A new study on focal surface of a given surface

Abstract

Focal surfaces are special cases of line congruences. With the aid of the definiton of a focal surface of a given surface M, we obtain a new type of focal surface in Galilean 3-space G_3. We show that the focal surface we found is not the same type of surface as the given surface. We present the visualizations of the focal surface and the given surface with an example. Lastly, by searching the curvature functions, we give the minimality conditions of the focal surface.

Keywords

• Line congruence
• focal surface
• tubular surface
• Galilen Space

References

1. [1] Ali AT. Position vectors of curves in the Galilean space G3. Mat. Vesn. 64(3), 2012, 200–210.
2. [2] Aydın ME, Külahçı MA, Öğrenmiş¸ AO. Constant curvature translation surfaces in Galilean 3-space. International Electronic Journal of Geometry. 12(1), 2019, 9–19.
3. [3] Aydın ME, Öğrenmiş¸ AO. Spherical product surface in the Galilean space. Konuralp Journal of Mathematics. 4(2), 2019, 290-298.
4. [4] Bulca B, Arslan K, Bayram B, Öztürk G. Canal surfaces in 4-dimensional Euclidean space. Libertas Mathematica. 32, 2012, 1–13.
5. [5] Dede M. Tubular surfaces in Galilean space. Math. Commun. 18, 2013, 209–217.
6. [6] Dede M, Ekici C, Çöken AC. On the parallel surfaces in the Galilean space. Hacettepe Journal of Mathematics and Statistics. 42(6), 2013, 605615.
7. [7] Hagen H, Hahmann S. Generalized Focal Surfaces: A New Method for Surface Interrogation. Proceedings Visualization’92, Boston; 1992, 70–76.
8. [8] Hagen H, Pottmann H, Divivier A. Visualization functions on a surface. Journal of Visualization and Animation. 2, 1991, 52–58.

9. [9] Kamenarovic I. Existence theorems for ruled surfaces in the Galilean space G3. Rad Hazu Math. 456(10), 1991, 183–196.
10. [10] Kişi İ, Öztürk G. A new approach to canal surface with parallel transport frame. International Journal of Geometric Methods in Modern Physics. 14, 2017, 1–16.
11. [11] Kişi İ, Öztürk G. A new type of tubular surface having pointwise 1-type Gauss map in Euclidean 4-space E4. J. Korean Math. Soc. 55, 2018, 923–938.
12. [12] Kişi İ, Öztürk G. Spherical product surface having pointwise 1-type Gauss map in Galilean 3-space G3. International Journal of Geometric Methods in Modern Physics. 16(12), 2019, 1–10.
13. [13] Kişi İ, Öztürk G. Tubular surface having pointwise 1-type Gauss map in Euclidean 4-space. International Electronic Journal of Geometry. 12, 2019, 202–209.
14. [14] Kişi İ, Öztürk G, Arslan K. A new type of canal surface in Euclidean 4-space E4. Sakarya University Journal of Science. 23, 2019, 801–809.
15. [15] Özdemir B. A characterization of focal curves and focal surfaces in E4. PhD Thesis, Uludag˘ University, Bursa, Turkey, 2008.
16. [16] Özdemir B, Arslan K. On generalized focal surfaces in E3. Rev. Bull. Calcutta Math. Soc. 16(1), 2008, 23–32.
17. [17] Öztürk G, Arslan K. On focal curves in Euclidean n-space Rn. Novi Sad J. Math. 48(1), 2016, 35–44.
18. [18] Öztürk G, Bulca B, (Kılıc) Bayram B, Arslan K. On canal surfaces in E3. Selc¸uk J. Appl. Math. 11, 2010, 103–108.
19. [19] Pavkovic BJ, Kamenarovic I. The equiform dierential geometry of curves in the Galilean space G3. Glasnik Matematicki. 22(42), 1987, 449–457.
20. [20] Röschel O. Die Geometrie Des Galileischen Raumes. Forschungszentrum Graz Research Centre, Austria, 1986.
21. [21] Sipus ZM. Ruled Weingarten surfaces in the Galilean space. Periodica Mathematica Hungarica. 56(2), 2008, 213–225.
22. [22] Sipus ZM, Divjak B. Surfaces of constant curvature in the pseudo-Galilean space. International Journal of Mathematics and Mathematical Sciences. 12, 2012, 1–28.
23. [23] Shepherd MD. Line congruences as surfaces in the space of lines. Dierential Geometry and its Applications. 10, 1999, l–26.
24. [24] Yaglom IM. A Simple Non-Euclidean Geometry and Its Physical Basis. Springer- Verlag Inc., New York, 1979.