Araştırma Makalesi

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

Yıl 2023, Cilt: 5 Sayı: 2, 6 - 17, 30.12.2023

### Öz

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.

### Kaynakça

• Kim, Y.H., Liu, H., & Qian, J. (2016). Some characterizations of canal surfaces. Bulletin of the Korean Mathematical Society, 53(2), 461-477.
• 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.
• 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.
• Aydın Şekerci, G., & Çimdiker, M. (2019). Bonnet canal surfaces. Dokuz Eylül Üniversitesi Mühendislik Fakültesi Fen ve Mühendislik Dergisi, 21(61), 195-200.
• Dede, M., Ekici, C., & Tozak, H. (2015). Directional tubular surfaces. International Journal of Algebra, 9(12), 527-535.
• 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.
• 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.
• Kaymanlı, G.U. (2020). Characterization of the evolute offset of ruled surfaces with B-Darboux frame. Journal of New Theory, 33, 50-55.
• 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.
• Ravani, B., & Ku, T.S. (1991). Bertrand offsets of ruled surface and developable surface. Computer-Aided Design, 23(2), 145-152.
Yıl 2023, Cilt: 5 Sayı: 2, 6 - 17, 30.12.2023

### Kaynakça

• Kim, Y.H., Liu, H., & Qian, J. (2016). Some characterizations of canal surfaces. Bulletin of the Korean Mathematical Society, 53(2), 461-477.
• 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.
• 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.
• Aydın Şekerci, G., & Çimdiker, M. (2019). Bonnet canal surfaces. Dokuz Eylül Üniversitesi Mühendislik Fakültesi Fen ve Mühendislik Dergisi, 21(61), 195-200.
• Dede, M., Ekici, C., & Tozak, H. (2015). Directional tubular surfaces. International Journal of Algebra, 9(12), 527-535.
• 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.
• 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.
• Kaymanlı, G.U. (2020). Characterization of the evolute offset of ruled surfaces with B-Darboux frame. Journal of New Theory, 33, 50-55.
• 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.
• Ravani, B., & Ku, T.S. (1991). Bertrand offsets of ruled surface and developable surface. Computer-Aided Design, 23(2), 145-152.