The Ruled Surfaces Generated By Quasi-Vectors in E^4 Space

Abstract

In this article, firstly, it is aimed to introduce the ruled surfaces, which is generated by quasi-vectors, by using the relationship between the Frenet frame and the quasi-frame, the quasi-equations, the quasi-curvatures in the spaces $\mathbb{E}^{3}$ and $\mathbb{E}^{4}$. Calculating the coefficients of the first fundamental form, Gaussian and mean curvatures of ruled surfaces, which are generated by quasi vectors are obtained in $4$-dimensional Euclidean space. In addition to these, the relation between the Gaussian and mean curvatures of the ruled surfaces is given. Then, some geometric properties such as developability, minimality and striction line for those surfaces are investigated. Also, an example of surface curvatures by using the coefficients of fundamental form is obtained and the shapes of the ruled surface sample in projection spaces are plotted.

Keywords

• Frenet frame
• Quasi-frame
• Ruled surface
• Curvatures

References

1. Kim, Y.H., Liu, H., & Qian, J. (2016). Some characterizations of canal surfaces. Bulletin of the Korean Mathematical Society, 53(2), 461-477.
2. Xu, Z., Feng, R., & Sun, J.G. (2006). Analytic and algebraic properties of canal surfaces. Journal of Computational and Applied Mathematics, 195(1-2), 220-228.
3. Dogan, F., & Yayli, Y. (2017). The relation between parameter curves and lines of curvature on canal surfaces. Kuwait Journal of Science, 44(1), 29-35.
4. Aydın Şekerci, G., & Çimdiker, M. (2019). Bonnet canal surfaces. Dokuz Eylül Üniversitesi Mühendislik Fakültesi Fen ve Mühendislik Dergisi, 21(61), 195-200.
5. Dede, M., Ekici, C., & Tozak, H. (2015). Directional tubular surfaces. International Journal of Algebra, 9(12), 527-535.
6. Dogan, F., & Yayli, Y. (2011). On the curvatures of tubular surface with Bishop frame. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 60(1), 59-69.
7. Ekici, C., Kaymanlı G.U., & Okur, S. (2021). A new characterization of ruled surfaces according to q-frame vectors in Euclidean 3-space. International Journal of Mathematical Combinatorics, 3, 20-31.
8. Kaymanlı, G.U. (2020). Characterization of the evolute offset of ruled surfaces with B-Darboux frame. Journal of New Theory, 33, 50-55.

