Normal Surfaces Defined by Smarandache Curves According to Frenet Frame in the Euclidean 3-space

Year 2025, Volume: 8 Issue: 2, 104 - 119, 19.10.2025

Amine Yılmaz

https://doi.org/10.33773/jum.1591062

Abstract

In this paper, we introduced original definitions of special normal surfaces defined by Smarandache curves according to Frenet frame in Euclidean space. We investigate theorems that give us necessary and sufficient conditions for those normal surfaces to be developable and minimal and give examples with illustrations.

Keywords

Smarandache curves , Mean curvature , Gaussian curvature , Normal surfaces

Öklid Uzayında Frenet Çatısına göre Smarandache Eğrileri ile Tanımlanan Normal Yüzeyler

Abstract

Bu makalede, Öklid uzayında Frenet çatısına göre Smarandache eğrileri tarafından tanımlanan özel normal yüzeylerin orijinal tanımlarını tanıttık. Bu normal yüzeylerin açılabilir ve minimal olması için gerekli ve yeterli koşulları veren teoremleri araştırıyoruz ve örnekler veriyoruz.

Keywords

Normal yüzeyler , Smarandache eğrileri , Gauss eğriliği , Ortalama eğrilik

References

• Ali, A. T. (2010). Special Smarandache curves in the Euclidean space. International Journal of Mathematical Combinatorics, 2, 30–36.
• Gray, A. (1998). Modern differential geometry of curves and surfaces. CRC Press.
• Gray, A. (2006). Modern differential geometry of curves and surfaces with Mathematica. CRC Press, Inc.
• Hananoi, S., & Izumiya, S. (2017). Normal developable surfaces of surfaces along curves. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 147 (1), 177–203.
• Heckman, G. (2017). Classical differential geometry: Curves and surfaces in Euclidean space. Radboud University Nijmegen.
• Ibrahim, A., & Solouma, E. M. (2021). Characteristic properties of type-2 Smarandach-ruled surface according to the type-2 Bishop frame in E3. Advances in Mathematical Physics, 1–7.
• Izumiya, S., & Takeuchi, N. (2004). New special curves and developable surfaces. Turkish Journal of Mathematics, 28, 153–163.
• Izumiya, S., & Takeuchi, N. (2003). Geometry of ruled surfaces. In J. C. Misra (Ed.), Applicable mathematics in the golden age (pp. 305–338). Narosa Publishing House.
• O’Neill, B. (1996). Elementary differential geometry. Academic Press, Inc.
• Ouarab, S., Chahdi, A. O., & Izid, M. (2020). Ruled surface generated by a curve lying on a regular surface and its characterizations. Journal for Geometry and Graphics, 24 (2), 257–267.
• Ouarab, S. (2021). Smarandache ruled surfaces according to Frenet-Serret frame of a regular curve in E3. Abstract and Applied Analysis, 2021, Article ID 5526536, 8 pages.
• Ouarab, S. (2021). Smarandache-ruled surface according to Darboux frame in E3. Journal of Mathematics, 1–10.
• Ouarab, S. (2021). NC-Smarandache ruled surface and NW-Smarandache ruled surface according to alternative moving frame in E3. Journal of Mathematics, 1–6.
• Rashad, A., & Hnazra, S. (2019). Normal ruled surface of a surface along a curve in Euclidean 3-space. International Journal of Analysis and Applications, 17 (4), 559–577.
• Suleyman, S., Davut, C., & Kebire, H. A. (2021). Smarandache ruled surface according to Bishop frame in E3. arXiv. https://doi.org/10.48550/arXiv.2112.05530
• Suleyman, S., Davut, C., & Elif, C. (2022). Some special Smarandache-ruled surface by Frenet frame in E3-I. Journal of Mathematics, 7 (1), 31–42.
• Suleyman, S., Davut, C., Elif, C., & Sumeyye, G. M. (2022). Some special Smarandache-ruled surface by Frenet frame in E3-II. Honam Mathematical Journal, 44 (4), 594–617.
• Turgut, M., & Yılmaz, S. (2008). Smarandache curves in Minkowski space-time. International Journal of Mathematical Combinatorics, 3, 51–55.
• Yılmaz, A. (2018). Doğru kongrüansları ve geometrik modellemesi [Doktora Tezi, Ege Üniversitesi].
• Yılmaz, A., & Şahin, B. (2018). On geodesics of the tangent and normal surfaces defined by TNSmarandache curve according to Frenet frame. 16th International Geometry Symposium (pp. 1–10). Manisa, Turkey.
• Yılmaz, A., & Şahin, B. (2019). On geodesics of the binormal surface defined by Smarandache curve [Paper presentation]. 3rd International Students Science Congress, Izmir, Turkey.