Approximations of Parallel Surfaces Along Curves

Year 2023, Volume: 16 Issue: 2, 715 - 726, 29.10.2023

Büşra Köse Yusuf Yaylı

https://doi.org/10.36890/iejg.1362590

Abstract

In this paper, we study developable surfaces which are flat and normal approximation of parallel surfaces along curves associated with three special vector fields. It is known that a surface whose points are at a constant distance along the normal of the surface is called a parallel surface. We investigate singularities of such developable surfaces. We show that under what conditions the approach surfaces are parallel. Also, we show that the approach surfaces are constant angle ruled surfaces if the curves selected on the surfaces are isophote, relatively normal-slant helix and helix.

Keywords

Parallel surfaces, developable surfaces, curves on surfaces, flat approximations, normal approximations.

References

• [1] Doğan, F., Yaylı, Y.: On isophote curves and their characterizations. Turkish Journal of Mathematics, 39, 650–664 (2015). https://doi.org/10.3906/mat-1410-4
• [2] Düldül, M., Macit, N.: Relatively normal-slant helices lying on a surface and their characterizations. Hacettepe Journal of Mathematics and Statistics, 46, 397–408 (2017).
• [3] Hananoi, S., Ito, N., Izumiya, S.: Spherical Darboux images of curves on surfaces. Beitr. Algebra Geom., 56, 575–585 (2015). https://doi.org/10.1007/s13366-015-0240-z
• [4] Hananoi, S., Izumiya, S.: Normal developable surfaces of surfaces along curves. Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 147, 177–203 (2017). https://doi.org/10.1017/S030821051600007X
• [5] Izumiya, S., Otani, S.: Flat Approximations of Surfaces Along Curves. Demonstratio Mathematica, 48, 217–241 (2015). https://doi.org/10.1515/dema-2015-0018
• [6] Kızıltuğ, S., Tarakcı, Ö., Yaylı, Y.: On the curves lying on parallel surfaces in the Euclidean 3-space E3. Journal of Advanced Research in Dynamical and Control Systems, 5, 26–35 (2013).
• [7] Özkaldı, S., Yaylı, Y.: Constant Angle Surfaces and Curves in E3. International Electronic Journal of Geometry, 4, 70–78 (2011).
• [8] Sabuncuoğlu, A.: Diferensiyel Geometri. Nobel Akademik Yayıncılık. Ankara (2016).