Directional Tube Surface in Euclidean 4-Space

Öz

The aim of this paper is to study characterization of tube surfaces (called directional tube surfaces) with respect to the q-frame in Euclidean $4$-space $\mathbb{E}^{4}$. First, a parametrization of these directional tube surfaces in $\mathbb{E}^{4}$ is established. Then, the normals of the directional tube surfaces, denoted as $\mathbf{U}_{1}$ and $\mathbf{U}_{2}$, are determined respectively. Furthermore, the Gaussian curvature $K$ and the mean curvature $H$ of the directional tube surfaces are investigated. Subsequently, an example of a directional tube surface is given in $\mathbb{E}^{4}$, together with visual representations of this tube surfaces in projection space.

Anahtar Kelimeler

• quasi-frame
• curvatures
• Euclidean space
• Tube surface

Kaynakça

1. 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.
2. Maekawa, T., Patrikalakis, N. M., Sakkalis, T., & Yu, G. (1998). Analysis and applications of pipe surfaces. Computer Aided Geometric Design, 15(5), 437-458.
3. Bloomenthal, J. (1990). Calculation of reference frames along a space curve. Graphics Gems, 1, 567-571.
4. Blaga, P. A. (2005). On tubular surfaces in computer graphics. Studia Universitatis Babes-Bolyai Informatica, L(2), 81-90.
5. Doğan, F., & Yaylı, Y. (2012). Tubes with Darboux frame. International Journal of Contemporary Mathematical Sciences, 7(16), 751-758.
6. Bishop, R. L. (1975). There is more than one way to frame a curve. The American Mathematical Monthly, 82(3), 246-251.
7. Alghanemi, A. (2016). On the singularities of the D-Tubular surfaces. Journal of Mathematical Analysis, 7(6), 97-104.
8. Karacan, M. K., Es, H., & Yaylı, Y. (2006). Singular points of tubular surface in Minkowski 3-space. Sarajevo Journal of Mathematics, 2(14), 73-82

9. Karacan, M. K., & Bukcu, B. (2007). An alternative moving frame for tubular surface around the spacelike curve with a spacelike binormal in Minkowski 3-space. Mathematica Moravica, 11, 47-54.
10. Dede, M. (2013). Tubular surfaces in Galilean space. Mathematical Communications, 18(1), 209-217.
11. Kızıltuğ S., Çakmak A., & Kaya S. (2013). Timelike tubes around a spacelike curve with Darboux Frame of Weingarten type in $E^3_1$ . International Journal of Physical and Mathematical Sciences, 4(1), 9-17.
12. Kızıltuğ, S., & Yaylı, Y. (2013). Timelike tubes with Darboux frame in Minkowski 3-space. International Journal of Physical Sciences, 8(1), 31-36.
13. Coquillart, S. (1987). Computing offsets of B-spline curves, Computer-Aided Design, 19(6), 305-309.
14. 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.
15. Dede, M., Ekici, C., & Tozak, H. (2015). Directional tubular surfaces. International Journal of Algebra, 9(12), 527-535.
16. Ekici, C., Tozak, H., & Dede, M. (2017). Timelike directional tubular surface. Journal of Mathematical Analysis, 8(5), 1-11.
17. Gezer, B., & Ekici, C. (2023). On space curve with quasi frame in E^4. 4th International Black Sea Modern Scientific Research Congress (p. 1951-1962).
18. Gluck, H. (1966). Higher curvatures of curves in Euclidean space. The American Mathematical Monthly, 73(7), 699-704.
19. Alessio, O. (2009). Differential geometry of intersection curves in R4 of three implicit surfaces. Computer Aided Geometric Design, 26(4), 455-471.
20. Elsayied, H. K., Tawfiq, A. M., & Elsharkawy, A. (2021). Special Smarandach curves according to the quasi frame in 4-dimensional Euclidean space E4. Houston Journal of Mathematics, 74(2), 467-482.
21. Gökçelik, F., Bozkurt, Z., Gök, İ., Ekmekçi, N., & Yaylı, Y. (2014). Parallel transport frame in 4-dimensional Euclidean space. The Caspian Journal of Mathematical Sciences, 3(1), 91-103.
22. Öztürk, G., Gürpinar, S., & Arslan, K. (2017). A new characterization of curves in Euclidean 4-space $E^4$. Buletinul Academiei de Stiinte a Republicii Moldova, Matematica, 1(83), 39-50.
23. Bayram, K. B., Bulca, B., Arslan, K., & Öztürk, G. (2009). Superconformal ruled surfaces in $E^4 $. Mathematical Communications, 14(2), 235-244.
24. 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).
25. Oláh-Gál, R., & Pál, L. (2009). Some notes on drawing twofolds in 4-Dimensional Euclidean space. Acta Universitatis Sapientiae, Informatica, 1(2), 125-134.
26. Chen, B.Y. (1976). Total mean curvature of immerseds Surface in $E^m$. Transactions of the American Mathematical Society, 218, 333-341.
27. Kişi, İ. (2018). Some characterizatıons of canal surfaces in the four dimensional Euclidean space. Kocaeli University, Phd Thesis, p. 94.
28. 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.
29. Kaymanlı G. U., Ekici, C., & Dede, M. (2018). Directional canal surfaces in $E^3$. 5th International Symposium on Multidisciplinary Studies (p.90-107).
30. Kim, Y. H., Liu, H., & Qian, J. (2016). Some characterizations of canal surfaces. Bulletin of the Korean Mathematical Society, 53(2), 461-477.
31. Uçum, A., & İlarslan, K. (2016). New types of canal surfaces in Minkowski 3-space. Advances in Applied Clifford Algebras, 26, 449-468.
32. Doğan, F., & Yaylı, Y. (2017). The relation between parameter curves and lines of curvature on canal surfaces. Kuwait Journal of Science, 44(1), 29-35.
33. Coşkun Ekici, A., & Akça, Z. (2023). The ruled surfaces generated by quasi-vectors in $E^4$ space. Hagia Sophia Journal of Geometry, 5(2), 6-17.
34. Yüce, S. (2019). Weingarten map of the hypersurface in Euclidean 4-space and its applications. Hagia Sophia Journal of Geometry, 1(1), 1-8.
35. Mello, L. F. (2009). Orthogonal asymptotic lines on surfaces immersed in $R^4$. The Rocky Mountain Journal of Mathematics, 39(5), 1597-1612.
36. Yağbasan, B., & Ekici, C. (2023). Tube surfaces in 4 dimensional Euclidean space. 4th International Black Sea Modern Scientific Research Congress, (p.1951-1962).
37. 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).
38. Chen, B. Y. (1984). Total mean curvature and submanifolds of finite type. Series in Pure Mathematics: Volume 1, World Scientific Publishing, Singapore.