SHAPE OPERATORS OF A DIRECTIONAL TUBULAR SURFACE IN 4-SPACE

Year 2025, Volume: 13 Issue: 2 , 109 - 121 , 25.08.2025

Başak Yağbasan , Cumali Ekici

https://doi.org/10.20290/estubtdb.1673455

https://izlik.org/JA56CG55CB

Abstract

This paper examines a tubular surface, a specific example of a canal surface, in 4-dimensional Euclidean space. In the plane stretched by the quasi-frame vectors B_q and C_q, this surface is established by the motion of a circle with a constant radius that uses each point on the curve a(t) as its center. Using the general equation provided in Euclidean 4-space, the first and second partial derivatives are determined. The Gram-Schmidt technique was used to derive the surface's first unit normal vector field U_1, and second unit normal vector field U_2, using the acquired partial derivatives. Using quasi-vectors, the tubular surface's first and second fundamental form coefficients were found. Furthermore, the shape operator matrices for the tubular surface's the unit normal vector fields U_1 and U_2 were acquired. We have found algebraic invariants of the shape operator, Gaussian curvature, and mean curvature. For a thorough understanding of the obtained theoretical calculations, an example of a directional tubular surface, the equation of the tubular surface has been parametrized using quasi-frame vectors and quasi-frame curvatures for a given space curve in 4-dimensional Euclidean space.

Keywords

Euclidean Space , Quasi-frame , Tubular Surface , Shape Operator

References

• [1] Monge G. Application de l’analyse a la géométrie. Bachelier. 1850.
• [2] Xu Z, Feng R, Sun JG. Analytic and algebraic properties of canal surfaces. Journal of Computational and Applied Mathematics, 2006; 195(1-2): 220-228.
• [3] Maekawa T, Patrikalakis NM, Sakkalis T, Yu G. Analysis and applications of pipe surfaces. Computer Aided Geometric Design, 1998; 15(5), 437-458.
• [4] Bloomenthal J. Calculation of reference frames along a space curve. Graphics Gems, 1990; 1: 567-571.
• [5] Doğan F, Yaylı Y. Tubes with Darboux frame. International Journal of Contemporary Mathematical Sciences, 2012; 7(16): 751-758.
• [6] Bishop RL. There is more than one way to frame a curve. The American Mathematical Monthly, 1975; 82(3): 246-251.
• [7] Coquillart S. Computing offsets of B-spline curves, Computer-Aided Design, 1987; 19(6): 305-309.
• [8] Dede M, Ekici C, Görgülü A. Directional q-frame along a space curve. International Journal of Advanced Research in Computer Science and Software Engineering, 2015; 5(12): 775-780.
• [9] Dede M. Tubular surfaces in Galilean space. Mathematical Communications, 2013; 18(1): 209-217.
• [10] Dede M, Ekici C, Tozak H. Directional tubular surfaces. International Journal of Algebra, 2015; 9(12): 527-535.
• [11] Ekici C, Tozak H, Dede M. Timelike directional tubular surface. Journal of Mathematical Analysis, 2017; 8(5): 1-11.
• [12] Gezer B, Ekici C. On space curve with quasi frame in E4. 4th International Black Sea Modern Scientific Research Congress; 6-7 June 2023; Rize, Turkey, 1951-1962.
• [13] Gluck H. Higher curvatures of curves in Euclidean space. The American Mathematical Monthly, 1966; 73(7): 699-704.
• [14] Alessio O. Differential geometry of intersection curves in R4 of three implicit surfaces. Computer Aided Geometric Design, 2009; 26(4): 455-471.
• [15] Elsayied HK, Tawfiq AM, Elsharkawy A. Special Smarandach curves according to the quasi frame in 4-dimensional Euclidean space E4. Houston Journal of Mathematics, 2021; 74(2): 467-482.
• [16] Gökçelik F, Bozkurt Z, Gök İ, Ekmekçi N, Yaylı Y. Parallel transport frame in 4-dimensional Euclidean space. The Caspian Journal of Mathematical Sciences, 2014; 3(1): 91-103.
• [17] Öztürk G, Gürpınar S, Arslan K. A new characterization of curves in Euclidean 4-space E4. Buletinul Academiei de Stiinte a Republicii Moldova, Matematica, 2017; 1(83), 39-50.
• [18] Bayram KB, Bulca B, Arslan K, Öztürk G. Superconformal ruled surfaces in E4. Mathematical Communications, 2009; 14(2), 235-244.
• [19] Ekici A, Akça Z, Ekici C. The ruled surfaces generated by quasi-vectors in E4 space. 7. International Biltek Congress on Current Developments in Science, Technology and Social Sciences, 2023; 400-418.
• [20] Ol´ah-G´al R, P´al L. Some notes on drawing twofolds in 4-Dimensional Euclidean space. Acta Universitatis Sapientiae, Informatica, 2009; 1(2), 125-134.
• [21] Chen BY. Total mean curvature of immerseds Surface in Em. Transactions of the American Mathematical Society, 1976; 218: 333-341.
• [22] Kişi İ. Some characterizatıons of canal surfaces in the four dimensional Euclidean space. Kocaeli University, Kocaeli, Türkiye, 2018.
• [23] Bulca B, Arslan K, Bayram B, Öztürk G. Canal surfaces in 4-dimensional Euclidean space. An International Journal of Optimization and Control: Theories & Applications, 2017; 7(1): 83-89.
• [24] Kaymanlı GU, Ekici C, Dede M. Directional canal surfaces in E3. 5th International Symposium on Multidisciplinary Studies, 2018; 90-107.
• [25] Kim YH, Liu H, Qian J. Some characterizations of canal surfaces. Bulletin of the Korean Mathematical Society, 2016; 53(2): 461-477.
• [26] Uçum A, İlarslan K. New types of canal surfaces in Minkowski 3-space. Advances in Applied Clifford Algebras, 2016; 26: 449-468.
• [27] Doğan F, Yaylı Y. The relation between parameter curves and lines of curvature on canal surfaces. Kuwait Journal of Science, 2017; 44(1): 29-35.
• [28] Coşkun Ekici A, Akça Z. The ruled surfaces generated by quasi-vectors in E4 space. Hagia Sophia Journal of Geometry, 2023; 5(2): 6-17.
• [29] Mello LF. Mean directionally curved lines on surfaces immersed in R4. Publicacions Matemàtiques, 2003; 47(2): 415-440.
• [30] Yağbasan B, Ekici C. Tube surfaces in 4 dimensional Euclidean space. 4th International Black Sea Modern Scientific Research Congress; 6-7 June 2023; Rize, Türkiye, 1951-1962.
• [31] Yağbasan B, Tozak H, Ekici C. The curvatures of the tube surface in 4 dimensional Euclidean space. VII.International Biltek Congress On Current Developments In Science, Technology And Social Sciences; 26-27 May 2023; Ankara, Türkiye 419-436.
• [32] Yağbasan B, Ekici C, Tozak H. Directional tube surface in Euclidean 4-space. Hagia Sophia Journal of Geometry, 2023; 5(2): 18-30.
• [33] Gray A. Modern differential geometry of curves and surface. CRS Press, Inc. 1993.
• [34] Bulca B. A characterization of a surface in E4. Uludağ University, Bursa, Türkiye, 2012.