Focal Surfaces Connected with VFF for Type 1-PAF in $\mathbb{E}{^{3}}$

Year 2025, Volume: 7 Issue: 1, 1 - 14, 30.06.2025

Esin Şahin , Şerife Nevin Gürbüz

Abstract

In this paper, we obtain vortex filament flow (VFF) surfaces for three classes according to Type 1-PAF in Euclidean 3-space $\mathbb{E}{^{3}}$. Additionally, we obtain the first and second fundamental forms, Gauss and mean curvatures of focal surfaces generated by vortex filament flow surfaces in Euclidean
3-space $\mathbb{E}{^{3}}$. Furthermore , we present focal surfaces generated by vortex filament flow for three classes according Type 1-PAF in Euclidean 3-space $\mathbb{E}{^{3}}$. Finaly, we provide examples related to focal surfaces generated by vortex filament flow surfaces for three classes according Type 1-PAF in Euclidean 3-space $\mathbb{E}{^{3}}$.

Keywords

VFF , Type 1-PAF , Euclidean space

References

• Özen, K. E., & Tosun, M. (2021). A new moving frame for trajectories with non-vanishing angular momentum. Journal of Mathematical Sciences and Modelling, 4(1), 7–18.
• Özen, K. E., Tosun, M., & Avcı, K. (2022). Type 2-positional adapted frame and its application to Tzitzeica and Smarandache curves. Karatekin University Journal of Science, 1(1), 42–53.
• Solouma, E. M. (2021). Characterization of Smarandache trajectory curves of constant mass point particles as they move along the trajectory curve via PAF. Bulletin of Mathematical Analysis and Applications, 13(4), 14–30.
• Gürbüz, N. E. (2022). The evolution an electric field with respect to the type 1-PAF and the PAFORS frames in R^3_1. Optik, 250(1), 168285.
• Gürbüz, N. E., & Yoon, D. W. (2024). The evolution of the electric field along optical fiber with respect to the type-2 and 3 PAFs in Minkowski 3-space. Tamkang Journal of Mathematics, 55(2), 113–128.
• Hasimoto, H. (1971). Motion of a vortex filament and its relation to elastica. J. Phys. Soc. Jpn., 31, 293–294.
• Hasimoto, H. (1972). A soliton on a vortex filament. J. Fluid. Mech. 51 (3), 477–485.
• Abdel-All, N. H., Hussien, R., & Taha, Y. (2012). Hasimoto surfaces. Life Science Journal, 9(3), 556–560.
• Lamb Jr, G. L. (1977). Solitons on moving space curves. Journal of Mathematical Physics, 18(8), 1654–1661.
• Lakshmanan, M. (1977). Continuum spin system as an exactly solvable dynamical system. Phys. Lett. A. 61(1), 53–54.
• Hagen, H., Hahmann, S., & Schreiber, T. (1995). Visualization and computation of curvature behaviour freeform curves and surfaces. Computer-Aided Design, 27 (7), 545–552.
• Hagen, H., Müller, H., & Nielson, G. M. (1993). Focus on scientific visualization. SpringerVerlag, Berlin Heidelberg, 252–258.
• Hagen, H., & Hahmann, S. (1993). Generalized focal surfaces: a new method for surface interrogation. IEEE Proceedings Visualization, 70–76.
• Honda. S., & Takahashi, M. (2020). Evolutes and focal surfaces of framed immersions in the Euclidean space. Proceedings of the Royal Society of Edinburgh Section A:Mathematics, 150 (1), 497–516.
• Güler, F. (2022). The focal surfaces of offset surface. Optik, 271, 170053.
• Al-Dayel, I., Solouma, E., & Khan, M. (2022). On geometry of focal surfaces due to B-Darboux and type-2 Bishop frames in Euclidean 3-space. AIMS Mathematics, 7, 13454–13468.
• Da Rios, L. S. (1906). Sul moto d’un liquido indefinito con un filetto vorticoso di forma qualunque. Rend. Circ. Matem. Palermo, 22 (1), 117–135.
• Şahin, E. (2023). Focal surfaces according to positional adapted frames (in Turkish). Master’s Thesis, Eskişehir Osmangazi University, Eskişehir.