A new approach to revolution surface with its focal surface in the Galilean 3-space $\mathbb{G}_{3}$

Year 2021, , 1722 - 1737, 14.12.2021

İlim Kişi

https://doi.org/10.15672/hujms.905636

Abstract

In this paper, we handle focal surfaces of surface of revolution in Galilean 3-space $\mathbb{G}_{3}$. We define the focal surfaces of surface of revolution and we obtain some results for these types of surfaces to become flat and minimal. Also, by giving some examples to these surfaces, we present the visualizations of each type of focal surface of surface of revolution in $\mathbb{G}_{3}$.

Keywords

line congruence, focal surface, surface of revolution

References

• [1] D. Aberre and K. Agraval, Surfaces of revolution in n dimensions, Int. J. Math. Educ. Sci. Technol. 38, 843-852, 2009.
• [2] K. Arslan, B. Kılıç Bayram, B. Bulca and G. Öztürk, Generalized rotation surfaces in $\mathbb{E}^4$, Results. Math. 61, 315-327, 2012.
• [3] K. Arslan, B. Bayram, B. Bulca, D. Kosova and G. Öztürk, Rotational surfaces in higher dimensional Euclidean spaces, Rend. Circ. Mat. Palermo (2) 67, 59-66, 2018.
• [4] M.E. Aydın, M.A. Külahçı and A.O. Öğrenmiş, Constant curvature translation surfaces in Galilean 3-space, Int. Electron. J. Geom. 12, 9-19, 2019.
• [5] D.V. Cuong, Surfaces of revolution with constant Gaussian curvature in four-space, Asian-Eur. J. Math. 6, 1350021-1–1350021-7, 2013.
• [6] M. Dede, C. Ekici and A.C. Çöken, On the parallel surfaces in the Galilean space, Hacet. J. Math. Stat. 42, 605-615, 2013.
• [7] M. Dede, C. Ekici and W. Goemans, Surfaces of revolution with vanishing curvature in Galilean 3-space, Zh. Mat. Fiz. Anal. Geom. 14, 141-152, 2018.
• [8] W. Goemans, Flat double rotational surfaces in Euclidean and Lorentz-Minkowski 4-space, Publ. Inst. Math. 103, 61-68, 2018.
• [9] A. Gray, E. Abbana and S. Salamon, Modern Differential Geometry of Curves and Surfaces with Mathematica, (3rd edition), Studies in Advanced Mathematics, Chapman and Hall/CRC, Boca Raton, FL, 2006.
• [10] H. Hagen and S. Hahmann, Generalized Focal Surfaces: A New Method for Surface Interrogation, Proceedings Visualization’92, Boston, 70-76, 1992.
• [11] H. Hagen, H. Pottmann and A. Divivier, Visualization functions on a surface, J. Visual. Comput. Anim. 2, 52-58, 1991.
• [12] A. Kazan and H. Karadağ, A classification of surfaces of revolution in Lorentz- Minkowski space, Int. J. Contemp. Math. Sci. 6, 1915-1928, 2011.
• [13] K. Kenmotsu, Surfaces of revolution with prescribed mean curvature, Tohoku Math. J. 32, 147-153, 1980.
• [14] K. Kenmotsu, Surfaces of revolution with periodic mean curvature, Osaka J. Math. 40, 687-696, 2003.
• [15] B. Özdemir, A characterization of focal curves and focal surfaces in $\mathbb{E}^4$, Ph.D. Thesis, Uludağ University, 2008.
• [16] B. Özdemir and K. Arslan, On generalized focal surfaces in $\mathbb{E}^3$, Rev. Bull. Calcutta Math. Soc. 16, 23-32, 2008.
• [17] G. Öztürk and K. Arslan, On focal curves in Euclidean n-space $\mathbb{R}^n$, Novi Sad J. Math. 48, 35-44, 2016.
• [18] B.J. Pavkovic and I. Kamenarovic, The equiform differential geometry of curves in the Galilean space $\mathbb{G}^3$, Glas. Mat. Ser. III 22, 449-457, 1987.
• [19] D. Pei and T. Sano, The focal developable and the binormal indicatrix of a nonlightlike curve in Minkowski 3-space, Tokyo J. Math. 23, 211-225, 2000.
• [20] O. Röschel, Die Geometrie Des Galileischen Raumes, Forschungszentrum Graz Research Centre, Austria, 1986.
• [21] Z.M. Sipus, Ruled Weingarten surfaces in the Galilean space, Period. Math. Hungar. 56, 213-225, 2008.
• [22] Z.M. Sipus and B. Divjak, Surfaces of constant curvature in the pseudo-Galilean space, Int. J. Math. Math. Sci. 12, 1-28, 2012.
• [23] M.D. Shepherd, Line congruences as surfaces in the space of lines, Differential Geom. Appl. 10, l-26, 1999.
• [24] I.M. Yaglom, A Simple Non-Euclidean Geometry and Its Physical Basis, Springer- Verlag Inc., New York, 1979.
• [25] D.W. Yoon, Surfaces of revolution in the three dimensional pseudo-Galilean space, Glas. Mat. Ser. III 48, 415-428, 2013.
• [26] J. Yu, X. Yin, X. Gu, L. McMillan and S. Gortler, Focal surfaces of discrete geometry, Eurographics Symposium on Geometry Processing, 2007.