9. Kılıçoğlu, S., Şenyurt, S., & Çalışkan, A. (2016). On the striction curves along the involutive and Bertrandian Darboux ruled surfaces based on the tangent vector fields. New Trends in Mathematical Sciences, 4(4), 128-136.
10. Ravani, B., & Ku, T.S. (1991). Bertrand offsets of ruled surface and developable surface. Computer-Aided Design, 23(2), 145-152.
11. Sarioglugil, A., & Tutar, A. (2007). On ruled surface in Euclidean space E3. Int. J. Contemp. Math. Sci., 2(1), 1-11.
12. Şentürk, G.Y., & Yüce, S. (2015). Characteristic properties of the ruled surface with Darboux frame in E3. Kuwait Journal of Science, 42(2), 14-33.
13. Ünlütürk, Y., Çimdiker, M., & Ekici, C. (2016). Characteristic properties of the parallel ruled surfaces with Darboux frame in Euclidean 3-space. Communication in Mathematical Modeling and Applications, 1(1), 26-43.
14. Aydemir, ̇I., & Orbay, K. (2009). The ruled surfaces generated by Frenet vectors of timelike ruled surface in the Minkowski space R^3_1. World Applied Science Journal, 6(5), 692-696.
15. Çimdiker, M., & Ekici, C. (2017). On the spacelike parallel ruled surfaces with Darboux frame. International Journal of Mathematical Combinatorics, 2, 60-69.
16. Kaymanlı, G.U., Ekici, C. & Dede, M. (2020). Directional evolution of the ruled surfaces via the evolution of their directrix using q-frame along a timelike space curve. The European Journal of Science and Technology, 20, 392-396.
17. Şentürk, G.Y., & Yüce, S. (2020). On ruled non-degenerate surfaces with Darboux frame in Minkowski 3-space. TWMS Journal of Applied and Engineering Mathematics, 10(2), 499-511.
18. Ekici, C., Körpınar, T., & Ünlütürk, Y. (2023). An approach to characterizations of null curves lying in timelike ruled surfaces. Soft Computing, 27(5), 2159-2169.
19. Orbay, K., & Aydemir, İ. (2010). The ruled surfaces generated by Frenet vectors of a curve in R^3_1. Celal Bayar University Journal of Science, 6(2), 155-160.
20. Bishop, R.L. (1975). There is more than one way to frame a curve. The American Mathematical Monthly, 82(3), 246-251.
21. Dede, M., Ekici, C., & Görgülü, A. (2015). Directional q-frame along a space curve. International Journal of Advanced Research in Computer Science and Software Engineering, 5(12), 775-780.
22. Dede, M., Ekici, C., & Güven İ.A. (2018). Directional Bertrand curves. Gazi University Journal of Science, 31(1), 202-211.
23. Elsayied, H.K., Tawfiq, A.M., & Elsharkawy, A. (2021). Special Smarandache curves according to the quasi frame in 4-dimensional Euclidean space E4. Houston J. Math, 74(2), 467-482.
24. Gezer, B., & Ekici, C. (2023). On space curve with quasi frame in E4. 4th International Black Sea Modern Scientific Research Congress (p. 1951-1962).
25. Alessio, O. (2009). Differential geometry of intersection curves in R4 of three implicit surfaces. Computer Aided Geometric Design, 26(4), 455-471.
26. Bloomenthal, J. (1990). Calculation of reference frames along a space curve. Graphics Gems, 1, 567-571.
27. Çelik, T., Bozkurt, Z., & Gök, ̇I. (2014). Parallel transport frame in 4-dimensional Euclidean space. Caspian Journal of Mathematical Sciences, 3(1), 91-103.
28. Do-Carmo, M.P. (1976). Differential geometry of curves and surfaces. Prentice Hall, Englewood Cliffs, New Jersey.
29. Gluck, H. (1966). Higher curvatures of curves in Euclidean space. The American Mathematical Monthly, 73(7), 699-704.
30. Gray, A., Abbena, E., & Salamon, S. (2006). Modern differential geometry of curves and surfaces with mathematica. Chapman & Hall, CRC press.
31. Öztürk, G., Gürpinar, S., & Arslan, K. (2017). A new characterization of curves in Euclidean 4-space E^4. Buletinul Academiei de S ̧ tiint ̧e a Republicii Moldova, Matematica, 83(1), 39-50.
32. Bayram, K., B., Bulca, B., Arslan, K., & Öztürk, G. (2009). Superconformal ruled surfaces in E^4. Mathematical Communications, 14(2), 235-244.
33. Bulca, B., Arslan, K., Bayram, B., & Öztürk, G. (2017). Canal surfaces in 4-dimensional Euclidean space. An International Journal of Optimization and Control: Theories & Applications, 7(1), 83-89.
34. Mello, L.F. (2003). Mean directionally curved lines on surfaces immersed in R^4. Publicacions matematiques, 47(2), 415-440.
35. Ekici A., Akça, Z., & Ekici, C. (2023). The ruled surfaces generated by quasi-vectors in E^4 space. 7. International Biltek Congress on Current Developments in Science, Technology and Social Sciences (p. 400-418).
36. Odabaşı, Ç. Z. (2019). Dört boyutlu Öklid uzayında regle yüzeyler, Yüksek Lisans Tezi, Erciyes Üniversitesi, Fen Bilimleri Enstitüsü.
37. Otsuki, T., & Shiohama, K. (1967). A theory of ruled surfaces in E^4. Kodai Mathematical Seminar Reports, 19(3), 370-380.
38. Yağbasan, B., & Ekici, C. (2023). Tube surfaces in 4 dimensional Euclidean space. 4th International Black Sea Modern Scientific Research Congress (p. 1951-1962).
39. Yağbasan, B., Tozak, H., & Ekici, C. (2023). The curvatures of the tube surface in 4 dimensional Euclidean space. 7. International Biltek Congress on Current Developments in Science, Technology and Social Sciences (p. 419-436).
40. Yüce, S. (2019). Weingarten map of the hypersurface in Euclidean 4-space and its applications. Hagia Sophia Journal of Geometry, 1(1), 1-8.