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Journals / Hagia Sophia Journal of Geometry / 2026 / Cilt: 7 Sayı: 1

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

Esin Şahin(ESKİŞEHİR OSMANGAZİ ÜNİVERSİTESİ, FEN-EDEBİYAT FAKÜLTESİ)0000-0001-8169-6465
Şerife Nevin Gürbüz(ESKISEHIR OSMANGAZI UNIVERSITY)0000-0003-3959-1779
Pages
1–14
Publication Date
2025-06-30
DOI
—
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

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 $\mathbb{E}{^{3}}$.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, we obtain vortex filament flow (VFF) surfaces for three classes according to Type 1-PAF in Euclidean 3-space $\mathbb{E}{^{3}}$.
📊
Key Findings & Data: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 $\mathbb{E}{^{3}}$.
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Conclusion & Impact: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 $\mathbb{E}{^{3}}$.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:VFF · Type 1-PAF · Euclidean space

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 $\mathbb{E}{^{3}}$.

Özet

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 $\mathbb{E}{^{3}}$.

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Cite This Article

Şahin, E., Gürbüz, Ş. N. (2025). Focal Surfaces Connected with VFF for Type 1-PAF in $\mathbb{E}{^{3}}$. Hagia Sophia Journal of Geometry, Cilt: 7 Sayı: 1, 1–14. https://ajindex.com/articles/1490424

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