfaces Using a Common Geodesic Curve With an Alternative Moving Frame in The 3-Dimensional Lie Group
Abstract
Our purpose in this research is to use an alternative moving frame in the 3-dimensional Lie group to construct the problem of how to characterize a surface family and derive the conditions from a given common geodesic curve as an isoparametric curve. We also derive the relation about developability along the common geodesic of a ruled surface as a member of the surface family. Finally, we will give some examples to show some applications of the method.
References
- [1] Bozkurt Z., Gök I, Okuyucu O. Z., Ekmekci F. N., Characterizations of rectifying, normal and osculating curves in three dimensional compact Lie groups, Life Sci. J., 10(3) (2013) 819-823.
- [2] Çiftçi Ü. , A generalization of Lancret’s theorem, J. Geom. Phys., 59(12) (2009) 1597-1603.
- [3] Degirmen C., On curves in three dimensional compact Lie Groups, Master’s Thesis, Bilecik Seyh Edebali University, 2017.
- [4] do Esprito-Santo N., Fornari S., Frensel K., Ripoll J., Constant mean curvature hypersurfaces in a Lie group with a bi-invariant metric, Manuscripta Math., 111(4) (2003) 459-470.
- [5] Mak M., Natural and Conjugate Mates of Frenet Curves in Three-Dimensional Lie Group, arXiv preprint arXiv:2008.05831 (2020).
- [6] Okuyucu O. Z., Gök İ., Yayli Y., Ekmekci N., Slant helices in three dimensional Lie groups, Appl. Math. Comput., 221 (2013) 672-683.
- [7] Yoon D. W., General helices of AW (k)-type in the Lie group, J. Appl. Math., (2012) ID 535123.
- [8] Yoon D. W., Tuncer Y., Karacan M. K., On curves of constant breadth in a 3-dimensional Lie group, Acta Math. Univ. Comenian., 85(1) (2016) 81-86.
- [9] Jiang X., Jiang P., Meng J., Wang K., Surface pencil pair interpolating Bertrand pair as common asymptotic curves in Galilean space G 3, Int. J. Geom. Methods Mod. Phys., 18(7) (2021) 114-459.
- [10] Kasap E., Family of surface with a common null geodesic, Int. J. Phys. Sci., 4(8) (2009) 428–433.
- [11] Kasap E., Akyildiz F.T., Surfaces with a common geodesic in Minkowski 3-space, Appl. Math. Comp. , 177 (2006) 260–270.
- [12] Küçükarslan Yüzbaşı Z., Bektaş M., On the construction of a surface family with common geodesic in Galilean space Open Phys., 14 (2016) 360–363.
- [13] Küçükarslan Yüzbaşı Z., Yoon D. W., On constructions of surfaces using a geodesic in Lie group, J. Geo., 110(2) (2019) 1-10.
- [14] [Tuncer O.O., Surface pencil with a common isophote curve, Master’s Thesis, Ankara University, 2016.
- [15] Wang G. J., Tang K., Tai C. L., Parametric representation of a surface pencil with a common spatial geodesic, Comput. Aided Des., 36 (2004) 447–459.
- [16] Altin M., Kazan, A., Karadag, H. B., Hypersurface families with Smarandache curves in Galilean 4-space, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 70 (2021) 744-761.
- [17] Ergun E., Bayram E., Surface family with a common natural asymptotic or geodesic lift of a spacelike curve with spacelike binormal in Minkowski 3-space, Konuralp J. Math., 8 (2020) 7-13
Year 2022,
Volume: 43 Issue: 1, 77 - 81, 30.03.2022
Abstract
References
- [1] Bozkurt Z., Gök I, Okuyucu O. Z., Ekmekci F. N., Characterizations of rectifying, normal and osculating curves in three dimensional compact Lie groups, Life Sci. J., 10(3) (2013) 819-823.
- [2] Çiftçi Ü. , A generalization of Lancret’s theorem, J. Geom. Phys., 59(12) (2009) 1597-1603.
- [3] Degirmen C., On curves in three dimensional compact Lie Groups, Master’s Thesis, Bilecik Seyh Edebali University, 2017.
- [4] do Esprito-Santo N., Fornari S., Frensel K., Ripoll J., Constant mean curvature hypersurfaces in a Lie group with a bi-invariant metric, Manuscripta Math., 111(4) (2003) 459-470.
- [5] Mak M., Natural and Conjugate Mates of Frenet Curves in Three-Dimensional Lie Group, arXiv preprint arXiv:2008.05831 (2020).
- [6] Okuyucu O. Z., Gök İ., Yayli Y., Ekmekci N., Slant helices in three dimensional Lie groups, Appl. Math. Comput., 221 (2013) 672-683.
- [7] Yoon D. W., General helices of AW (k)-type in the Lie group, J. Appl. Math., (2012) ID 535123.
- [8] Yoon D. W., Tuncer Y., Karacan M. K., On curves of constant breadth in a 3-dimensional Lie group, Acta Math. Univ. Comenian., 85(1) (2016) 81-86.
- [9] Jiang X., Jiang P., Meng J., Wang K., Surface pencil pair interpolating Bertrand pair as common asymptotic curves in Galilean space G 3, Int. J. Geom. Methods Mod. Phys., 18(7) (2021) 114-459.
- [10] Kasap E., Family of surface with a common null geodesic, Int. J. Phys. Sci., 4(8) (2009) 428–433.
- [11] Kasap E., Akyildiz F.T., Surfaces with a common geodesic in Minkowski 3-space, Appl. Math. Comp. , 177 (2006) 260–270.
- [12] Küçükarslan Yüzbaşı Z., Bektaş M., On the construction of a surface family with common geodesic in Galilean space Open Phys., 14 (2016) 360–363.
- [13] Küçükarslan Yüzbaşı Z., Yoon D. W., On constructions of surfaces using a geodesic in Lie group, J. Geo., 110(2) (2019) 1-10.
- [14] [Tuncer O.O., Surface pencil with a common isophote curve, Master’s Thesis, Ankara University, 2016.
- [15] Wang G. J., Tang K., Tai C. L., Parametric representation of a surface pencil with a common spatial geodesic, Comput. Aided Des., 36 (2004) 447–459.
- [16] Altin M., Kazan, A., Karadag, H. B., Hypersurface families with Smarandache curves in Galilean 4-space, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 70 (2021) 744-761.
- [17] Ergun E., Bayram E., Surface family with a common natural asymptotic or geodesic lift of a spacelike curve with spacelike binormal in Minkowski 3-space, Konuralp J. Math., 8 (2020) 7-13
There are 17 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Natural Sciences |
| Authors | |
| Publication Date | March 30, 2022 |
| Submission Date | July 28, 2021 |
| Acceptance Date | December 15, 2021 |
| Published in Issue | Year 2022Volume: 43 Issue: 1 |
